On the 430-cap of $\textrm{PG}(6,4)$ having two intersection sizes with respect to hyperplanes

Abstract

Let ${\mathcal {C}}$ be a 430-cap of $\textrm{PG}(6,4)$ having two intersection sizes with respect to hyperplanes. We show that no hyperplane of $\textrm{PG}(6,4)$ intersects ${\mathcal {C}}$ in a Hill 78-cap. So if it can be shown that the Hill 78-cap of $\textrm{PG}(5,4)$ is projectively unique, then such a 430-cap does not exist, or equivalently, a two-weight $[430,7]_{\mathbb {F}_4}$ linear code with dual weight at least 4, does not exist.

1 Introduction

Uniformly packed codes generalise perfect codes, and the 1-error correcting examples have connections to strongly regular graphs, partial quadrangles, and two-character sets in finite projective spaces (see [3]). An e-error correcting code C is uniformly packed if spheres of radius $e+1$ about codewords cover the whole space, and vectors at distance e from the C are in $\lambda +1$ spheres while vectors at distance $e+1$ from the code are in $\mu$ spheres. For the case that $e=1$, a code C is 1-error correcting if and only if the dual code $C^\perp$ has two nonzero weights. If C has minimum distance at least 3, then $C^\perp$ is projective, and gives rise to a two-character set of a projective space: a set of points S such that there are only two values for the possible intersection size of a hyperplane with S.

There is also a connection with finite partial quadrangles. Partial quadrangles were introduced by Cameron [4] as a (finite) geometry of points and lines such that every two points are on at most one line, there are $s+1$ points on a line, every point is on $t+1$ lines and satisfying the following two important properties: (i) for every point P and every line $\ell$ not incident with P, there is at most one point on $\ell$ collinear with P; (ii) there is a constant $\mu$ such that for every pair of non-collinear points (X, Y) there are precisely $\mu$ points collinear with X and Y. The collinearity graph of a partial quadrangle is strongly regular. The only known partial quadrangles, that are not generalised quadrangles, are: triangle-free strongly regular graphs, obtained by removing points from a generalised quadrangle of order $(s,s^2)$, or one of three exceptional examples arising from the linear representation of one of the Coxeter 11-cap of $\textrm{PG}(4,3)$, the Hill 56-cap of $\textrm{PG}(5,3)$ or the Hill 78-cap of $\textrm{PG}(5,4)$.

A k-cap of a projective space $\textrm{PG}(n,q)$ is a set of k points with no three collinear. Calderbank [2] proved using number-theoretic arguments that if a partial quadrangle is a linear representation of a k-cap then $q\geqslant 5$ or it is isomorphic to the linear representation of one of the following: (i) an ovoid of $\textrm{PG}(3,q)$; (ii) the Coxeter 11-cap of $\textrm{PG}(4,3)$; (iii) the Hill 56-cap of $\textrm{PG}(5,3)$; (iv) a 78-cap of $\textrm{PG}(5,4)$; (v) a 430-cap of $\textrm{PG}(6,4)$. Tzanakis and Wolfskill [8] proved that if $q\geqslant 5$, then the first case applies. It is still not known if case (v) occurs; that is, whether there is a 430-cap of $\textrm{PG}(6,4)$ such that every hyperplane intersects it in 78 or 110 elements. If a hyperplane intersects it in 78 elements, then it is a two-character 78-cap of $\textrm{PG}(5,4)$ (see Lemma 5.1). This leaves two open problems:

• 1. Does there exist a two-character 78-cap of $\textrm{PG}(5,4)$ projectively inequivalent to Hill’s cap?

• 2. Does there exist a two-character 430-cap of $\textrm{PG}(6,4)$?

These problems are of interest to finite geometry and coding theory alike, and have been open for over 40 years, since at least [2]. We show in this note that a negative solution to the first problem implies a negative solution to the second problem.

Theorem 1.1

Let ${\mathcal {C}}$ be a 430-cap of $\textrm{PG}(6,4)$ having two intersection sizes with respect to hyperplanes. Then no hyperplane of $\textrm{PG}(6,4)$ intersects ${\mathcal {C}}$ in a cap projectively equivalent to the Hill 78-cap.

The basic argument proceeds as follows. Suppose H is the Hill 78-cap of $\textrm{PG}(5,4)$ and embed $\textrm{PG}(5,4)$ as a hyperplane $\Pi$ of $\textrm{PG}(6,4)$. Let ${\mathcal {Q}}$ be the partial quadrangle arising from the linear representation of H, and let $\Gamma$ be its collinearity graph. Then $\Gamma$ is a strongly regular graph with parameters (4096, 234, 2, 14). Now the affine points are the points of ${\mathcal {Q}}$, and the affine lines meeting $\Pi$ in a point of H are the lines of ${\mathcal {Q}}$. Let ${\mathcal {C}}$ be a 430-cap of $\textrm{PG}(6,4)$ containing H. So the affine points $\bar{{\mathcal {C}}}:={\mathcal {C}}\backslash \Pi$ of ${\mathcal {C}}$ form a set of points of size 352 of ${\mathcal {Q}}$ such that every line of ${\mathcal {Q}}$ intersects it in at most one point. Moreover, $\bar{{\mathcal {C}}}$ forms a Delsarte coclique for $\Gamma$; a coclique that has size attaining the Delsarte/Hoffman bound. We will show that $\Gamma$ does not have a Delsarte coclique, which then shows that the Hill 78-cap does not extend to a 430-cap of $\textrm{PG}(6,4)$. To do this, we take the Schurian scheme for the automorphism group of $\Gamma$, which is a 9-class fission scheme for the natural 2-class scheme arising from $\Gamma$. We then use another 2-class fusion of this Schurian scheme to yield information on the inner distribution of a putative Delsarte coclique.

2 Some background

Let $\Omega$ be a set, and let $A_0, A_1, \ldots , A_d$ be symmetric $\{0,1\}$-matrices with rows and columns indexed by $\Omega$. Then ${\mathcal {A}}=(\Omega , \{A_0, A_1, \ldots , A_d\})$ is a d-class association scheme if the following conditions hold:

• 1. $A_0$ is the identity matrix I,

• 2. $\sum _{i=0}^d A_i$ is the matrix with every entry equal to 1,

• 3. There exist constants $p_{ij}^k$ depending only on i, j, and k, such that $A_i A_j = \sum _{k=0}^d p_{ij}^k A_k$.

The matrices $A_0$, $A_1$, $\ldots$, $A_d$ are the adjacency matrices of ${\mathcal {A}}$, and indeed, each $A_i$ is the adjacency matrix of an undirected graph. A strongly regular graph $\Delta$ is essentially equivalent to a 2-class association scheme, where $A_1$ and $A_2$ are the adjacency matrices for $\Delta$ and its complement.

It is well known that $\mathbb {R}^\Omega$ decomposes into $d+1$ simultaneous eigenspaces for the adjacency matrices of ${\mathcal {A}}$. Moreover, there are projection matrices $E_0, E_1, \ldots , E_d$ (the minimal idempotents) onto each of these eigenspaces, such that

$$\begin{aligned} E_i = \sum _{j=0}^d Q_{ji} A_j, \end{aligned}$$

where Q is called the matrix of dual eigenvalues. If C is a subset of $\Omega$, then its inner distribution is the vector $a = (a_0,a_1,\ldots ,a_d)$ defined by

$$\begin{aligned} a_i= \frac{1}{|C|}\mathbb {1}_C A_i \mathbb {1}_C^\top . \end{aligned}$$

where we use $\mathbb {1}_C$ to denote the characteristic function of C in $\Omega$. If Q is the matrix of dual eigenvalues of ${\mathcal {A}}$, then

$$\begin{aligned} (a Q)_j = \frac{|\Omega |}{|C|} \mathbb {1}_C E_j \mathbb {1}_C^\top \end{aligned}$$

for all $j\geqslant 0$. The vector aQ is sometimes known as the MacWilliams transform of C, and it follows from the fact that the $E_j$ are positive semidefinite, that each entry of aQ is non-negative.

The dual degree set of C is the set of nonzero indices j for which the j-th coordinate of its MacWilliams transform is nonzero. Two subsets of $\Omega$ are design-orthogonal if their dual degree sets are disjoint. In this case, we have the following elementary result, due at least to Roos.

Theorem 2.1

([7, Corollary 3.3]) If $S,T\subset \Omega$ are design-orthogonal, then $|S\cap T|=\dfrac{|S|\cdot |T|}{|\Omega |}$.

The outer distribution B of S is the $|\Omega |\times d$ matrix, with rows indexed by $\Omega$ and columns indexed by the $R_i$, defined by

$$\begin{aligned} B_{x,i}=|\{ y\in S: (x,y)\in R_i\}|=\mathbb {1}_{\{x\}}A_i\mathbb {1}_S^\top . \end{aligned}$$

A transitive group G acting on $\Omega$ is generously transitive if for any distinct pair $(\alpha ,\beta )$ of elements of $\Omega$, there is some $g\in G$ such that $\alpha ^g=\beta$ and $\beta ^g=\alpha$. If a finite group G acts generously transitively on a set $\Omega$, then the orbits of G on unordered pairs of $\Omega$ give rise to an association scheme: a Schurian association scheme. An association scheme is called a translation scheme if there an abelian group of automorphisms acting regularly on its vertices. If there is an ordering of the relations and minimal idempotents such that the matrix of eigenvalues P is equal to the matrix of dual eigenvalues Q, then we say the association scheme is formally dual. We refer the reader to [6] or [9] for more information on association schemes.

3 A Schurian scheme and some interesting subsets

The following cyclotomic construction of $\Gamma$ can be found as [3, Example FE3]. Let z be a primitive element of $\mathbb {F}_{4^6}$. Let O be $\langle z^{35}\rangle \cup \langle z^{35}\rangle z^7$. Then $\Gamma$ is isomorphic to the Cayley graph $\textrm{Cay}(V,O)$ where V is the additive group of $\mathbb {F}_{4^6}$, and it is a strongly regular graph with parameters (4096, 234, 2, 14). Note that we can also view O as the set of underlying vectors of the Hill 78-cap (n.b., $234=3\times 78$), represented as elements of $\mathbb {F}_{4^6}$.

Some of the details below were aided by computer, and in particular, the AssociationScheme package [1] in GAP. The code can be found in the Appendix. The automorphisms of $\Gamma$ are generated by the translations (addition by elements of V), multiplication by $z^{35}$, and the map $\rho :x\mapsto z^{42} x^4$ (of order 6). So $\textrm{Aut}(\Gamma )$ is isomorphic to $C_2^{12}:(C_{117}:C_6)$. Indeed, the stabiliser of 0 is generated by $z^{35}$ and $\rho$, and these automorphisms act on O. Moreover, $\textrm{Aut}(\Gamma )$ acts generously transitively on the points of $\Gamma$.

Let $v:=4096$, the number of vertices of $\Gamma$. Take the Schurian association scheme ${\mathcal {A}}$ for $\textrm{Aut}(\Gamma )$, which is a fission scheme for the 2-class association scheme ${\mathcal {G}}$ associated to the original strongly regular graph $\Gamma$. Then the valencies of ${\mathcal {A}}$ are (in order) 1, 117, 234, 234, 351, 351, 702, 702, 702, 702 with $R_2$ being the adjacency relation for $\Gamma$ (and the $R_i$ are indexed with $i\in \{0,\ldots ,9\}$). In fact, ${\mathcal {A}}$ is a translation scheme and it is formally dual. The matrix P of eigenvalues, and the matrix Q of dual eigenvalues, for ${\mathcal {A}}$ are:

$$\begin{aligned} P=Q= \begin{bmatrix} 1&{} 117&{} 234&{} 234&{} 351&{} 351&{} 702&{} 702&{} 702&{} 702 \\ 1&{} -27&{} 10&{} 10&{} 15&{} 63&{} 30&{} 30&{} -66&{} -66 \\ 1&{} 5&{} -22&{} 10&{} 15&{} -33&{} 30&{} 30&{} 30&{} -66 \\ 1&{} 5&{} 10&{} -22&{} 15&{} -33&{} 30&{} 30&{} -66&{} 30 \\ 1&{} 5&{} 10&{} 10&{} 47&{} -1&{} -34&{} -34&{} -2&{} -2 \\ 1&{} 21&{} -22&{} -22&{} -1&{} 31&{} -2&{} -2&{} -2&{} -2 \\ 1&{} 5&{} 10&{} 10&{} -17&{} -1&{} 30&{} -34&{} -2&{} -2 \\ 1&{} 5&{} 10&{} 10&{} -17&{} -1&{} -34&{} 30&{} -2&{} -2 \\ 1&{} -11&{} 10&{} -22&{} -1&{} -1&{} -2&{} -2&{} 30&{} -2 \\ 1&{} -11&{} -22&{} 10&{} -1&{} -1&{} -2&{} -2&{} -2&{} 30 \end{bmatrix} \end{aligned}$$

We note that there is an involution acting as automorphisms on the association scheme. It is induced by the following semilinear map of order 12:

$$\begin{aligned} \tau :x\mapsto z^{14} x^2 \end{aligned}$$

and it interchanges relations of the association scheme in the following way: $R_2^\tau =R_3$, $R_6^\tau =R_7$, and $R_8^\tau =R_9$.

There are some unions of relations in ${\mathcal {A}}$ that yield interesting graphs.

• (i) $R_2$ is the original strongly regular graph $\Gamma$. Moreover, the non-principal minimal idempotents for $\Gamma$ are $\sum _{j\in \{1, 3, 4, 6,7, 8\}}E_j$ and $\sum _{j\in \{2,5,9\}}E_j$, where $E_j$ is the j-th minimal idempotent for ${\mathcal {A}}$.

• (ii) $R_2\cup R_7\cup R_8$ yields a strongly regular Cayley graph ${\mathcal {F}}$ that will feature in our proof of Theorem 1.1. The elements of the subfield $\mathbb {F}_{4^3}$ form a maximal clique of ${\mathcal {F}}$.

Below we list some interesting (Delsarte) designs for ${\mathcal {A}}$. We denote by $V_j$ the j-th eigenspace for ${\mathcal {A}}$, for which the minimal idempotent $E_j$ projects to.

3.1 Example 1: a subfield design

Consider the elements U of V that lie in the subfield $\mathbb {F}_{4^3}$. It turns out that the inner distribution of U is (1, 0, 9, 0, 0, 0, 0, 27, 27, 0), and so its MacWilliams transform is (64, 0, 576, 0, 0, 0, 0, 1728, 1728, 0). Therefore, $\mathbb {1}_U\in V_0\perp V_2\perp V_7\perp V_8$.

3.2 Example 2: a Delsarte coclique

The complement of $\Gamma$ is k-regular with $k=3861$, and it has least eigenvalue $\tau :=-11$. The Delsarte bound for the size of a coclique of $\Gamma$ is then $1-k/\tau =352$. Suppose there exists a coclique S of $\Gamma$ of size 352. Then S is a Delsarte coclique, and $\mathbb {1}_SE=0$ where $E:=\sum _{j\in \{1, 3, 4, 6,7, 8\}}E_j$ (see [6, Corollary 3.7.2]). (Note that $P_{i2}=-22$ for $i\in \{2,5,9\}$.) Recall that if $E:=\sum _{j\in \{1, 3, 4, 6,7, 8\}}E_j$, then $\mathbb {1}_SE=0$. Consider $E_j$ where $j\in \{1, 3, 4, 6,7, 8\}$. Then $E_j=EE_j$ and so $\mathbb {1}_SE_j=\mathbb {1}_SEE_j=0$. So we have $(aQ)_j=0$ for $j\in \{1, 3, 4, 6,7, 8\}$, or in other words,

$$\begin{aligned} \mathbb {1}_S\in V_0\perp V_2\perp V_5\perp V_9. \end{aligned}$$

If we apply the involution $\tau$, we find that $\mathbb {1}_{S^\tau }\in V_0\perp V_3\perp V_5\perp V_8$.

Now consider the vector $v_P:=22\mathbb {1}_{\{P\}}+\mathbb {1}_{P^\perp }$ where P is a point of ${\mathcal {Q}}$, and $P^\perp$ is the set of points adjacent to P. Notice that $v_P=\mathbb {1}_{\{P\}}(A_2+22I)$ and so $v_PE_j=0$ for $j\in \{2,5,9\}$. In particular, $v_P$ is design-orthogonal¹ to S and so $\mathbb {1}_S\cdot v_P=22$. It follows that $|P^\perp \cap S|$ is equal to 22 when $P\notin S$, but equal to 0 when $P\in S$.

4 Proof of Theorem 1.1

Proof

Let S be a Delsarte coclique for $\Gamma$ and let B be the outer distribution of S. By a theorem of Delsarte [5, Theorem 3.1], for all vertices x of $\Gamma$, and for all $j\in \{1, 3, 4, 6, 7, 8\}$,

$$\begin{aligned} \sum _{i=0}^d \frac{P_{ji}}{P_{0i}} B_{x,i}=0. \end{aligned}$$

(1)

Fix an element $x\notin S$. Recall that there are 22 elements of S adjacent to x (see Sect. 3.2 above), and so we can write $B_{x,i}=(0, y_1, 22, y_3, y_4, y_5, y_6, y_7, y_8, 330 - y_1 - y_3 - y_4 - y_5 - y_6 - y_7 - y_8)$ for some $y_i$. Then, we have the following equations (arising from (1)):

j	Equation
1	$220 + y_1 - (y_3 + y_4 + 2 y_5 + y_6 + y_7) = 0$
3	$y_3 + y_5 + y_8 = 110$
4	$y_1 + y_3 + 3 y_4 - y_6 - y_7=0$
6	$y_1 + y_3 + y_6 - y_4 - y_7=0$
7	$y_1 + y_3 + y_7 - y_4 - y_6=0$
8	$2 (y_1 + y_3) - y_8=0$

These equations reduce: $2y_4=y_6=y_7 = y_8$,   $y_3 =110 - y_5 - y_8$,    $y_1 = y_5 + \tfrac{3}{2} y_8-110$. So

$$\begin{aligned} B_{x,i}=(0, y_5 + \tfrac{3}{2} y_8-110, 22, 110 - y_5 - y_8, \tfrac{1}{2}y_8, y_5, y_8, y_8, y_8, 330 - y_5 - 4 y_8). \end{aligned}$$

Let a be the inner distribution of S. Now $A_2$ is the adjacency matrix of $\Gamma$, and so $\mathbb {1}_{S}A_2\mathbb {1}_{S}^\top =0$. Hence we can write the inner distribution of S as $a=(1,x_1,0,x_3,x_4,x_5,x_6,x_7,x_8,351-x_1-x_3-x_4-x_5-x_6-x_8)$, where the $x_i$ are indeterminate. Now multiply by Q to yield the MacWilliams transform of S:

$$\begin{aligned} aQ=32(&11, \quad \tfrac{1}{2}(- x_1 + x_3 + x_4 + 2 x_5 + x_6 + x_7-234), \quad x_1 + x_3 + x_4 + x_6 + x_7 + x_8-234, \\&117 - x_3 - x_5 - x_8,\quad \tfrac{1}{2}(x_1 + x_3 + 3 x_4 - x_6 - x_7), \quad 2 x_1 - x_3 + x_5,\\&x_1 + x_3 - x_4 + x_6 - x_7, \quad x_1 + x_3 - x_4 - x_6 + x_7,\quad -2 (x_1 + x_3) + x_8, \\&351 - 3 x_1 - x_4 - x_5 - x_6 - x_7 - x_8). \end{aligned}$$

Recall that if $E:=\sum _{j\in \{1, 3, 4, 6, 7, 8\}}E_j$, then $\mathbb {1}_SE=0$. Consider $E_j$ where $j\in \{1, 3, 4, 6, 7, 8\}$. Then $E_j=EE_j$ and so $\mathbb {1}_SE_j=\mathbb {1}_SEE_j=0$. So we have $(aQ)_j=0$ for $j\in \{1, 3, 4, 6, 7, 8\}$, and hence

$$\begin{aligned} 2x_4=x_6=x_7 = x_8, \quad x_3 = 117 - x_5 - x_8,\quad x_5 = x_1-\tfrac{3}{2} x_8+117. \end{aligned}$$

Therefore,

$$\begin{aligned} a&=\left( 1,x_1,0,\frac{x_8}{2}-x_1,\frac{x_8}{2},x_1-\frac{3 x_8}{2}+117,x_8,x_8,x_8,-x_1-\frac{5 x_8}{2}+234\right) ,\\ aQ&=\left( 352, 0, 64 ( 2 x_8-117), 0, 0, 32 (117 + 4 x_1 - 2 x_8), 0, 0, 0, 64 (117-2 x_1 - x_8)\right) . \end{aligned}$$

We now take a different fusion scheme yielding a strongly regular graph ${\mathcal {F}}$. Let $A= \sum _{i \in \{2,7,8\}}A_i$ where the $A_i$ are adjacency matrices of ${\mathcal {A}}$, ordered according to the matrix P above. Then A is the adjacency matrix of a strongly regular graph ${\mathcal {F}}$ with parameters (4096, 1638, 662, 650). The matrix of eigenvalues for ${\mathcal {F}}$ is

$$\begin{aligned} P_{\mathcal {F}}=\begin{bmatrix} 1&{}1638&{}2457\\ 1&{}38&{}-39\\ 1&{}-26&{}25 \end{bmatrix} \end{aligned}$$

and the matrix of dual eigenvalues $Q_{\mathcal {F}}$ is exactly the same as $P_{\mathcal {F}}$. From the inner distribution a for S, it follows that $|{\mathcal {F}}(v)\cap S|=2x_8$ for all $v\in S$, where ${\mathcal {F}}(v)$ denotes the neighbourhood of v in ${\mathcal {F}}$. From the outer distribution of S, it follows that $|{\mathcal {F}}(v)\cap S|=2y_8$ for all $v\notin S$. Therefore,

$$\begin{aligned} \mathbb {1}_SA=2x_8\mathbb {1}_S +2y_8 (\mathbb {1}-\mathbb {1}_S) \end{aligned}$$

where $\mathbb {1}$ is the ‘all ones’ vector, and so $(2x_8-2y_8-1638)\mathbb {1}_S+2y_8\mathbb {1}$ is an eigenvector for A. In particular, $(2x_8-2y_8-1638)\mathbb {1}_S+2y_8\mathbb {1}$ is annihilated by one of the non-principal idempotents D of ${\mathcal {F}}$, and so $\mathbb {1}_SD=0$ as $\mathbb {1}D=0$. The inner distribution for S, with respect to ${\mathcal {F}}$, is $a_{\mathcal {F}}:=(1,2x_8,351-2x_8)$ and therefore, its MacWilliams transform is

$$\begin{aligned} a_{\mathcal {F}}Q_{\mathcal {F}}=\left( 352, 64 (2x_8-117), 32(351 - 4 x_8)\right) . \end{aligned}$$

Since $\mathbb {1}_S$ is annihilated by one of the non-principal minimal idempotents, $(a_{\mathcal {F}}Q_{\mathcal {F}})_{j}=0$ for either $j=1$ or $j=2$. So there are two cases to consider.

Case 1: $(a_{\mathcal {F}}Q_{\mathcal {F}})_{1}=0$:

Here we have $x_8=117/2$ and so

$$\begin{aligned} a&=\left( 1, x_1, 0, \tfrac{117}{4} - x_1, \tfrac{117}{4}, \tfrac{117}{4} + x_1, \tfrac{117}{2}, \tfrac{117}{2},\tfrac{117}{2}, \tfrac{351}{4} -x_1\right) ,\\ aQ&=(352, 0, 0, 0, 0, 128 x_1,0,0,0, 32 (117 - 4 x_1)). \end{aligned}$$

This implies that S is design-orthogonal to the subfield design given in Sect. 3.1. So by Roos’ Theorem 2.1,

$$\begin{aligned} |S\cap \mathbb {F}_{4^3}|=\frac{|S||\mathbb {F}_{4^3}|}{|\Gamma |}=\frac{352\cdot 64}{4096}=\frac{11}{2}. \end{aligned}$$

This is a contradiction as $|S\cap \mathbb {F}_{4^3}|$ is an integer.

Case 2: $(a_{\mathcal {F}}Q_{\mathcal {F}})_{2}=0$:

Here we have $x_8=351/4$ and

$$\begin{aligned} a&=\left( 1, x_1, 0, \tfrac{351}{8} - x_1, \tfrac{351}{8}, x_1-\tfrac{117}{8}, \tfrac{351}{4}, \tfrac{351}{4}, \tfrac{351}{4}, \tfrac{117}{8} - x_1\right) ,\\ aQ&=\left( 352, 0, 3744, 0, 0, 32 (4x_1-\tfrac{117}{2}),0,0,0,32(\tfrac{117}{2}-4x_1)\right) . \end{aligned}$$

In particular, $aQ\geqslant 0$ implies that $x_1=\tfrac{117}{8}$ and hence

$$\begin{aligned} aQ=(352, 0, 3744, 0, 0, 0, 0, 0, 0, 0) \end{aligned}$$

and we have $\mathbb {1}_S\in V_0\perp V_2$. So S is design-orthogonal to $S^\tau$, and so by Roos’ Theorem 2.1,

$$\begin{aligned} |S\cap S^\tau |=\frac{|S||S^\tau |}{|\Gamma |}=\frac{352\cdot 352}{4096}=\frac{121}{4}. \end{aligned}$$

This is a contradiction as $|S\cap S^\tau |$ is an integer.

Both cases lead to a contradiction, and so there is no Delsarte coclique. $\square$

5 Conclusion

The existence of a two-character 430-cap would not only yield a new 78-cap of $\textrm{PG}(5,4)$, but it would yield a two-character cap (see Lemma 5.1), and hence a new uniformly packed code with parameters [78, 6] (over $\mathbb {F}_4$). So indeed, if it can be shown that the Hill 78-cap of $\textrm{PG}(5,4)$ is the only two-character cap in this space (up to projectivity), then a two-character 430-cap does not exist. The following is well-known, but the author cannot find it in print, and so it is proved here.

Lemma 5.1

Let ${\mathcal {C}}$ be a 430-cap of $\textrm{PG}(6,4)$, such that every hyperplane intersects it in 78 or 110 elements. Let $\Sigma$ be a hyperplane intersecting ${\mathcal {C}}$ in 78 elements. Then $\Sigma \cap {\mathcal {C}}$ is a cap (of a copy of $\textrm{PG}(5,4)$) such that every hyperplane of $\Sigma$ intersects it in 14 or 22 elements.

Proof

Let $\mu _i$ be the number of elements of $\Sigma \cap {\mathcal {C}}$ in the i-th hyperplane of $\Sigma$. Then the usual counting arguments show that

$$\begin{aligned} \sum _{i}\mu _i&=78 \cdot \genfrac[]{0.0pt}{}{5}{1}_{\mathbb {F}_4}=78\times 341,\\ \sum _{i}\mu _i(\mu _i-1)&=78 \cdot 77\cdot \genfrac[]{0.0pt}{}{4}{1}_{\mathbb {F}_4}=78\times 6545,\\ \sum _{i}\mu _i(\mu _i-1)(\mu _i-2)&=78 \cdot 77\cdot 76 \cdot \genfrac[]{0.0pt}{}{3}{1}_{\mathbb {F}_4}=78\times 122892. \end{aligned}$$

Take a hyperplane H of $\Sigma$. Then it is on five hyperplanes, each meeting the 430-cap ${\mathcal {C}}$ in at most 110 points. Since each point of the cap is in at least one of these hyperplanes (n.b., the span of a point and H is a hyperplane of $\textrm{PG}(6,4)$), we have

$$\begin{aligned} 4(110-| H\cap {\mathcal {C}} |) + 78\geqslant 430 \end{aligned}$$

and hence $| H\cap {\mathcal {C}} |\leqslant 22$. So $(\mu _i - 14)^2 (22-\mu _i)\geqslant 0$ for all i. From the displayed equations above, we have

$$\begin{aligned} \sum _i (\mu _i - 14)^2 (22-\mu _i )&= \sum _i \left( -\mu _i(\mu _i-1)(\mu _i-2)+47\mu _i(\mu _i-1) -763\mu _i+4312\right) \\&= 78\left( -122892+47\times 6545-763\times 341+75460\right) \\&=0. \end{aligned}$$

Therefore, $\mu _i\in \{14,22\}$ as required. $\square$

We remark that the largest coclique of $\Gamma$ that we have been able to find by computation has size 119, but no well established technique that bounds the size of a coclique seemed to eliminate 352 cocliques immediately. This includes eigenvalue bounds, spherical code bounds, the No-Homomorphism Lemma, and the Clique-Adjacency polynomial.

Appendix A: GAP code

The following GAP code was used for computing the matrix of eigenvalues of our translation scheme ${\mathcal {A}}$.