1 Introduction

This paper is concerned with optimal arrangements of unit vectors in Euclidean space. Let \(d,m,s\ge 1\) be integers, let \(\mathbb {R}^d\) denote the d-dimensional Euclidean space with standard inner product \(\langle \,{\cdot }\,,\,{\cdot }\,\rangle \), and let \(\mathscr {X}\subset \mathbb {R}^d\) be a set of unit vectors with the associated set of inner products \(A(\mathscr {X}):=\{\langle x,x'\rangle :x\ne x',\,x,x'\in \mathscr {X}\}\). The following two concepts are central to this paper: \(\mathscr {X}\) forms a spherical s-distance set [5, 34, 37, 40] if \(|A(\mathscr {X})|\le s\); and \(\mathscr {X}\) spans a system of m-angular lines (passing through the origin in the direction of \(x\in \mathscr {X}\)), if \(-1\notin A(\mathscr {X})\) and \(|\{\gamma ^2: \gamma \in A(\mathscr {X})\}|\le m\). With this terminology a system of m-angular lines can be considered as the switching class of certain spherical 2m-distance set without antipodal vectors. If the parameters s and m are not specified, then we talk about few-distance sets [9], and multiangular lines, respectively. The fundamental question of interest concerns the maximum cardinality and structure of the largest sets \(\mathscr {X}\) and their corresponding \(A(\mathscr {X})\). In particular, one is interested in finding the correct asymptotic growth rate.

Equiangular lines (i.e., the case \(m=1\)) are well-known combinatorial objects [19, 32, 33], receiving considerable recent attention, see e.g., [3, 24]. Biangular lines correspond to the case \(m=2\), which have also been the subject of both classical [15, 43], as well as more recent studies [7, 8, 14, 28, 41]. In addition, they have been investigated from the viewpoint of tight frames [23, 45]. Our motivation for studying these objects is fueled by their intrinsic connection to kissing arrangements [16, 18, 36]. In particular, we hope that the techniques and results described in this paper will eventually contribute to a deeper understanding of low-dimensional sphere packings. The goal of this paper, which heavily builds on the theory set forth earlier in [44], is to describe a systematic approach to the study of multiangular lines, focusing in particular on the biangular case.

The outline of this paper is as follows: in Sect. 2 we give various constructions of biangular lines, showing that their maximum number is at least \(2(d-1)(d-2)\) in \(\mathbb {R}^d\) for every \(d\ge 3\). In Sect. 3 we set up a general computational framework for exhaustively generating all (sufficiently large) biangular line systems, and in Sect. 4 we leverage on these ideas to classify the largest sets in \(\mathbb {R}^d\) for every \(d\le 6\). In Sect. 5 we present our results on multiangular lines. In Sect. 6 we conclude our manuscript with a selection of open problems. To improve the readability, a technical part on graph representation was moved to Appendix A, along with a few rather large matrices displayed in Appendix B.

For a convenient reference, we display here in Table 1 the best known lower bounds on the maximum number of biangular lines in \(\mathbb {R}^d\) (where entries marked by \(*\) are exact). All these numbers are new, except for the well-known cases in dimensions 2, 3, 22, and 23.

Table 1 Lower bounds on the maximum number of biangular lines in \(\mathbb {R}^d\)

2 Constructions of Biangular Lines

The goal of this section is to give various explicit constructions of large biangular line systems in low dimensional spaces.

Let \(\mathscr {X}\subset \mathbb {R}^d\) be a set of unit vectors, spanning biangular lines, and let O be an orthogonal matrix representing an isometry of \(\mathbb {R}^d\). Since for every \(x\in \mathscr {X}\) the sets \(\mathscr {X}':=(\mathscr {X}\setminus \{x\})\cup \{-x\}\) and \(\mathscr {X}'':=\{Ox:x\in \mathscr {X}\}\) span the same system of biangular lines as \(\mathscr {X}\), we may replace any \(x\in \mathscr {X}\) with its negative or apply O whenever it is necessary. Throughout this section we represent biangular line systems with a (conveniently chosen) corresponding set of unit vectors, and uniqueness is understood up to these operations. First, we give an elementary proof to the following trivial warm-up result.

Lemma 2.1

The five lines passing through the antipodal vertices of the convex regular decagon form the unique maximum biangular line system in \(\mathbb {R}^2\).

Proof

Let \(n\ge 1\), let \(\alpha ,\beta \in \mathbb {R}\) be such that \(0\le \alpha< \beta <1\), and assume that \(\mathscr {X}:=\{x_i: i\in \{1,\dots ,n\}\}\) spans a maximum biangular line system in \(\mathbb {R}^2\) with corresponding set of inner products \(A(\mathscr {X})\subseteq \{\pm \alpha , \pm \beta \}\). We may assume without loss of generality that \(x_1=[1,0]\). Since for \(i\in \{2,\dots ,n\}\) we have \(\langle x_1,x_i\rangle \in A(\mathscr {X})\), it immediately follows that

$$\begin{aligned} x_i\in \bigl \{\big [\alpha ,\sqrt{1-\alpha ^2}\big ],\big [\alpha ,-\sqrt{1-\alpha ^2}\big ],\big [\beta ,\sqrt{1-\beta ^2}\big ],\big [\beta ,-\sqrt{1-\beta ^2}\big ]\bigr \}, \end{aligned}$$

after replacing \(x_i\) by \(-x_i\) if it is necessary. Therefore \(n\le 5\), and the claimed configuration is indeed a largest possible example.

To see uniqueness, let us use the notation \(x_2=[\alpha ,\sqrt{1-\alpha ^2}]\), \(x_3=[\alpha \), \(-\sqrt{1-\alpha ^2}]\), \(x_4=[\beta ,\sqrt{1-\beta ^2}]\), and \(x_5=[\beta ,-\sqrt{1-\beta ^2}]\). Since \(\langle x_2,x_3\rangle =2\alpha ^2-1\), \(\langle x_4,x_5\rangle =2\beta ^2-1\), and \(\langle x_2,x_4\rangle +\langle x_2,x_5\rangle =2\alpha \beta \), the following system of polynomial equations in the variables \(\alpha \) and \(\beta \) must hold:

$$\begin{aligned} {\left\{ \begin{array}{ll} ((2\alpha ^2-1)^2-\alpha ^2)((2\alpha ^2-1)^2-\beta ^2)=0,\\ ((2\beta ^2-1)^2-\alpha ^2)((2\beta ^2-1)^2-\beta ^2)=0,\\ \alpha \beta ((\alpha \beta )^2-\alpha ^2)((\alpha \beta )^2-\beta ^2)((2\alpha \beta )^2-(\alpha +\beta )^2)((2\alpha \beta )^2-(\alpha -\beta )^2)=0. \end{array}\right. } \end{aligned}$$

This admits the unique feasible solution \(\alpha =(-1+\sqrt{5})/4\) and \(\beta =(1+\sqrt{5})/4\). \(\square \)

An alternate proof can be given to Lemma 2.1 by using more sophisticated tools, such as Theorem 3.1 and [19, Remark 6.2]. We will show yet another proof later in Sect. 4.

Recall that a binary code of length d with minimum distance \(\varDelta \) is a set \(\mathscr {B}\subseteq \mathbb {F}_2^d\) such that \({{\,\mathrm{dist}\,}}(b,b')\ge \varDelta \) for every distinct \(b,b'\in \mathscr {B}\) where \({{\,\mathrm{dist}\,}}(\,{\cdot }\,,\,{\cdot }\,)\) denotes the Hamming distance [21]. Let us define the following function:

$$\begin{aligned} \varSigma :\mathbb {F}_2\rightarrow \mathbb {R},\qquad \varSigma (0)=\frac{1}{\sqrt{d}},\qquad \varSigma (1)=-\frac{1}{\sqrt{d}}, \end{aligned}$$

and extend it coordinate-wise to a function from \(\mathbb {F}_2^d\) by writing \(\varSigma (b):=[\varSigma (b_1),\dots ,\) \(\varSigma (b_d)]\in \mathbb {R}^d\). This yields a spherical embedding of the codewords.

Lemma 2.2

Let \(d\ge 2\), and let \(\varDelta _1,\varDelta _2\in \{1,\dots ,d-1\}\). Let \(\mathscr {B}\) be a binary code of length d, such that \({{\,\mathrm{dist}\,}}(b,b')\in \{\varDelta _1,\varDelta _2,d-\varDelta _1,d-\varDelta _2\}\) for every distinct \(b,b'\in \mathscr {B}\). Then \(\mathscr {X}:=\{\varSigma (b):b\in \mathscr {B}\}\) spans a system of biangular lines with \(A(\mathscr {X})\subseteq \{\pm (1-2\varDelta _1/d), \pm (1-2\varDelta _2/d)\}\).

Proof

For every \(b,b'\in \mathscr {B}\) we have \(\langle \varSigma (b),\varSigma (b')\rangle = 1-2{{\,\mathrm{dist}\,}}(b,b')/d>-1\). \(\square \)

For terminology and basic facts on lattices we refer the reader to the textbook [18]. It is well known (see [15, 18, p. 117]) that the shortest vectors of the \(D_d\) lattices give rise to biangular line systems.

Lemma 2.3

Let \(d\ge 2\), and let \(\mathscr {X}\subset \mathbb {R}^d\) be the subset of all permutations of the unit vectors \([\pm 1,\pm 1,0,\dots ,0]/\sqrt{2}\) whose first nonzero coordinate is positive. Then \(\mathscr {X}\) spans \(|\mathscr {X}|=d(d-1)\) biangular lines with \(A(\mathscr {X})\subseteq \{0,\pm 1/2\}\).

Proof

For distinct \(x,x'\in \mathscr {X}\) the inner product \(\langle x,x'\rangle \) depends on the number of positions where the nonzero coordinates of x and \(x'\) overlap. If there is no overlap, or there are exactly two overlaps, then \(\langle x,x'\rangle =0\). Otherwise, if there is a single overlap, then \(\langle x,x'\rangle =\pm 1/2\). \(\square \)

Remark 2.4

We remark that for \(d\in \{6,7,8\}\) the sets of (nonantipodal) shortest vectors of the exceptional lattices \(E_d\) give rise to biangular line systems in \(\mathbb {R}^d\) with inner product set \(\{0,\pm 1/2\}\) formed by 36, 63, and 120 lines, respectively [18, p. 120].

Let \(0<h<1\). Starting from a spherical 2-distance set \(\mathscr {X}\subset \mathbb {R}^d\), one may obtain a family of biangular line systems in \(\mathbb {R}^{d+1}\), where the vectors \(x\in \mathscr {X}\) are rescaled by a factor of \(\sqrt{1-h^2}\) and translated along the \((d+1)\)st coordinate to height h. In a similar spirit, the six diagonals of the icosahedron can be continuously twisted in \(\mathbb {R}^3\), yielding a family of biangular lines [14].

Proposition 2.5

(infinite families) Let \(\mathscr {X}\subset \mathbb {R}^d\) be a spherical 2-distance set with \(A(\mathscr {X})\subseteq \{\alpha ,\beta \}\), with \(\alpha ,\beta \ge -1\) and \(\alpha ,\beta <1\). Let \(0<h <1\). Then

$$\begin{aligned} \mathscr {Y}(h):=\bigl \{\bigl [h,x\sqrt{1-h^2}\bigr ]:x\in \mathscr {X}\bigr \} \end{aligned}$$

spans a system of biangular lines in \(\mathbb {R}^{d+1}\) with \(A(\mathscr {Y}(h))\subseteq \{h^2+(1-h^2)\alpha ,h^2+(1-h^2)\beta \}\).

Proof

For every \(y,y'\in \mathscr {Y}(h)\) we have \(\langle y,y'\rangle =h^2+(1-h^2)\langle x,x'\rangle \) for some \(x,x'\in \mathscr {X}\). Furthermore, \(-1\notin A(\mathscr {Y}(h))\) by our assumptions on h. \(\square \)

Since the midpoints of the edges of the regular simplex in \(\mathbb {R}^d\) form a spherical 2-distance set of size \(d(d+1)/2\), biangular lines of this cardinality are abundant in \(\mathbb {R}^{d+1}\). Translation to a well-chosen height yields the following.

Proposition 2.6

(lifting) Let \(\mathscr {X}\subset \mathbb {R}^d\) be a spherical 4-distance set with \(A(\mathscr {X})\subseteq \{\alpha ,\beta ,\gamma \), \(\alpha +\beta -\gamma \}\), with \(\alpha ,\beta ,\gamma \ge -1\) and \(\alpha ,\beta ,\gamma <1\), and assume that \(\alpha +\beta <0\). Then \(\mathscr {Y}:=\{[\sqrt{-\alpha -\beta },x\sqrt{2}]/\sqrt{2-\alpha -\beta }:x\in \mathscr {X}\}\) spans a system of biangular lines in \(\mathbb {R}^{d+1}\) with

$$\begin{aligned}A(\mathscr {Y})\subseteq \biggl \{\pm \frac{\alpha -\beta }{2-\alpha -\beta },\pm \frac{2\gamma -\alpha -\beta }{2-\alpha -\beta }\biggr \}.\end{aligned}$$

Proof

For every \(y,y'\in \mathscr {Y}\) we have \(\langle y,y'\rangle =(-(\alpha +\beta )+2\langle x,x'\rangle )/(2-\alpha -\beta )\) for some \(x,x'\in \mathscr {X}\). Furthermore, \(-1\notin A(\mathscr {Y})\) by our assumptions on \(\alpha ,\beta ,\gamma \). \(\square \)

Remark 2.7

Given a spherical 3-distance set \(\mathscr {X}\) with \(A(\mathscr {X})\subseteq \{\alpha ,\beta ,\gamma \}\), it might happen that \(\alpha +\beta <0\), \(\alpha +\gamma <0\), and \(\beta \ne \gamma \). When this occurs, lifting via Proposition 2.6 could result in nonisometric biangular line systems.

The main utility of Proposition 2.6 is that antipodal vectors (which span exactly the same line) can be split into two nonantipodal vectors in the space one dimension higher. It immediately follows that any equiangular line system leads to twice as many biangular lines in the space one dimension higher.

Theorem 2.8

(cf. Theorem 3.1) For every \(d\ge 3\), there exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning \(|\mathscr {X}|=2(d-1)(d-2)\) biangular lines with \(A(\mathscr {X})\subseteq \{\pm 1/5,\pm 3/5\}\).

Proof

Take all \(2(d-1)(d-2)\) vectors in \(\mathbb {R}^{d-1}\) forming a spherical 4-distance set \(\mathscr {X}\) with \(A(\mathscr {X})\subseteq \{-1,-1/2,0,1/2\}\) in Lemma 2.3, and then use Proposition 2.6 to get the claimed biangular line systems. \(\square \)

A further application of Proposition 2.6 is the following.

Corollary 2.9

For \(d\in \{4,\dots ,16\}\) there exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning \(|\mathscr {X}|=\left( {\begin{array}{c}d\\ 3\end{array}}\right) \) biangular lines. There exists a set \(\mathscr {X}\subset \mathbb {R}^{17}\) spanning \(|\mathscr {X}|=\left( {\begin{array}{c}18\\ 3\end{array}}\right) \) biangular lines.

Proof

Consider the ‘canonical’ spherical 3-distance set \(\mathscr {X}\) of cardinality \(\left( {\begin{array}{c}d\\ 3\end{array}}\right) \) which can be obtained from

$$\begin{aligned}\sqrt{\frac{d-3}{3d}}\biggl [1,1,1,-\frac{3}{d-3},\ldots ,-\frac{3}{d-3}\biggr ]\in \mathbb {R}^{d}\end{aligned}$$

by permuting the coordinates. It is easily seen that

$$\begin{aligned}A(\mathscr {X})\subseteq \biggl \{-\frac{3}{d-3}, \frac{d-9}{3(d-3)},\frac{2d-9}{3(d-3)}\biggr \}.\end{aligned}$$

Since for every \(x\in \mathscr {X}\), \(\langle x,[1,1,\dots ,1]\rangle =0\), \(\mathscr {X}\) is embedded into \(\mathbb {R}^{d-1}\). Consequently, if \(d=18\) then \(\mathscr {X}\) spans a biangular line system in \(\mathbb {R}^{17}\). If \(d\le 16\), then since \(({d-9})/({3(d-3)})-{3}/({d-3})<0\), Proposition 2.6 yields the claimed configurations in \(\mathbb {R}^d\). \(\square \)

Finally, a rather surprising consequence of Proposition 2.6 is the following: the biangular line systems mentioned in Remark 2.4 are not the best possible in their respective dimension.

Corollary 2.10

There exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning biangular lines with \(A(\mathscr {X})\subseteq \{\pm 1/5,\pm 3/5\}\) for

$$\begin{aligned} (d,|\mathscr {X}|)\in \{(3,4),(4,12),(5,24),(6,40),(7,72),(8,126), (9,240)\}. \end{aligned}$$

Proof

The cases \(d\in \{3,4,5,6\}\) follow from Theorem 2.8. To see the remaining cases, combine Proposition 2.6 with the exceptional configurations mentioned in Remark 2.4. \(\square \)

Later (see Sect. 4) we will show that Theorem 2.8 gives rise to a largest possible line system for \(d\in \{4,5,6\}\), and we tend to believe that Corollary 2.10 gives the best configurations for \(d\in \{7,8,9\}\) as well.

Next we prove a preliminary technical result. Following the terminology of [32], we denote by \(N_{1/3}(d)\) the maximum number of equiangular lines in \(\mathbb {R}^d\) where the set of inner products is a subset of \(\{\pm 1/3\}\). Recall that \(N_{1/3}(0)=0\).

Proposition 2.11

For \(m\ge 1\) and \(d\ge m\), there exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning \(|\mathscr {X}|=2m\cdot N_{1/3}(d-m)\) biangular lines with \(A(\mathscr {X})\subseteq \{\pm 1/5,\pm 3/5\}\).

Proof

Let \(\mathscr {E}\) denote the set of canonical basis vectors of \(\mathbb {R}^{m}\), and consider a maximum set \(\mathscr {Y}\subset \mathbb {R}^{d-m}\) spanning \(N_{1/3}(d-m)\) equiangular lines with \(A(\mathscr {Y})\subseteq \{\pm 1/3\}\). We claim that the following set \(\mathscr {X}\subset \mathbb {R}^d\) spans a biangular line system:

$$\begin{aligned} \mathscr {X}:=\biggl \{\frac{[y\sqrt{6},2e]}{\sqrt{10}}:y\in \mathscr {Y},\, e\in \mathscr {E}\biggr \} \cup \biggl \{\frac{[y\sqrt{6},-2e]}{\sqrt{10}}: y\in \mathscr {Y}, \,e\in \mathscr {E}\biggr \}. \end{aligned}$$

Indeed, as for \(x,x'\in \mathscr {X}\), we have \(\langle x,x'\rangle = 3\langle y,y'\rangle /5\pm 2\langle e,e'\rangle /5\) for some (not necessarily distinct) \(e,e'\in \mathscr {E}\) and \(y,y'\in \mathscr {Y}\). Since \(\langle e,e'\rangle \in \{0,1\}\) and \(\langle y,y'\rangle \in \{\pm 1/3,1\}\), the claim follows. \(\square \)

We note the following.

Corollary 2.12

There exists a set \(\mathscr {X}\subset \mathbb {R}^{14}\) spanning \(|\mathscr {X}|=392\) biangular lines with \(A(\mathscr {X})\subseteq \{\pm 1/5,\pm 3/5\}\).

Proof

It follows from Proposition 2.11 by setting \(m=7\) and \(d=7\), and by recalling from [32] that \(N_{1/3}(7)=28\). \(\square \)

It turns out that one may combine certain line systems described in Proposition 2.11 with the 256 lines spanned by the ‘even half’ of the 10-dimensional hypercube. This yields improved results for \(d\in \{10,11,12,13,15,16\}\) and gives the same number of biangular lines for \(d=17\) as Corollary 2.9.

Theorem 2.13

For \(d\ge 10\), there exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning \(|\mathscr {X}|=256+20N_{1/3}(d-10)\) biangular lines with \(A(\mathscr {X})\subseteq \{\pm 1/5,\pm 3/5\}\).

Proof

Let \(\mathscr {B}\subset \mathbb {F}_2^{10}\) be the binary code of length 10 formed by codewords of even weight, such that the first coordinate of every \(b\in \mathscr {B}\) is 0. By Lemma 2.2 the set \(\mathscr {Z}:=\{\varSigma (b):b\in \mathscr {B}\}\subset \mathbb {R}^{10}\) spans a system of 256 biangular lines with \(A(\mathscr {Z})\subseteq \{\pm 1/5,\pm 3/5\}\). Next, we consider a maximum set \(\mathscr {Y}\subset \mathbb {R}^{d-10}\) spanning \(N_{1/3}(d-10)\) equiangular lines with \(A(\mathscr {Y})\subseteq \{\pm 1/3\}\). Let \(\mathscr {E}\) denote the set of canonical basis vectors of \(\mathbb {R}^{10}\), and let \(o\in \mathbb {R}^{d-10}\) denote the zero vector. We claim that the following set \(\mathscr {X}\subset \mathbb {R}^{d}\) spans a biangular line system:

$$\begin{aligned} \mathscr {X}:=&\bigl \{\bigl [\sqrt{6}y,2e\bigr ]{/}\sqrt{10}:y\in \mathscr {Y},\, e\in \mathscr {E}\bigr \}\\&\cup \,\bigl \{\bigl [\sqrt{6}y,-2e\bigr ]/\sqrt{10}: y\in \mathscr {Y},\, e\in \mathscr {E}\bigr \}\cup \{[o,z]:z\in \mathscr {Z}\}. \end{aligned}$$

Indeed, for \(x,x'\in \mathscr {X}\), we have

$$\begin{aligned} \langle x,x'\rangle \,\in \,\bigl \{3\langle y,y'\rangle /5\pm 2\langle e,e'\rangle /5,\,\pm 2\langle e,z\rangle /\sqrt{10},\langle z,z'\rangle \bigr \} \end{aligned}$$

for some (not necessarily distinct) \(e,e'\in \mathscr {E}\), \(y,y'\in \mathscr {Y}\), and \(z,z'\in \mathscr {Z}\). Since \(\langle e,e'\rangle \in \{0,1\}\), \(\langle e,z\rangle \in \{\pm 1/\sqrt{10}\}\), \(\langle y,y'\rangle \in \{\pm 1/3,1\}\), and \(\langle z,z'\rangle \in \{\pm 1/5, \pm 3/5,1\}\), the claim follows. \(\square \)

Corollary 2.14

There exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning biangular lines with \(A(\mathscr {X})\subseteq \{\pm 1/5,\pm 3/5\}\) for

$$\begin{aligned} (d,|\mathscr {X}|)\in & {} \{(10,256), (11,276), (12,296), (13,336), \\&\qquad \qquad \qquad \ (15,456), (16,576), (17,816)\}. \end{aligned}$$

Proof

Combine Theorem 2.13 with [32, Thm. 4.5]. \(\square \)

Finally, we note that various cross-sections of the Leech lattice \(\varLambda _{24}\) (see [18, p. 133] for how to construct its shortest vectors from the extended binary Golay code [12] in explicit form) give rise to biangular line systems with inner product set \(\{0,\pm 1/3\}\). Such line systems were investigated in [43].

Theorem 2.15

There exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning biangular lines with \(A(\mathscr {X})\subseteq \{0,\pm 1/3\}\) for \((d,|\mathscr {X}|)\in \{(21,896),(22,1408),(23,2300)\}\).

Proof

Let \(\mathscr {L}\subset \mathbb {R}^{24}\), \(|\mathscr {L}|=196560\), be the set of shortest vectors of \(\varLambda _{24}\), where the vectors are normalized so that \(\langle \ell ,\ell \rangle =1\) for every \(\ell \in \mathscr {L}\). With this convention, \(\langle \ell , \ell '\rangle \in \{0,\pm 1/4,\pm 1/2,\pm 1\}\) for every \(\ell ,\ell '\in \mathscr {L}\). Now let \(\ell \in \mathscr {L}\) be fixed. It is well known (see [18, p. 264]) that the subset \(\mathscr {Y}=\{y:\langle \ell ,y\rangle = 1/2,\, y\in \mathscr {L}\}\) contains 4600 vectors, independently of the choice of \(\ell \). Note that for \(y\in \mathscr {Y}\) we have \(\ell -y\in \mathscr {Y}\) and therefore the set \(\mathscr {Z}:=\{(2y-\ell )/\sqrt{3}:y\in \mathscr {Y}\}\) is antipodal, and \(\langle \ell ,z\rangle =0\) for every \(z\in \mathscr {Z}\). Finally, let \(\mathscr {X}\subset \mathscr {Z}\) with \(|\mathscr {X}|=2300\) so that \(\mathscr {Z}=\{x:x\in \mathscr {X}\}\cup \{-x:x\in \mathscr {X}\}\). Now \(\mathscr {X}\) spans the claimed biangular line system in \(\mathbb {R}^{23}\), since \(\langle y,y'\rangle \notin \{-1/4,-1\}\) and therefore for \(x,x'\in \mathscr {X}\) we have \(\langle x,x'\rangle =(4\langle y,y'\rangle -1)/3\in \{0,\pm 1/3,1\}\). Let \(x,x'\in \mathscr {X}\) be such that \(\langle x,x'\rangle = 0\). Then \(\mathscr {U}:=\{u:\langle u,x\rangle = 0,\, u\in \mathscr {X}\}\), \(\mathscr {V}:=\{v:\langle v,x\rangle = \langle v,x'\rangle = 0,\,v\in \mathscr {X}\}\) span the claimed biangular line systems in dimension 22 and 21, respectively. \(\square \)

Another way to get biangular lines with the set of inner products \(\{0,\pm 1/3\}\) is the following.

Lemma 2.16

Let \(w\equiv 3\ ({{\,\mathrm{mod}\,}}4)\) and \(d\ge 2w+1\) be positive integers. Let \(\mathscr {B}\subset \mathbb {F}_2^d\) be a binary constant weight code of length d, weight w, and minimum distance \(2w-2\), and assume that there exists a Hadamard matrix of order \(w+1\). Then there exists a set \(\mathscr {X}\subset \mathbb {R}^d\) with \(|\mathscr {X}|=(w+1)|\mathscr {B}|\) spanning a biangular line system with \(A(\mathscr {X})\subseteq \{0,\pm 1/w\}\).

Proof

Recall that a Hadamard matrix H of order \(w+1\) is a \((w+1)\times (w+1)\) orthogonal matrix with entries \(\pm 1/\sqrt{w+1}\). Let \(H'\) be the matrix obtained from H after removing its first column and renormalizing its rows. Let \(\mathscr {H}\subset \mathbb {R}^{w}\) be the set of rows of \(H'\). Clearly, \(\langle h,h'\rangle \in \{\pm 1/w,1\}\) for \(h,h'\in \mathscr {H}\). Now \(\mathscr {X}\) can be obtained by replacing each codeword \(b\in \mathscr {B}\) with a set of \(w+1\) real vectors where the support of b (i.e., coordinates with binary 1) are replaced by the entries of \(h\in \mathscr {H}\), and coordinates with binary 0 are replaced by \(0\in \mathbb {R}\). Since \(d\ge 2w+1\), there are no two codewords at Hamming distance d, and therefore the claim follows. \(\square \)

Corollary 2.17

For \(d\ge 7\) there exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning \(|\mathscr {X}|{=}4\lceil (d-1) (d-2)/6\rceil \) biangular lines with \(A(\mathscr {X})\subseteq \{0,\pm 1/3\}\). Furthermore, there exists a set \(\mathscr {Y}\subset \mathbb {R}^{d+1}\) spanning \(|\mathscr {X}|\) biangular lines with \(A(\mathscr {Y})\subseteq \{\pm 1/7,\pm 3/7\}\).

Proof

Indeed, this is a specialization of Lemma 2.16 for \(w=3\) and using constant weight codes coming from the averaging argument in [13, Thm. 14]. The second part of the claim is an immediate consequence of Proposition 2.6. \(\square \)

While Corollary 2.17 is weaker than Theorem 2.8, it can be used in two ways. First, one may embed the 2300 biangular lines from Theorem 2.15 into \(\mathbb {R}^{23+d}\), and extend this configuration with an additional \(4\lceil (d-1)(d-2)/6\rceil \) vectors (for \(d\ge 7\)). Secondly, it may happen that these configurations can be further extended to a spherical 4-distance set with inner products \(\{-2/3,-1/3,0,1/3\}\), and then an application of Proposition 2.6 would immediately yield biangular lines with inner products \(\{\pm 1/7,\pm 3/7\}\) in \(\mathbb {R}^{24+d}\). The following result shows that the two largest sets mentioned in Theorem 2.15 are inextendible.

Theorem 2.18

(the relative bound, [10, 19]) Let \(d\ge 3\) and assume that \(\mathscr {X}\subset \mathbb {R}^d\) spans a biangular line system with \(A(\mathscr {X})\subseteq \{\pm \alpha ,\pm \beta \}\), \(0\le \alpha ,\beta <1\). Assume that \(\alpha ^2+\beta ^2\le 6/(d+4)\) and \(3-(d+2)(\alpha ^2+\beta ^2)+d(d+2)\alpha ^2\beta ^2>0\). Let \(n_{\alpha }:=|\{[x,x']:\langle x,x'\rangle ^2=\alpha ^2,\,x,x'\in \mathscr {X}\}|\). Then

$$\begin{aligned} |\mathscr {X}|\le \frac{d(d+2)(1-\alpha ^2)(1-\beta ^2)}{3-(d+2)(\alpha ^2+\beta ^2)+d(d+2)\alpha ^2\beta ^2}. \end{aligned}$$
(1)

Equality holds if and only if

$$\begin{aligned} \displaystyle \biggl (\frac{6}{d+4}-\alpha ^2-\beta ^2\biggr )&\biggl ((\alpha ^2-\beta ^2)n_\alpha +|\mathscr {X}|(|\mathscr {X}|-1)\beta ^2+|\mathscr {X}|-\frac{|\mathscr {X}|^2}{d}\biggr )=0\ \ \text {and}\\ \displaystyle \biggl (\frac{6}{d+4}-\alpha ^2-\beta ^2\biggr )&(\alpha ^2-\beta ^2)n_\alpha \\&=\frac{|\mathscr {X}|(d^2+3|\mathscr {X}|-4)}{(d+2)(d+4)}-|\mathscr {X}|(|\mathscr {X}|-1)\beta ^2\biggl (\frac{6}{d+4}-\beta ^2\biggr ). \end{aligned}$$

Remark 2.19

For \(d\ge 3\) and \(i\in \{2,4\}\) let \(C_i^{((d-2)/2)}(z)\) denote the Gegenbauer polynomials in the sense of [20]. In Theorem 2.18 the equality holds if and only if

$$\begin{aligned} \biggl (\frac{6}{d+4}-\alpha ^2-\beta ^2\biggr )\sum _{x,x'\in \mathscr {X}}\!C_2^{((d-2)/2)}(\langle x,x'\rangle )\,=\!\sum _{x,x'\in \mathscr {X}}\!C_4^{((d-2)/2)}(\langle x,x'\rangle )=0.\end{aligned}$$

In particular, if \(\mathscr {X}\subset \mathbb {R}^d\) forms a spherical 4-design [4] such that \(A(\mathscr {X})\subseteq \{\pm \alpha ,\pm \beta \}\), then the equality holds in (1).

Remark 2.20

If there is equality in (1), then the quantity \(n_{\alpha }\) as defined in Theorem 2.18 is a nonnegative integer. The failure of this condition could be used to show the nonexistence of various hypothetical configurations, e.g., in \(\mathbb {R}^8\) there does not exist 50 biangular lines with the set of inner products \(\{\pm 1/4,\pm 1/2\}\).

In Table 2 we display data on the known biangular line systems meeting the relative bound, and later in Corollary 4.12 we prove that this list is (essentially) complete for \(d\le 6\). The canonical examples are mutually unbiased bases (MUBs) [29], spanning \(2^{4i-1}+2^{2i}\) biangular lines in dimension \(d=4^{i}\) with inner products \(\{0,\pm 2^{-i}\}\), \(i\ge 1\). We note the following two examples.

Example 2.21

(36 biangular lines in \(\mathbb {R}^7\), see [17, 22])  Let U be the \(7\times 7\) circulant matrix with first row [0, 1, 0, 0, 0, 0, 0]. Let

$$\begin{aligned} \mathscr {Y}&:=\{[-7,1,1,1,1,1,1],[-1, 3, 3, -3, 3, -3, -3]\}\quad \text {and}\\ \mathscr {Z}&:=\{[1, -1, -3, 3, 3, -3,-3],[1, 3, -1, -3,-3, -3, 3],\\&\qquad \qquad [-1, 3, -3, 1, -3, 3, -3]\}. \end{aligned}$$

Then, the set

$$\begin{aligned} \mathscr {X}=\bigl \{[-7,1,1,1,1,1,1,1]/\sqrt{56}\bigr \}&\cup \bigl \{[1,yU^i]/\sqrt{56}:i\in \{0,1,\dots ,6\},\,y\in \mathscr {Y}\bigr \}\\&\cup \bigl \{[3,zU^i]/\sqrt{56}:i\in \{0,1,\dots ,6\},\,z\in \mathscr {Z}\bigr \} \end{aligned}$$
Table 2 Biangular line systems meeting the relative bound

spans 36 biangular lines in \(\mathbb {R}^7\) with \(A(\mathscr {X})\subseteq \{\pm 1/7,\pm 3/7\}\). Indeed, all vectors are orthogonal to \([1,\dots ,1]\in \mathbb {R}^8\). The parameters of this line system meet the relative bound.

Example 2.22

(256 biangular lines in \(\mathbb {R}^{16}\), see [21, p. 486], [38, 43]) Consider a biplane [31] of order 4, that is, a \(16\times 16\) square \(\{0,1\}\)-matrix H with constant row and column sum 6, such that \(HH^T=4I_{16}+2J_{16}\). We may simply take \(H:=(J_4-I_4)\otimes I_4+I_4\otimes (J_4-I_4)\), and let \(\mathscr {H}\subset \mathbb {R}^{16}\) be the set of rows of H. Let \(\mathscr {B}\subset \mathbb {F}_2^{6}\) be a binary code of length 6 formed by codewords of even weight, such that the first coordinate of every \(b\in \mathscr {B}\) is 0. By Lemma 2.2 the set \(\mathscr {Z}:=\{\varSigma (b):b\in \mathscr {B}\}\subset \mathbb {R}^{6}\) spans a system of 16 biangular lines with \(A(\mathscr {Z})\subseteq \{\pm 1/3\}\). Replacing each codeword \(b\in \mathscr {B}\) with a set of 16 real vectors where the support of b (i.e., coordinates with binary 1) are replaced by the entries of \(h\in \mathscr {H}\), and coordinates with binary 0 are replaced by \(0\in \mathbb {R}\), spans the claimed 256 biangular lines in \(\mathbb {R}^{16}\) with the set of inner products \(\{0,\pm 1/3\}\). Further nonisometric examples can be constructed by, e.g., choosing H in a different way. The parameters of these line systems meet the relative bound.Footnote 1

Finally, we note the following (almost immediate) consequence of [39, Theorem 5.2 and 5.3].

Theorem 2.23

(see [39]) Let \(d\ge 5\) and let \(\mathscr {X}\subset \mathbb {R}^d\) span a maximum biangular line system with \(A(\mathscr {X})\subseteq \{\pm \alpha ,\pm \beta \}\), \(0\le \alpha< \beta < 1\). Then \(z:=(1-\alpha ^2)/(\beta ^2-\alpha ^2)\) is an integer. Furthermore,

$$\begin{aligned} z\le \biggl \lfloor \frac{1}{2}+\sqrt{\frac{(d^2+d+2)(d^2+d-1)}{4d^2+4d-8}}\biggr \rfloor . \end{aligned}$$

Proof

The statement is a reformulation of [39, Thms. 5.2 and 5.3] and it holds whenever \(|\mathscr {X}|\ge d(d+1)\). This in turn holds by Theorem 2.8 for maximum biangular line systems whenever \(d\ge 7\). For \(d\in \{5,6\}\) the set of inner products of (the unique) maximum biangular line systems is \(\{\pm 1/5,\pm 3/5\}\) (see Theorems 4.7 and 4.9), and therefore in these cases \(z=3\) is indeed an integer below the claimed bound. \(\square \)

3 Computational Framework

In this section, following ideas developed in [44], we set up a framework for systematically generating biangular lines. We will leverage on this newly established theory in Sect. 4 where we demonstrate how to use this approach in practice. In particular, we will determine the size of the largest biangular line systems in dimension \(d\le 6\) by using supercomputational resources, and classify the maximum cases. We remark that this framework carries over to the multiangular setting after minor technical changes (see Sect. 5 and Appendix A).

3.1 A High Level Overview

Let \(d,n\ge 1\), and let \(\mathscr {X}=\{x_1,\dots ,x_n\}\subset \mathbb {R}^d\) be a set of unit vectors, spanning a system of n biangular lines. From here on, we will represent \(\mathscr {X}\) by its Gram matrix \(G:=[\langle x_i,x_j\rangle ]_{i,j=1}^n\). Conveniently, the matrix G is invariant up to change of basis, and has the following combinatorial properties: G is \(n\times n\), \(G=G^T\), \(G_{ii}=1\) for every \(i\in \{1,\dots ,n\}\), and \(G_{ij}\in A(\mathscr {X})\) for distinct \(i,j\in \{1,\dots ,n\}\). Furthermore, it has the following algebraic properties: G is positive-semidefinite and \({\text {rank}}G\le d\). Conversely, given any matrix G with these properties, one may reconstruct (uniquely, up to change of basis), via the Cholesky decomposition, an \(n\times {\text {rank}}G\) matrix F such that \(FF^T=G\) holds [27].

Our aim is to find a way for generating all (sufficiently large) \(n\times n\) Gram matrices of biangular line systems in a fixed dimension d. It follows from Ramsey theory that n is bounded in terms of d, and we recall here the following explicit upper bound.

Theorem 3.1

(absolute bound, [19], cf. Theorem 2.8) Let \(\mathscr {X}\subset \mathbb {R}^d\) span a biangular line system. Then \(|\mathscr {X}|\le \left( {\begin{array}{c}d+3\\ 4\end{array}}\right) \).

We say that a permutation \(\sigma \) of the set \(\varGamma =\{\alpha ,\beta ,-\alpha ,-\beta \}\) is a relabeling if \(\sigma (\gamma )=-\sigma (-\gamma )\) for every \(\gamma \in \varGamma \). The following is central to this paper.

Definition 3.2

Let \(C(\alpha ,\beta )\) be an \(n\times n\) symmetric matrix with constant diagonal 1 over the polynomial ring \(\mathbb {Q}[\alpha ,\beta ]\) whose off-diagonal entries are \(\{0,\pm \alpha ,\pm \beta \}\). Two such matrices, \(C_1\) and \(C_2\), are called equivalent, if \(C_1(\alpha ,\beta )=PC_2(\sigma (\alpha ),\sigma (\beta ))P^T\) for some signed permutation matrix P and relabeling \(\sigma \). A representative of this matrix equivalence class is called a candidate Gram matrix.

Candidate Gram matrices capture the combinatorial structure of Gram matrices. Since our focus is on the biangular case, we will assume in the following that

$$\begin{aligned} \alpha \beta (\alpha ^2-\beta ^2)(\alpha ^2-1)(\beta ^2-1)\ne 0. \end{aligned}$$
(2)

Furthermore, at most two out of the three symbols \(0, \pm \alpha ,\pm \beta \) can appear as a matrix entry in \(C(\alpha ,\beta )\). Clearly, if G is a Gram matrix of a biangular line system, then there exists a candidate Gram matrix \(C(\alpha ,\beta )\) such that \(G=C(\alpha ^*,\beta ^*)\) for some \(\alpha ^*,\beta ^*\in \mathbb {R}\), subject to (2). In particular, \({\text {rank}}C(\alpha ^*,\beta ^*)\le d\) should hold.

Example 3.3

(candidate Gram matrices of order 3)

$$\begin{aligned}\begin{bmatrix} 1 &{} 0 &{} 0\\ 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 1 \end{bmatrix},\quad \begin{bmatrix} 1 &{} 0 &{} 0\\ 0 &{} 1 &{} \alpha \\ 0 &{} \alpha &{} 1 \end{bmatrix}, \quad \begin{bmatrix} 1 &{} 0 &{} \alpha \\ 0 &{} 1 &{} \alpha \\ \alpha &{} \alpha &{} 1 \end{bmatrix}, \quad \begin{bmatrix} 1 &{} \alpha &{} \alpha \\ \alpha &{} 1 &{} \alpha \\ \alpha &{} \alpha &{} 1 \end{bmatrix}, \quad \begin{bmatrix} 1 &{} \alpha &{} \alpha \\ \alpha &{} 1 &{} \beta \\ \alpha &{} \beta &{} 1 \end{bmatrix} \end{aligned}$$

Note that at most two symbols appear (whose values are unspecified) within the off-diagonal positions, signifying distinct inner products.

The main advantage of using candidate Gram matrices is that in this way we are transforming the problem of ‘infinitely many \(n\times n\) Gram matrices’ to the conceptually simpler ‘finite list of \(n\times n\) candidate Gram matrices’ (where n itself is bounded by Theorem 3.1). Then, one should decide whether a candidate Gram matrix actually represents a Gram matrix via a spectral analysis, as illustrated below.

Example 3.4

(Petersen graph, cf. Proposition 2.5) Consider the following example of a candidate Gram matrix of order 10:

$$\begin{aligned} C(\alpha ,\beta )=\begin{bmatrix} 1 &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} \beta &{} \beta &{} \beta \\ \alpha &{} 1 &{} \alpha &{} \alpha &{} \alpha &{} \beta &{} \beta &{} \alpha &{} \alpha &{} \beta \\ \alpha &{} \alpha &{} 1 &{} \alpha &{} \beta &{} \alpha &{} \beta &{} \alpha &{} \beta &{} \alpha \\ \alpha &{} \alpha &{} \alpha &{} 1 &{} \beta &{} \beta &{} \alpha &{} \beta &{} \alpha &{} \alpha \\ \alpha &{} \alpha &{} \beta &{} \beta &{} 1 &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} \beta \\ \alpha &{} \beta &{} \alpha &{} \beta &{} \alpha &{} 1 &{} \alpha &{} \alpha &{} \beta &{} \alpha \\ \alpha &{} \beta &{} \beta &{} \alpha &{} \alpha &{} \alpha &{} 1 &{} \beta &{} \alpha &{} \alpha \\ \beta &{} \alpha &{} \alpha &{} \beta &{} \alpha &{} \alpha &{} \beta &{} 1 &{} \alpha &{} \alpha \\ \beta &{} \alpha &{} \beta &{} \alpha &{} \alpha &{} \beta &{} \alpha &{} \alpha &{} 1 &{} \alpha \\ \beta &{} \beta &{} \alpha &{} \alpha &{} \beta &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} 1 \end{bmatrix}. \end{aligned}$$

Here \(C(0,1)-I_{10}\) is the adjacency matrix of the Petersen graph. Using standard spectral graph theory, one may find that for every \(\alpha ^*, \beta ^*\in \mathbb {R}\) we have \(\varLambda (C(\alpha ^*,\beta ^*))=\{[1+6\alpha ^*+3\beta ^*]^1,[1+\alpha ^*-2\beta ^*]^4,[1-2\alpha ^*+\beta ^*]^5\}\). Therefore \({\text {rank}}C(\alpha ^*,2\alpha ^*-1)\le 5\). Furthermore, for \(\alpha ^*\ge 1/6\), \(\alpha ^*<1\) the matrix \(C(\alpha ^*,2\alpha ^*-1)\) is positive semidefinite. The matrix \(C(1/6,-2/3)\) on the boundary describes the Petersen code [2], which corresponds to the midpoints of the regular simplex in \(\mathbb {R}^4\). We remark that since the Petersen code is a 2-distance set, Proposition 2.5 can be applied which gives rise to the family of biangular lines \(C(\alpha ^*,2\alpha ^*-1)\) in \(\mathbb {R}^5\).

However, computing the spectrum of a candidate Gram matrix without any apparent structure is a delicate task, and instead we will rely on the following key technical result.

Proposition 3.5

(strong Gröbner test, cf. Corollary 3.8) Let \(d\ge 2\) be fixed and let \(C(\alpha ,\beta )\) be a candidate Gram matrix of order \(n\ge d+1\). Let \(\mathscr {M}\) denote the set of all \((d+1)\times (d+1)\) submatrices of C. Let \(\omega \) be an auxiliary variable. If the following system of polynomial equations,

$$\begin{aligned} {\left\{ \begin{array}{ll} \det M(\alpha ,\beta )=0 \qquad \text {for all }M\in \mathscr {M},\\ \omega \alpha \beta (\alpha ^2-\beta ^2)(\alpha ^2-1)(\beta ^2-1)+1=0, \end{array}\right. } \end{aligned}$$
(3)

has no solutions in \(\mathbb {C}^3\), then \({\text {rank}}C(\alpha ^*,\beta ^*)\le d\) cannot hold for any \(\alpha ^*,\beta ^*\in \mathbb {R}\) subject to (2).

Proof

Indeed, if \({{\,\mathrm{rank}\,}}C(\alpha ^*,\beta ^*)\le d\) for some \(\alpha ^*,\beta ^*\in \mathbb {C}\) subject to (2), then necessarily all \((d+1)\times (d+1)\) minors of \(C(\alpha ^*,\beta ^*)\) are vanishing. In particular, there exists an \(\omega ^*\in \mathbb {C}\) such that \((\alpha ^*,\beta ^*,\omega ^*)\in \mathbb {C}^3\) is a solution of the system of equations (3). \(\square \)

We remark that one can decide whether a system of polynomial equations with rational coefficients has any complex solutions by computing a Gröbner basis [6].

Based on these concepts, we now may classify biangular line systems in the following way. First, we fix \(d\ge 2\) and \(n=\left( {\begin{array}{c}d+3\\ 4\end{array}}\right) \). Secondly, we generate (by computers, say) all \(n\times n\) candidate Gram matrices. Thirdly, for each candidate Gram matrix \(C(\alpha ,\beta )\) generated, we attempt to determine, via solving the system of equations (3) the (not necessarily finite) set of all real matrices \(\{C(\alpha _i^*,\beta _i^*):{\text {rank}}C(\alpha _i^*,\beta _i^*)\le d,\, i\in \mathscr {I}\}\). Finally, we keep only those which are positive semidefinite. When no such matrices are found, then we decrease n by one and repeat the same procedure.

There are several weak points of this naive method restricting heavily its utility. First of all, the bound on n, stipulated by Theorem 3.1, is rather crude, and there is no way to generate all candidate Gram matrices of that size. Secondly, when the solution set of (3) is infinite, then it is a very delicate task to parametrize the matrices \(C(\alpha _i^*,\beta _i^*)\), \(i\in \mathscr {I}\), and to describe which of these are positive semidefinite. We overcome these difficulties by sophisticated matrix generation techniques, and using Proposition 3.5 for discarding a large fraction of small candidate Gram matrices. We discuss these efforts in the next subsection.

3.2 The Framework in Detail

In this subsection we describe in more detail how to generate candidate Gram matrices in an equivalence-free exhaustive manner. The main technical tool is canonization, see [30, Section 4.2.2] and [42]. The vectorization of a candidate Gram matrix C of order n is the vector \({\text {vec}}C:=[C_{21},C_{31},C_{32},\dots ,C_{n1},\dots \), \(C_{n,n-1}]\). We say that a candidate Gram matrix \(C(\alpha ,\beta )\) is in canonical form, if it holds that

$$\begin{aligned} \begin{aligned} {\text {vec}}C(\alpha ,\beta ):=\min {\{{\text {vec}}(PC(\sigma (\alpha ), \sigma (\beta ))P^T):P\text { is a signed}}\\ \text {permutation matrix,}\,\,{\sigma }\,\text {is a relabeling}\}, \end{aligned} \end{aligned}$$
(4)

where comparison of vectors is done lexicographically (one may assume, e.g., that the entries are ordered as \(0\prec \alpha \prec -\alpha \prec \beta \prec -\beta )\). One particularly attractive feature of the above canonical form is that the leading principal submatrices of canonical matrices are themselves canonical. Therefore canonical matrices can be generated inductively, using smaller canonical matrices as ‘seeds’. This method is usually called ‘orderly generation’.

Lemma 3.6

The number of \(n\times n\) canonical candidate Gram matrices with entries \(\{0,\pm \alpha ,\pm \beta \}\) (in which all three symbols do not appear simultaneously) is given in Table 3 for \(n\in \{1,\dots ,8\}\).

Table 3 The number of candidate Gram matrices up to equivalence

Proof

Case \(n=1\) is [1], case \(n=2\) are

$$\begin{aligned} \begin{bmatrix}1 &{} 0\\ 0 &{} 1\end{bmatrix}\quad \text {and}\quad \begin{bmatrix}1 &{} \alpha \\ \alpha &{} 1\end{bmatrix}.\end{aligned}$$

Case \(n=3\) is shown in Example 3.3. The remaining cases follow by computation. \(\square \)

As seen from Table 3, the number of \(n\times n\) candidate Gram matrices grows very rapidly. However, when \(d\ge 2\) is fixed and \(n=d+2\), then we may filter out a very large fraction of candidate Gram matrices with the aid of Proposition 3.5. Indeed, for a given candidate Gram matrix we can check whether (3) has any complex solutions by computing a degree reverse lexicographic reduced Gröbner basis [6], and keep only those candidate Gram matrices in a set \(\mathscr {C}_d(n)\) for which some solutions are found. We performed this step with the aid of the C++ library ‘CoCoA’ [1].

We proceed by augmenting each candidate Gram matrix \(C\in \mathscr {C}_d(n)\) with a new row (and column) whose prefix \([C_{n+1,1},C_{n+1,2},\dots ,C_{n+1,n-1}]\) is lexicographically larger than the respective prefix of the last row of C (cf. (4)), keeping only those canonical matrices which in addition survive the next computationally cheap test.

Lemma 3.7

(combinatorial test)   Let \(d\ge 2\) be fixed and let \(\mathscr {C}_d(n)\) be a set containing all pairwise inequivalent candidate Gram matrices of order n for which the system of equations (3) has a solution. Let C be a candidate Gram matrix of order \(n+1\). If C corresponds to a Gram matrix in \(\mathbb {R}^d\), then necessarily all its \(n+1\) principal submatrix of order n belong to the set \(\mathscr {C}_d(n)\), up to equivalence.

Proof

Indeed, if C corresponds to some Gram matrix, then there exist real numbers \(\alpha ^*,\beta ^*\) (subject to (2)) such that \({\text {rank}}C(\alpha ^*,\beta ^*)\le d\). Since the rank of submatrices cannot increase, this must be true for every principal submatrix of \(C(\alpha ^*,\beta ^*)\). Therefore they must be in the set \(\mathscr {C}_d(n)\), up to equivalence. \(\square \)

Since the \(n\times n\) principal submatrix of a candidate Gram matrix of order \(n+1\) must be compatible, we test them further with the following.

Corollary 3.8

(weak Gröbner test, cf. Proposition 3.5) Let \(d\ge 2\) be fixed, and let \(C(\alpha ,\beta )\) be a candidate Gram matrix of order \(n\ge d+1\). Let \(\mathscr {M}\) denote the set of all \((d+1)\times (d+1)\) principal submatrices of C. Let \(\omega \) be an auxiliary variable. If the following system of polynomial equations,

$$\begin{aligned} {\left\{ \begin{array}{ll} \det M(\alpha ,\beta )=0,\qquad \text {for all }M\in \mathscr {M},\\ \omega \alpha \beta (\alpha ^2-\beta ^2)(\alpha ^2-1)(\beta ^2-1)+1=0, \end{array}\right. } \end{aligned}$$

has no solutions in \(\mathbb {C}^3\), then \({\text {rank}}C(\alpha ^*,\beta ^*)\le d\) cannot hold for any \(\alpha ^*,\beta ^*\in \mathbb {R}\) subject to (2).

Proof

This is a variant of Proposition 3.5. \(\square \)

Finally, we store all surviving matrices in a set \(\mathscr {C}_d(n+1)\), and repeat this procedure as long as new matrices are discovered (but until n reaches the absolute bound from Theorem 3.1). Once the largest candidate Gram matrices are found, we use Proposition 3.5 to determine explicitly the matrices with rank at most d, and then by computing their characteristic polynomial (or eigenvalues, if it is possible) we determine the positive semidefinite matrices. We remark that the set of inner products of the maximum Gram matrices is a by-product of this procedure. We summarize our approach in the following ‘roadmap’ which we will frequently use as a convenient reference.

Roadmap 3.9

The following is our approach for generating and classifying biangular lines in \(\mathbb {R}^d\):

  • Fix the dimension \(d\ge 2\).

  • Generate all \(\{0,\pm \alpha ,\pm \beta \}\) canonical candidate Gram matrices (with at most two symbols) of size \(d+1\), and store them in a set \(\mathscr {C}_d(d+1)\).

  • Augment every \(C\in \mathscr {C}_d(d+1)\) with a new row and column in every possible way, and then test the canonical matrices by Proposition 3.5. Store the surviving matrices of size \(d+2\) in a set \(\mathscr {C}_d(d+2)\).

  • For every \(i\in \bigl \{d+2,\dots ,\left( {\begin{array}{c}d+3\\ 4\end{array}}\right) \bigr \}\) augment every \(C\in \mathscr {C}_d(i)\) with a new row and column in every possible way, and then test the canonical matrices by Lemma 3.7 and Corollary 3.8. Store the surviving matrices of size \(i+1\) in a set \(\mathscr {C}_d(i+1)\), and repeat this step.

  • For the largest candidate Gram matrices use Proposition 3.5 and in particular the solutions of the system of equations (3) to determine the real matrices of rank at most d.

  • Select from these the positive semidefinite matrices.

Remark 3.10

We observed that once the size n of candidate Gram matrices is large enough, say \(n\ge d+5\), then essentially all matrices survive Corollary 3.8. In these cases we solely rely on Lemma 3.7 for pruning. We believe that the reason for this phenomenon is related to the fact that the congruence order of \(\mathbb {R}^d\) is \(d+3\), see [34, Thm. 7.2].

Remark 3.11

Let \(d\ge 3\), \(n\ge d+1\), \(\alpha ^*,\beta ^*\in \mathbb {R}\) fixed, and let \(C(\alpha ^*,\beta ^*)\) be an \(n\times n\) Gram matrix with \({\text {rank}} C(\alpha ^*,\beta ^*)\le d-2\). Then for every \(v\in \mathbb {R}^n\),

$$\begin{aligned} {\text {rank}}\begin{bmatrix} C(\alpha ^*,\beta ^*) &{} v^T\\ v &{} 1\end{bmatrix}\le d\end{aligned}$$

by subadditivity. In particular, the tests described in Proposition 3.5 and Corollary 3.8 have no effect.

Remark 3.12

There are two major techniques for matrix canonization: one relies on formula (4) which nicely fits into the framework of ‘orderly generation’. The other possibility is to transform the problem of matrix canonization to graph canonization for which there are readily available efficient implementations, such as the ‘nauty’ software [35]. In Appendix A we describe a graph representation of candidate Gram matrices, which can be used in the framework of ‘canonical augmentation’. These two techniques are of similar efficiency, and we have used both of them to cross-check our results. We refer the reader to [11] and references therein.

4 Classification of Maximum Biangular Lines

We implemented the framework developed in Sect. 3 in C++ and used a computer cluster with 500 CPU cores for several weeks to obtain the following new classification results in \(\mathbb {R}^d\) for \(d\le 6\). For completeness, we begin our discussion with the case \(d=2\) by giving an independent, computational proof to Lemma 2.1.

Lemma 4.1

(equivalent restatement of Lemma 2.1) The maximum cardinality of a biangular line system in \(\mathbb {R}^2\) is 5. The unique configuration has candidate Gram matrix

$$\begin{aligned} C(\alpha ,\beta )=\begin{bmatrix} 1 &{} \alpha &{} \alpha &{} \beta &{} \beta \\ \alpha &{} 1 &{} \beta &{} \alpha &{} \beta \\ \alpha &{} \beta &{} 1 &{} \beta &{} \alpha \\ \beta &{} \alpha &{} \beta &{} 1 &{} \alpha \\ \beta &{} \beta &{} \alpha &{} \alpha &{} 1 \end{bmatrix} \end{aligned}$$
(5)

and Gram matrix \(C((\sqrt{5}-1)/4,(-\sqrt{5}-1)/4)\), describing the main diagonals of the convex regular decagon.

Table 4 \(\{0,\pm \alpha ,\pm \beta \}\) candidate Gram matrices in \(\mathbb {R}^2\)

Proof

The proof follows Roadmap 3.9 with \(d=2\). In Table 4 we display the number of surviving candidate Gram matrices, that is, the numbers \(|\mathscr {C}_2(n)|\) for \(n\in \{2,\dots ,6\}\). Since \(|\mathscr {C}_2(6)|=0\), it follows that \(|\mathscr {C}_2(n)|=0\) for every \(n\ge 6\). The unique maximum candidate Gram matrix of size 5 is shown in (5) from which the Gram matrices can be recovered by solving the system of equations (3). It follows that \(4\alpha ^2+2\alpha -1=0\) and \(\beta =-\alpha -1/2\). This yields two permutation equivalent, positive semidefinite solutions: \(C((\sqrt{5}-1)/4,(-\sqrt{5}-1)/4)\) and \(C((-\sqrt{5}-1)/4,(\sqrt{5}-1)/4)\), both corresponding to the main diagonals of the convex regular decagon. \(\square \)

Remark 4.2

The four lines, passing through the antipodal vertices of the convex regular octagon form the second largest, inextendible configuration of biangular lines in \(\mathbb {R}^2\) with the set of inner products \(\{0,\pm 1/\sqrt{2}\}\).

Theorem 4.3

The maximum cardinality of a biangular line system in \(\mathbb {R}^3\) is 10. The unique configuration has candidate Gram matrix

$$\begin{aligned} C(\alpha ,\beta )=\begin{bmatrix} 1 &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} \beta &{} \beta &{} \beta \\ \alpha &{} 1 &{} \alpha &{} -\alpha &{} -\alpha &{} \beta &{} -\beta &{} \alpha &{} -\alpha &{} \beta \\ \alpha &{} \alpha &{} 1 &{} \beta &{} -\beta &{} -\alpha &{} -\alpha &{} -\alpha &{} \alpha &{} \beta \\ \alpha &{} -\alpha &{} \beta &{} 1 &{} -\alpha &{} -\beta &{} \alpha &{} -\alpha &{} \beta &{} \alpha \\ \alpha &{} -\alpha &{} -\beta &{} -\alpha &{} 1 &{} \alpha &{} \beta &{} \beta &{} \alpha &{} -\alpha \\ \alpha &{} \beta &{} -\alpha &{} -\beta &{} \alpha &{} 1 &{} -\alpha &{} \beta &{} -\alpha &{} \alpha \\ \alpha &{} -\beta &{} -\alpha &{} \alpha &{} \beta &{} -\alpha &{} 1 &{} \alpha &{} \beta &{} -\alpha \\ \beta &{} \alpha &{} -\alpha &{} -\alpha &{} \beta &{} \beta &{} \alpha &{} 1 &{} \alpha &{} \alpha \\ \beta &{} -\alpha &{} \alpha &{} \beta &{} \alpha &{} -\alpha &{} \beta &{} \alpha &{} 1 &{} \alpha \\ \beta &{} \beta &{} \beta &{} \alpha &{} -\alpha &{} \alpha &{} -\alpha &{} \alpha &{} \alpha &{} 1 \end{bmatrix} \end{aligned}$$
(6)

and Gram matrix \(C(1/3,\sqrt{5}/3)\), corresponding to the main diagonals of the platonic dodecahedron.

Table 5 \(\{0,\pm \alpha ,\pm \beta \}\) candidate Gram matrices in \(\mathbb {R}^3\)

Proof

The proof follows Roadmap 3.9 with \(d=3\). In Table 5 we display the number of surviving candidate Gram matrices, that is, the numbers \(|\mathscr {C}_3(n)|\) for \(n\in \{2,\dots ,11\}\). Since \(|\mathscr {C}_3(11)|=0\), it follows that \(|\mathscr {C}_3(n)|=0\) for every \(n\ge 11\). The unique maximum candidate Gram matrix of size 10 is shown in (6). Equations (3) imply that \(\alpha =1/3\) and \(\beta ^2=5/9\). This yields two permutation equivalent, positive semidefinite solutions: \(C(1/3,\sqrt{5}/3)\) and \(C(1/3,-\sqrt{5}/3)\), both corresponding to the main diagonals of the platonic dodecahedron. \(\square \)

Remark 4.4

The second largest among inextendible examples in \(\mathbb {R}^3\) can be obtained by lifting the convex regular heptagon by Proposition 2.6 to two carefully chosen heights.

Theorem 4.5

The maximum cardinality of a biangular line system in \(\mathbb {R}^4\) is 12. There are four pairwise nonisometric maximum configurations: the shortest vectors of the \(D_4\) lattice; the shortest vectors of the \(D_3\) lattice after lifting; and two spherical 3-distance sets with the common candidate Gram matrix

$$\begin{aligned} C(\alpha ,\beta )=\begin{bmatrix} B(\alpha ,\beta )+I_6 &{} B(\beta ,\alpha )-\beta I_6\\ B(\beta ,\alpha )-\beta I_6 &{} B(\alpha ,\beta )+I_6 \end{bmatrix},\quad B(\alpha ,\beta )=\begin{bmatrix} 0 &{} \alpha &{} \alpha &{} \alpha &{} \alpha &{} \alpha \\ \alpha &{} 0 &{} \alpha &{} \beta &{} \beta &{} \alpha \\ \alpha &{} \alpha &{} 0 &{} \alpha &{} \beta &{} \beta \\ \alpha &{} \beta &{} \alpha &{} 0 &{} \alpha &{} \beta \\ \alpha &{} \beta &{} \beta &{} \alpha &{} 0 &{} \alpha \\ \alpha &{} \alpha &{} \beta &{} \beta &{} \alpha &{} 0 \end{bmatrix}, \end{aligned}$$
(7)

yielding nonisometric Gram matrices \(C((3-2\sqrt{5})/11,(4+\sqrt{5})/11)\) and \(C((3+2\sqrt{5})/11,(4-\sqrt{5})/11)\).

Table 6 \(\{0,\pm \alpha ,\pm \beta \}\) candidate Gram matrices in \(\mathbb {R}^4\)

Proof

The proof follows Roadmap 3.9 with \(d=4\). In Table 6 we display the number of surviving candidate Gram matrices, that is, the numbers \(|\mathscr {C}_4(n)|\) for \(n\in \{2,\dots ,13\}\). Since \(|\mathscr {C}_4(13)|=0\), it follows that \(|\mathscr {C}_4(n)|=0\) for every \(n\ge 13\). The candidate Gram matrices corresponding to the \(D_4\) and the lifted \(D_3\) lattice vectors are not shown here, as they can be easily recovered from Lemma 2.3 and Proposition 2.6, and one may check by solving (3) that these are the only solutions. Interestingly, the third candidate Gram matrix \(C(\alpha ,\beta )\) shown in (7) yields two nonisometric solutions, as the equations (3) imply that \(11\alpha ^2-6\alpha -1=0\) and \(\beta =\alpha /2-1/2\). \(\square \)

We note that since the candidate Gram matrix (7) describes a spherical 3-distance set, it has already been generated earlier in [44].

Remark 4.6

The Gram matrices obtained from (7) are contained in the Bose–Mesner algebra of a 3-class association scheme [25].

Theorem 4.7

The maximum cardinality of a biangular line system in \(\mathbb {R}^5\) is 24. The unique configuration can be obtained by lifting the shortest vectors of the \(D_4\) lattice.

Table 7 \(\{0,\pm \alpha ,\pm \beta \}\) candidate Gram matrices in \(\mathbb {R}^5\)

Proof

The proof follows Roadmap 3.9 with \(d=5\). In Table 7 we display the number of surviving candidate Gram matrices, that is, the numbers \(|\mathscr {C}_5(n)|\) for \(n\in \{6,\dots ,25\}\). Since \(|\mathscr {C}_5(25)|=0\), it follows that \(|\mathscr {C}_5(n)|=0\) for every \(n\ge 25\). The candidate Gram matrix corresponding to the lifted \(D_4\) lattice vectors is not shown here, as it can be easily recovered from Lemma 2.3 and Proposition 2.6, and one may check by solving (3) that it is the only maximum solution. \(\square \)

Remark 4.8

We remark that the Bose–Mesner algebra (see [25]) of a particular example of 4-class association schemes on 24 vertices contains the maximum Gram matrix G of biangular lines in \(\mathbb {R}^5\), up to equivalence. Furthermore, since \(G^2=(24/5)G\), G is a sporadic example of biangular tight frames [14].

The main computational result of this paper is the following.

Theorem 4.9

The maximum cardinality of a biangular line system in \(\mathbb {R}^6\) is 40. The unique configuration can be obtained by lifting the shortest vectors of the \(D_5\) lattice.

Table 8 \(\{0,\pm \alpha ,\pm \beta \}\) candidate Gram matrices in \(\mathbb {R}^6\)

Proof

The proof follows Roadmap 3.9 with \(d=6\). In Table 8 we display the number of surviving candidate Gram matrices, that is, the numbers \(|\mathscr {C}_6(n)|\) for \(n\in \{8,\dots ,41\}\). Since \(|\mathscr {C}_6(41)|=0\), it follows that \(|\mathscr {C}_6(n)|=0\) for every \(n\ge 41\). The candidate Gram matrix corresponding to the lifted \(D_5\) lattice vectors is not shown here, as it can be easily recovered from Lemma 2.3 and Proposition 2.6, and one may check by solving (3) that it is the only maximum solution. \(\square \)

In dimension 5 and 6 the largest biangular line systems with irrational angles consist of 20 and 24 lines respectively, each having exactly the same inner product set \(\{\pm (3-2\sqrt{5})/11,\pm (4+\sqrt{5})/11\}\) as one of the largest configurations in \(\mathbb {R}^4\) (cf. Theorem 4.5). Examples of these are shown in Appendix B.

Remark 4.10

In \(\mathbb {R}^6\) two \(27\times 27\) candidate Gram matrices were found corresponding to Gram matrices with angle set \(\{\pm 1/4,\pm 1/2\}\). It turns out, one of these is the largest spherical 2-distance set [34, 37], and the other one belongs to the Bose–Mesner algebra of a 4-class association scheme [25]. See Appendix B.

We note the following by-products of our classification.

Corollary 4.11

The largest infinite family of biangular lines in \(\mathbb {R}^d\) for \(d\in \{3,4,5,6\}\) is formed by 6, 6, 10, and 16 lines, respectively.

Proof

For \(d=3\) we have the twisted icosahedron [14]. For \(d\ge 4\), we can use Proposition 2.5 and well-known spherical 2-distance sets (see [34, 37], Examples 3.4 and B.3) in \(\mathbb {R}^{d-1}\) to establish the claimed lower bounds. To see that these are indeed the largest, one should inspect the candidate Gram matrices we generated. It is easy to see that if \(C(\alpha ,\beta )\) is a parametric family of biangular line systems, then the values of \(\alpha \) and \(\beta \) are not uniquely determined by any of its subsystems. Therefore it is enough to augment those (rather few) candidate Gram matrices for which the dimension of the ideal generated by (3) is positive (see [6]). \(\square \)

Corollary 4.12

The biangular line systems meeting the relative bound in dimension \(d\in \{3,4,5,6\}\) for \(\alpha ^2+\beta ^2<6/(d+4)\) are exactly those listed in Table 2.

Proof

Let \(\mathscr {X}\subset \mathbb {R}^d\) span a biangular line system meeting the relative bound (1). Since \(\alpha ^2+\beta ^2<6/(d+4)\), we have

$$\begin{aligned} \sum _{x,x'\in \mathscr {X}}\!C_2^{((d-2)/2)}(\langle x,x'\rangle )=0\quad \ \text {and}\quad \sum _{x,x'\in \mathscr {X}}\!C_4^{((d-2)/2)}(\langle x,x'\rangle )=0.\end{aligned}$$

In particular, the antipodal double \(\mathscr {Y}:=\{x:x\in \mathscr {X}\}\cup \{-x:x\in \mathscr {X}\}\) is a spherical 5-design [4, 10], and hence \(|\mathscr {X}|=|\mathscr {Y}|/2\ge d(d+1)/2\). For \(d=3\) the only tight spherical 5-design is the icosahedron [4, 20, Exam. 5.16]. For \(d\ge 4\) it follows from Corollary 4.11 that the number of Gram matrices of size \(|\mathscr {X}|\) is finite, therefore one may plug in the (finitely many) inner products \(\alpha ^*\) and \(\beta ^*\) into (1) to test equality. This yields Table 2 for \(d\le 6\). \(\square \)

Remark 4.13

If \(\alpha ^2+\beta ^2=6/(d+4)\) and there is equality in the relative bound (1), then necessarily

$$\begin{aligned} \frac{d^2+3|\mathscr {X}|-4}{(d+2)(d+4)}=(|\mathscr {X}|-1)\beta ^2\biggl (\frac{6}{d+4}-\beta ^2\biggr ).\end{aligned}$$

For fixed d and \(|\mathscr {X}|\) this in turn determines the possible inner products in \(A(\mathscr {X})\). Then one may go through all candidate Gram matrices and check which of these inner products are compatible with the solutions of (3). Since we tend to believe that for \(d\le 6\) there are no biangular lines of this type, we have not gone through the details of this lengthy and seemingly very tedious task.

5 Results on Multiangular Lines

The theory developed in Sect. 3 can be generalized to multiangular lines in a straightforward manner. The main challenge in our study is solving (the multiangular analogue of) the system of equations (3). Indeed, the efficiency of computing a Gröbner basis very much depends on the number of variables [6], and 4-angular line systems are the largest ones our methods can currently handle. In this section we briefly report on our computational results on multiangular lines.

5.1 Multiangular Lines in \(\mathbb {R}^3\)

It is well known that in \(\mathbb {R}^3\) the main diagonals of the platonic icosahedron form the largest equiangular line system, and we showed in Theorem 4.3 that the main diagonals of the platonic dodecahedron form the largest biangular line system. It is natural to ask what are the multiangular analogues of these objects. On the plane the maximum cardinality of m-angular lines is \(2m+1\), and an example is coming from the main diagonals of the convex regular \((4m+2)\)-gon [37].

Theorem 5.1

The maximum cardinality of a triangular line system in \(\mathbb {R}^3\) is 12. There are exactly two such configurations coming from the following candidate Gram matrix:

(8)

namely \(C((-7+4\sqrt{2})/17,(5+2\sqrt{2})/17,(-3-8\sqrt{2})/17)\) is the truncated cube and \(C((-7-4\sqrt{2})/17,(5-2\sqrt{2})/17,(-3+8\sqrt{2})/17)\) is the small rhombicuboctahedron.

Proof

The proof follows analogously to Roadmap 3.9 with \(d=3\). In Table 9 we display the number of surviving candidate Gram matrices with symbols \(\{0,\pm \alpha ,\pm \beta ,\pm \gamma \}\) (where at most three out of these four symbols appear), that is, the numbers \(|\mathscr {C}_3(n)|\) for \(n\in \{2,\dots ,13\}\). Since \(|\mathscr {C}_3(13)|=0\), it follows that \(|\mathscr {C}_3(n)|=0\) for every \(n\ge 13\). In addition, there is a unique maximum candidate Gram matrix of size 12, as shown in (8). Equations analogous to (3) imply the claimed solutions. \(\square \)

Theorem 5.2

The maximum cardinality of a 4-angular line system in \(\mathbb {R}^3\) is 15. There is a unique configuration coming from the following candidate Gram matrix:

(9)

namely \(C((1+\sqrt{5})/4,(1-\sqrt{5})/4,1/2)\) is the icosidodecahedron.

Table 9 \(\{0,\pm \alpha ,\pm \beta , \pm \gamma \}\) candidate Gram matrices in \(\mathbb {R}^3\)
Table 10 \(\{0,\pm \alpha ,\pm \beta , \pm \gamma ,\pm \delta \}\) candidate Gram matrices in \(\mathbb {R}^3\)

Proof

The proof follows analogously to Roadmap 3.9 with \(d=3\), with the following noted difference: first we generated all \(5\times 5\) candidate Gram matrices, and used Proposition 3.5 for filtering the \(6\times 6\) (and larger) matrices. In Table 10 we display the number of surviving candidate Gram matrices with symbols \(\{0,\pm \alpha ,\pm \beta ,\pm \gamma ,\pm \delta \}\) (where at most four out of these five symbols appear), that is, the numbers \(|\mathscr {C}_3(n)|\) for \(n\in \{2,\dots ,16\}\). Since \(|\mathscr {C}_3(16)|=0\), it follows that \(|\mathscr {C}_3(n)|=0\) for every \(n\ge 16\). In addition, there is a unique maximum candidate Gram matrix of size 15, as shown in (9). Equations analogous to (3) imply that \(4\alpha ^2-2\alpha -1=0\), \(\beta =1/2-\alpha \), \(\gamma =1/2\). This yields two equivalent, positive semidefinite solutions, both corresponding to the main diagonals of the icosidodecahedron. \(\square \)

Remark 5.3

It turns out that the icosidodecahedron is the largest 5-angular configuration in \(\mathbb {R}^3\) containing orthogonal lines. The search is completely analogous to what is described in Theorem 5.2 and its proof.

We refer the reader to [26] for further interesting arrangements in \(\mathbb {R}^3\).

5.2 Higher Dimensional Examples

In this section we report on our computational results on triangular line systems, where one of the three possible inner products is 0. On the plane, the unique maximum configuration is formed by the main diagonals of the convex regular 12-gon, and in dimension 3 these are once again the main diagonals of the dodecahedron. Both of these results can be concluded from inspecting the matrices we generated for the proof of Theorem 5.1 (see Table 9).

Theorem 5.4

The maximum cardinality of a triangular line system containing orthogonal lines in \(\mathbb {R}^4\) is 24. There is a unique configuration spanned by

$$\begin{aligned} \mathscr {X}= & {} \{[1,\pm 1,\pm 1,\pm 1]/2\}\cup \{[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]\}\nonumber \\&\quad \cup \,\bigl \{x:x\text { is a permutation of } [\pm 1,\pm 1,0,0]/\sqrt{2},\,\langle x,[4,3,2,1]\rangle >0\bigr \}, \end{aligned}$$

which describes the main diagonals of the 24-cell and its dual.

Table 11 \(\{0,\pm \alpha ,\pm \beta \}\) candidate Gram matrices in \(\mathbb {R}^4\)

Proof

The proof follows analogously to Roadmap 3.9 with \(d=4\). In Table 11 we display the number of surviving candidate Gram matrices with symbols \(\{0,\pm \alpha ,\pm \beta \}\), that is, the numbers \(|\mathscr {C}_4(n)|\) for \(n\in \{2,\dots ,25\}\). Since \(|\mathscr {C}_4(25)|=0\), it follows that \(|\mathscr {C}_4(n)|=0\) for every \(n\ge 25\). The unique largest candidate Gram matrix corresponding to this case can be easily recovered from \(\mathscr {X}\), and then solving (3) yields two equivalent solutions with the set of inner products \(\{0,\pm 1/2,\pm 1/\sqrt{2}\}\). \(\square \)

Remark 5.5

In \(\mathbb {R}^4\), the second largest inextendible configuration has cardinality 16, spanned by all permutations of \([\pm 1,\pm 1,\pm 1,0]/\sqrt{3}\) where the first nonzero entry is positive. The set of inner products of this configuration is \(\{0,\pm 1/3,\pm 2/3\}\).

Theorem 5.6

The maximum cardinality of a triangular line system containing orthogonal lines in \(\mathbb {R}^5\) is 40. This unique configuration is spanned by the set \(\mathscr {X}\) of all permutations of \([\pm 1,\pm 1,\pm 1,0,0]/\sqrt{3}\), where the first nonzero entry is positive.

Table 12 \(\{0,\pm \alpha ,\pm \beta \}\) candidate Gram matrices in \(\mathbb {R}^5\)

Proof

The proof follows analogously to Roadmap 3.9 with \(d=5\). In Table 12 we display the number of surviving candidate Gram matrices with symbols \(\{0,\pm \alpha ,\pm \beta \}\), that is, the numbers \(|\mathscr {C}_5(n)|\) for \(n\in \{7,\dots ,41\}\). Since \(|\mathscr {C}_5(41)|=0\), it follows that \(|\mathscr {C}_5(n)|=0\) for every \(n\ge 41\). The unique largest candidate Gram matrix corresponding to this case can be easily recovered from \(\mathscr {X}\), and then solving (3) yields a unique solution with the set of inner products \(\{0,\pm 1/3,\pm 2/3\}\). \(\square \)

6 Open Problems

We conclude this paper with the following set of problems.

Problem 6.1

(superquadratic lines, see [3]) Let \(c,\varepsilon >0\) be fixed. Find a construction of a series of biangular lines \(\mathscr {X}_d\subset \mathbb {R}^d\) such that \(|\mathscr {X}_d|\ge c\cdot d^{2+\varepsilon }\) holds for infinitely many \(d\ge 1\).

In particular, investigate if Proposition 2.6 can be applied to a suitable series of spherical 3-distance sets.

Problem 6.2

Find a series of spherical 3-distance sets \(\mathscr {X}_d\subset \mathbb {R}^d\) with \(A(\mathscr {X}_d)\subseteq \{\alpha _d,\beta _d,\gamma _d\}\) such that \(\alpha _d+\beta _d<0\) and \(|\mathscr {X}_d|\) is superquadratic (in the sense of Problem 6.1).

Problem 6.3

(see [14]) Find a series of biangular tight frames \(\mathscr {X}_d\subset \mathbb {R}^d\) such that \(|\mathscr {X}_d|>d^2\) for infinitely many \(d\ge 1\).

It is known that the twisted icosahedron [14] forms an infinite family of six biangular lines in \(\mathbb {R}^3\), which is one line larger compared to what Proposition 2.5 guarantees.

Problem 6.4

(cf. Corollary 4.11) Determine if there exists an infinite family of biangular lines \(\mathscr {X}(h)\subset \mathbb {R}^d\) such that \(|\mathscr {X}(h)|\) is larger than the one described in Proposition 2.5 for some \(d\ge 7\).

Problem 6.5

(see [32], cf. Example B.3) Determine if there exists an infinite family of 28 biangular lines \(\mathscr {X}(h)\subset \mathbb {R}^7\) such that \(\mathscr {X}(0)\) spans equiangular lines.

It would be also very interesting to see whether binary codes with four distinct distances lead to improved constructions in \(\mathbb {R}^d\) for some \(d\le 23\) or possibly beyond.

Problem 6.6

(see Lemma 2.2) For \(d\ge 2\) determine the maximum cardinality of binary codes of length d admitting at most four distinct Hamming distances \(\{\varDelta _1,\varDelta _2,d-\varDelta _1,d-\varDelta _2\}\), \(\varDelta _1,\varDelta _2\in \{1,\dots ,d-1\}\).

Problem 6.7

(cf. Theorem 2.18, Remark 4.13) Determine if there exists a set \(\mathscr {X}\subset \mathbb {R}^d\) spanning biangular lines with \(A(\mathscr {X})\subseteq \{\pm \alpha ,\pm \beta \}\), such that \(\alpha ^2+\beta ^2=6/(d+4)\), and there is equality in (1) for some \(d\ge 3\).

Problem 6.8

(see [2]) Complement Table 1 by using the semidefinite programming technique to establish (sharp) upper bounds for the maximum cardinality of biangular lines \(\mathscr {X}\subset \mathbb {R}^d\) with \(A(\mathscr {X})\subseteq \{\pm 1/5,\pm 3/5\}\) for \(d\le 23\).