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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 841–854 | Cite as

Linear codes close to the Griesmer bound and the related geometric structures

  • Assia RoussevaEmail author
  • Ivan Landjev
Article
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Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

In this paper, we study the behavior of the function \(t_q(k)\) defined as the maximal deviation from the Griesmer bound of the optimal length of a linear code with a fixed dimension k:
$$\begin{aligned} t_q(k)=\max _d(n_q(k,d)-g_q(k,d)), \end{aligned}$$
where the maximum is taken over all minimum distances d. Here \(n_q(k,d)\) is the shortest length of a q-ary linear code of dimension k and minimum distance d, \(g_q(k,d)\) is the Griesmer bound for a code of dimension k and minimum distance d. We give two equivalent geometric versions of this problem in terms of arcs and minihypers. We prove that \(t_q(k)\rightarrow \infty \) when \(k\rightarrow \infty \) which implies that the problem is non-trivial. We prove upper bounds on the function \(t_q(k)\). For codes of even dimension k we show that \(t_q(k)\le 2(q^{k/2}-1)/(q-1)-(k+q-1)\) which implies that \(t_q(k)\in O(q^{k/2})\) for all k. For three-dimensional codes and q even we prove the stronger estimate \(t_q(3)\le \log q-1\), as well as \(t_q(3)\le \sqrt{q}-1\) for the case q square.

Keywords

Griesmer bound Optimal linear codes Arcs Minihypers 

Mathematics Subject Classification

94B65 51E21 51E22 94B05 94B27 51E15 51E23 

1 Introduction

Let \(\mathbb {F}_q\) be the finite field with q elements and let \(\mathbb {F}_q^n\) denote the vector space of all ordered n-tuples over \(\mathbb {F}_q\). A linear code of length n and dimension k is a k-dimensional subspace of \(\mathbb {F}_q^n\). The minimum (Hamming) distance of a linear code C is the smallest of the distances between different codewords of C. A linear code over \(\mathbb {F}_q\) of length n, dimension k, and minimum distance d is referred to as an \([n,k,d]_q\)-code. For an in-depth introduction to coding theory we refer to [11, 15, 20, 21, 26].

The problem of optimizing one of the parameters of a linear code over \(\mathbb {F}_q\) given the other two is usually called the main problem in coding theory [11, 12]. In fact, one gets three problems: to minimize the length n for given dimension k and minimum distance d, to maximize the dimension k given the length n and the minimum distance d, and, finally, to maximize the minimum distance d for given length n and dimension k. The field \(\mathbb {F}_q\) is assumed to be fixed. The three problems are not equivalent in the sense that a code which is optimal with respect to one of the parameters might not be optimal with respect to the other two. It is easily checked that a code optimal with respect to the length is always optimal with respect to the dimension and the minimum distance. In this paper, we consider the problem of finding the minimal length n of a linear code over \(\mathbb {F}_q\) for a fixed dimension k and fixed minimum distance d. As usual, we denote this minimal value of n by \(n_q(k,d)\).

The Griesmer bound provides a natural lower bound on \(n_q(k,d)\):
$$\begin{aligned} n_q(k,d)\ge \sum _{i=0}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil . \end{aligned}$$
(1)
The right-hand side of this inequality is traditionally denoted by \(g_q(k,d)\). The bound (1) was proved originally for binary codes by Griesmer [8] and generalized for q-ary codes by Solomon and Stiffler [25]. An \([n,k,d]_q\)-code with \(n=g_q(k,d)\) is called a Griesmer code.

There exists an important connection of linear codes with multisets of points in finite projective geometries. Let \(\mathcal {P}\) be the set of points of the geometry \({{\mathrm{PG}}}(r,q)\). Every mapping \(\mathcal {K}:\mathcal {P}\rightarrow \mathbb {N}_0\) is called a multiset in \({{\mathrm{PG}}}(r,q)\). This mapping is extended to the subsets S of \(\mathcal {P}\) by \(\mathcal {K}(S)=\sum _{P\in S}\mathcal {K}(P)\). The integer \(\mathcal {K}(P)\), \(P\in \mathcal {P}\), is called the multiplicity of the point P, and \(n=\sum _{P\in \mathcal {P}}\mathcal {K}(P)\) is the cardinality of the multiset \(\mathcal {K}\). A multiset is called projective if the multiplicity of every point is 0 or 1. The characteristic function \(\chi _S\) of a pointset \(S\subseteq \mathcal {P}\) is defined by \(\chi _S(P)=1\) if \(P\in S\), and 0 – otherwise. Multisets can be viewed as arcs or minihypers.

A multiset \(\mathcal {K}\) in \({{\mathrm{PG}}}(r,q)\) is called an (nw)-arc if its cardinality is n, \(\mathcal {K}(H)\le w\) for every hyperplane H, and there exists a hyperplane \(H_0\) with \(\mathcal {K}(H_0)=w\). Similarly, a multiset \(\mathcal {K}\) in \({{\mathrm{PG}}}(r,q)\) is called an (nw)-minihyper if its cardinality is n, \(\mathcal {K}(H)\ge w\) for every hyperplane H, and there exists a hyperplane \(H_0\) with \(\mathcal {K}(H_0)=w\).

It has been known for a long time that arcs and linear codes are equivalent objects (cf. [6]). In particular, one can associate an \((n,n-d)\)-arc in \({{\mathrm{PG}}}(k-1,q)\) with every linear \([n,k,d]_q\) code C (generally, in a non-unique way) and vice versa every linear code with parameters \([n,k,d]_q\) gives rise to an \((n,n-d)\)-arc. If the arcs \(\mathcal {K}_1\) and \(\mathcal {K}_2\) in \({{\mathrm{PG}}}(k-1,q)\) are associated with the linear codes \(C_1\) and \(C_2\), respectively, then \(\mathcal {K}_1\) and \(\mathcal {K}_2\) are projectively equivalent if and only if the codes \(C_1\) and \(C_2\) are semilinearly isomorphic (cf. [6, 18]). Arcs associated with Griesmer codes are called Griesmer arcs.

A rather large class of Griesmer codes was presented by Belov, Logachov and Sandimirov in [3]. It is obtained by deleting copies of simplex codes of suitable dimensions from several concatenated copies of the k-dimensional simplex code. This result implies among other things that for a fixed dimension k and a fixed field of order q. Griesmer codes do exist for all sufficiently large values of d. In other words, there exists a positive integer \(d_0\) such that
$$\begin{aligned} n_q(k,d)-g_q(k,d)=0 \text { for all } \ {d\ge d_0}. \end{aligned}$$
We define the integer \(\delta (k,q)\) by the following two properties (see also [10]): (a) \(n_q(k,d)=g_q(k,d)\) for all \(d>\delta (k,q)\); (b) \(n_q(k,d)>g_q(k,d)\) for \(d=\delta (k,q)\). Hence \(\delta (k,q)\) is the smallest feasible \(d_0\) in the above definition. On the other hand, a less known result by Dodunekov [5] shows that for fixed d and q, and k going to infinity, the difference \(n_q(k,d)-g_q(k,d)\) can exceed any given number. In the above notation,In this paper, we investigate the problem of finding good estimates for the function \(t_q(k)\) defined as
$$\begin{aligned} t_q(k):=\max _d (n_q(k,d)-g_q(k,d)). \end{aligned}$$
By Dodunekov’s result we have that \(t_q(k)\rightarrow \infty \) if \(k\rightarrow \infty \), but we would like to have a more precise estimate for the rate of growth of this function. The same problem was also posed by by Ball [1] for 3-dimensional codes in the following way: For a fixed \(n-d\) is there always a 3-dimensional code meeting the Griesmer bound (maybe a constant or \(\log q\) away)? A related approach to the main coding theory problem was considered by Hamada and Maruta [9]. They proved the following theorem.

Theorem 1

[9] Let \((e_0,e_1,\ldots ,e_{k-2})\) be an ordered \((k-1)\)-tuple of integers with \(0\le e_i\le q-1\), and let \(d=q^{k-1}-\sum _{i=0}^{k-2} e_iq^{i}\) for a given integer \(k\ge 3\) and a prime power \(q\ge 3\). Then there exist non-negative functions \(h_{k,q}(e_0,\ldots ,e_{k-2})\), \(k=3,\ldots ,k\), such that \(n_q(k,d)-g_q(k,d)\) can be expressed as
$$\begin{aligned} n_q(k,d)-g_q(k,d)=\sum _{j=3}^k h_{j,q}(e_{k-j},\ldots ,e_{k-2}). \end{aligned}$$

Although Hamada and Maruta determine the exact values of the functions \(h_{k,q}(e_0,\ldots ,e_{k-2})\) for \(q=3\) and \(k=3,4,5\) (cf. [9]), it looks impossible to obtain exact results for large k and large q.

This paper is organized as follows. In Sect. 2 we give three equivalent formulations of our main problem in terms of linear codes, arcs and minihypers. Further we give a reconstruction of the proof of Dodunekov’s result stating that for large dimensions there exist values for d for which the best codes deviate infinitely from the Griesmer bound. In Sect. 3, we give a general construction for minihypers that is an analogue of Belov–Logachev–Sandimirov’s construction for Griesmer codes. We give several estimates for \(t_q(k)\) obtained by puncturing some good codes. We prove that for even dimensions \(t_q(k)\le 2(q^{k/2}-1)/(q-1)-(k+q-1)\) which yields a similar estimate for odd dimensions. Section 4 is devoted to 3-dimensional linear codes and plane arcs. We prove a result which is a partial answer to Ball’s question for 3-dimensional codes. We prove that in planes of even order there exists always an arc such that the associated linear code exceeds the Griesmer bound by at most \(\log _2q-1\). For planes of square order we prove the weaker estimate \(t_q(3)\le \sqrt{q}-1\).

2 The main problem

We start this section by stating our main problem. First we formulate it in terms of linear codes.

Problem A

Given the prime power q and the positive integer k, find the minimal value of t such that there exist \([g_q(k,d)+t,k,d]_q\)-codes for all d.

This minimal value of t will be denoted by \(t_q(k)\), or, in other words,
$$\begin{aligned} t_q(k):=\max _{1\le d\le \infty }(n_q(k,d)-g_q(k,d)). \end{aligned}$$
Problem A is very similar to the main problem of coding theory which asks to find \(n_q(k,d)\) for particular qk, and d. The task of determining \(n_q(k,d)\) for all qkd is hopeless in general. In this way we can get no idea of how good or bad the Griesmer bound is and how the best codes deviate from it. The most one can hope for is to find an exact solution for small q and small k for all d, which was the approach advocated by Hamada and Hill. Numerous results for small q and k were obtained in the papers of Hamada, Helleseth, Dodunekov, Hill, van Tilborg, Manev, Ytrehus, Maruta, Boukliev etc. (to mention but a few). From their results as well as from the results on optimal plane arcs (see [4, 7, 24] and the references there) one can deduce that
$$\begin{aligned} t_q(1)= & {} t_q(2)=0,\;\; \text { for all } q; \\ t_q(3)= & {} 1 \text { for all } q\le 19, t_q(3)\le 2 \text { for } q=23, 25, 27, 29; \\ t_2(4)= & {} 0, t_2(5)=1, t_2(6)=1, t_2(7)=2, t_2(8)=3; \\ t_3(4)= & {} 1, t_3(5)=2, t_3(6)=2 \text { or } 3; \\ t_4(4)= & {} 1, t_4(5)=2; t_5(4)=2, 2\le t_5(5)\le 5. \end{aligned}$$
In the early 70’s, Belov, Logachev and Sandimirov gave a sufficient condition for the existence of binary Griesmer codes which in the same time provides a rather general construction for such codes [3]. Later on, Hill showed how their result generalizes to codes over arbitrary fields [12]. In order to state this result, we introduce some notation which will also be used in the rest of the paper.
Let d and k be positive integers and let d be written in the following form:
$$\begin{aligned} d=sq^{k-1}-\lambda _{k-2}q^{k-2}-\cdots -\lambda _1q-\lambda _0, \end{aligned}$$
(2)
where \(0\le \lambda _i\le q-1\). It is easily checked that
$$\begin{aligned} g_q(k,d)=sv_k-\lambda _{k-2}v_{k-1}-\cdots \lambda _1v_2-\lambda _0v_1, \end{aligned}$$
(3)
where \(v_i=(q^i-1)/(q-1)\).

Let \(V_{i,q}\), \(i=1,\ldots ,k-1\), be the set of all i-dimensional subspaces of \(\mathbb {F}_q^k\) and denote by \(U_{i,q}^k\) the set of all matrices with k rows and \((q^i-1)/(q-1)\) non-zero columns from some subspace of \(V_{i,q}\) with first non-zero-coordinate 1. Clearly, the rows of each matrix from \(U_{i,q}^k\) generate a simplex code with parameters \(\displaystyle \Big [\frac{q^i-1}{q-1},i,q^{i-1}\Big ]_q\). Denote by \(S_{k,q}\) the matrix with columns all non-zero vectors of \(\mathbb {F}_q^k\) in which the first non-zero coordinate is equal to 1. If we concatenate s copies of \(S_{k,q}\) and delete from it \(\lambda _i\) matrices from \(U_{i,q}^k\), we get a Griesmer code (cf. [12]). Following MacWilliams and Sloane [21], we call a linear code constructed in this way, a code of type BV.

Now let us express d in a somewhat different form
$$\begin{aligned} d=sq^{k-1}-\sum _{i=1}^{l}q^{u_i-1}, \end{aligned}$$
where \(s=\left\lceil d/q^{k-1}\right\rceil \), \(k> u_1\ge u_2\ge \cdots \ge u_l\), and at most \(q-1\)\(u_i\)’s take the same value. The following theorem was proved in [12].

Theorem 2

[3, 12] There exists a \([g_q(k,d),k,d]_q\) code of type BV if and only if
$$\begin{aligned} \sum _{i=1}^{\min (s+1,l)} u_i\le sk. \end{aligned}$$

This theorem has the following important corollary.

Corollary 1

For given dimension k and prime power q, there exists an integer \(d_0\) such that \(n_q(k,d)=g_q(k,d)\) for all \(d\ge d_0\).

Proof

Take \(d_0=(k-2)q^{k-1}+1\). Then \(s\ge k-1\) and since \(u_i\le k-1\le s\) for all i, we have
$$\begin{aligned} \sum _{i=1}^{\min (s+1,l)}u_i\le \sum _{i=1}^{s+1}u_i\le (s+1)(k+1)=sk-(s+1-k)\le sk. \end{aligned}$$
\(\square \)

This corollary tells us that \(n_q(k,d)-g_q(k,d)=0\) if k is fixed and \(d\rightarrow \infty \). Furthermore, it tells us that \(\delta (k,q)\le (k-2)q^{k-1}\). The bound for \(d_0\) deduced from this corollary is not the best possible. There exist numerous improvements of this result [10, 16, 17, 22, 23]. In fact, there exists a hypothesis that \(\delta (k,q)=(k-2)q^{k-1}-(k-1)q^{k-2}\) which has been proved in some special cases [17, 22].

On the other side, the behavior of the difference \(n_q(k,d)-g_q(k,d)\) is quite different if we fix d and let k go to infinity. In the case when k gets large the length of the optimal codes deviates infinitely from the Griesmer bound. This observation was done by Dodunekov in [5].

Theorem 3

[5] For every two integers t and \(d\ge 3\), there exists an integer \(k_0\) such that for all \(k\ge k_0\), \(n_q(k,d)-g_q(k,d)\ge t\).

Proof

Let d be a fixed positive integer and let q be a fixed prime power. Set \(\rho =\lfloor (d-1)/2\rfloor \). Denote by \(B^n(\rho )\) a ball in \(\mathbb {F}_q^n\) with an unspecified center and radius \(\rho \). Obviously
$$\begin{aligned} |B^n(\rho )|=\sum _{i=0}^{\rho }{n\atopwithdelims ()i}(q-1)^{i}. \end{aligned}$$
If \(k\rightarrow \infty \) then \(g_q(k,d)\rightarrow \infty \) andThis implies also thatOn the other hand, consider an optimal \([n_q(k,d),k,d]_q\)-code. From the sphere-packing bound one gets
$$\begin{aligned} q^{n_q(k,d)}\ge & {} q^k\cdot \sum _{i=0}^\rho {n_q(k,d)\atopwithdelims ()i}(q-1)^i \\\ge & {} q^k\cdot \sum _{i=0}^\rho {g_q(k,d)\atopwithdelims ()i}(q-1)^i \\ \end{aligned}$$
which implies
$$\begin{aligned} n_q(k,d)-k\ge \log _q|B_{\rho }^{g_q(k,d)}(\rho )|\rightarrow \infty . \end{aligned}$$
(5)
From the Griesmer bound we obtain
$$\begin{aligned} g_q(k,d)= & {} d +\left\lceil \frac{d}{q}\right\rceil + \left\lceil \frac{d}{q^2}\right\rceil + \cdots + \left\lceil \frac{d}{q^{k-1}}\right\rceil \\< & {} d+ \frac{d}{q} + \frac{d}{q^2}+\cdots +\frac{d}{q^{k-1}}+k-1 \end{aligned}$$
whence
$$\begin{aligned} g_q(k,d)-k< d\frac{q^k-1}{q^k-q^{k-1}}-1. \end{aligned}$$
(6)
Now combining (5) and (6), we get
$$\begin{aligned} n_q(k,d)-g_q(k,d)>\log _q|B_q^{g_q(k,d)}(\rho )|-d\frac{q^k-1}{q^k-q^{k-1}}+1. \end{aligned}$$
From (4) and the obvious fact that \(\displaystyle \lim _{k\rightarrow \infty } \frac{q^k-1}{q^k-q^{k-1}}=1\) we get that\(\square \)

Remark 1

The above proof is our reconstruction of the original one which must have been similar.

Theorem 3 implies that for a fixed d one has \(n_q(k,d)-g_q(k,d)\rightarrow \infty \), and, consequently \(t_q(k)\rightarrow \infty \) when q is fixed and k goes to infinity. Now the question is to find a suitable function that is an upper bound of \(t_q(k)\).

Before we proceed further, we give two equivalent geometric versions of Problem A. Let again kd and q be fixed, and let d be written in the form (2). Then \(g_q(k,d)\) is given by (3). Define \(w_q(k,d):=g_q(k,d)-d\). Now from (2) and (3), using the obvious \(v_i=v_{i-1}+q^{i-1}\), we get
$$\begin{aligned} w_q(k,d)=sv_{k-1}-\lambda _{k-2}v_{k-2}-\cdots -\lambda _1v_1. \end{aligned}$$
(7)
Since \([n,k,d]_q\)-codes exist if and only if there exist \((n,n-d)\)-arcs in \({{\mathrm{PG}}}(k-1,q)\), Problem A can be stated as follows.

Problem B

Given the prime power q and the positive integer k, find the minimal value of t such that there exist \((g_q(k,d)+t,w_q(k,d)+t)\)-arcs in \({{\mathrm{PG}}}(k-1,q)\) for all d.

We mentioned already that the problem of finding \(t_q(k)\) for particular values of q and k is a finite problem. In fact,
$$\begin{aligned} t_q(k)=\max _{1\le d\le \delta (k,q)}(n_q(k,d)-g_q(k,d)), \end{aligned}$$
where \(\delta (k,q)\) was defined as the largest value of d for which a Griesmer code does not exist (k and q are fixed). We can make the interval for d even smaller by the following simple observation.

Lemma 1

If \(n_q(k,d)=g_q(k,d)+t\), then \(n_q(k,d+q^{k-1})\le g_q(k,d+q^{k-1})+t\).

Proof

Let \(\mathcal {K}\) be an (nw)-arc in \({{\mathrm{PG}}}(k-1,q)\) with \(n=n_q(k,d)+t\) and \(w=w_q(k,d)+t\). Define the arc \(\mathcal {K}'\) by increasing the multiplicity of each point of \({{\mathrm{PG}}}(k-1,q)\) by 1, i.e. \(\mathcal {K}'=\mathcal {K}+\chi _{\mathcal {P}}\), where \(\chi _{\mathcal {P}}\) is the characteristic function of the pointset \(\mathcal {P}\) of \({{\mathrm{PG}}}(k-1,q)\). Obviously, \(\mathcal {K}'\) is an \((n+v_k,w+v_{k-1})\)-arc. On the other hand, we have
$$\begin{aligned} g_q(k,d+q^{k-1})=\sum _{i=0}^{k-1} \left\lceil \frac{d+q^{k-1}}{q^i}\right\rceil =g_q(k,d)+v_k, \end{aligned}$$
which completes the proof.\(\square \)
Lemma 1 implies that for fixed k and q it is enough to consider only values of d that are smaller than \(q^{k-1}\):
$$\begin{aligned} t_q(k)=\max _{1\le d\le q^{k-1}} (n_q(k,d)-g_q(k,d)). \end{aligned}$$
The interval for d can be made even smaller by observing that Griesmer codes always do exist for \(d=q^{k-1}\) (take the simplex codes or geometrically all points of \({{\mathrm{PG}}}(k-1,q)\)), and for \(d=q^{k-1}-q^{k-2}\) (take the MacDonald codes or, equivalently, all points of \({{\mathrm{AG}}}(k-1,q)\)).

We proceed by formulating Problem B in terms of minihypers. Let \(\mathcal {K}\) be a \((g_q(k,d)+t,w_q(k,d)+t)\)-arc in \({{\mathrm{PG}}}(k-1,q)\). We denote the maximal point multiplicity in \(\mathcal {K}\) by \(s_0\). If d is given by (2) then \(s_0\le s+t\). This is due to the following lemma known as a folklore.

Lemma 2

Let \(d=sq^{k-1}-\lambda _{k-2}q^{k-2}-\cdots -\lambda _0\) and let \(\mathcal {K}\) be a \((g_q(k,d)+t,w_q(k,d)+t)\)-arc in \({{\mathrm{PG}}}(k-1,q)\). Then for every j-dimensional subspace S in \({{\mathrm{PG}}}(k-1,q)\)
$$\begin{aligned} \mathcal {K}(S)\le t+\sum _{i=k-1-j}^{k-1}\left\lceil \frac{d}{q^i}\right\rceil . \end{aligned}$$

Proof

The statement is obviously true for hyperplanes, i.e. \(j=k-2\). If we assume that the result is true for subspaces of dimension \(j\le k-2\), then it is true also for subspaces of dimension \(j-1\). In order to see this, count the multiplicities of all j-dimensional subspaces through a fixed \(j-1\)-dimensional subspace of maximal multiplicity.\(\square \)

Define a multiset \(\mathcal {F}:=s_0\chi _{\mathcal {P}}-\mathcal {K}\). The multiset \(\mathcal {F}\) is a minihyper with parameters
$$\begin{aligned} (\sigma v_k+\lambda _{k-2}v_{k-1}+\cdots +\lambda _1v_2+\lambda _0v_1-t, \sigma v_{k-1}+\lambda _{k-2}v_{k-2}+\cdots +\lambda _1v_1-t), \end{aligned}$$
where
$$\begin{aligned} \sigma =\left\{ \begin{array}{ll} s_0-s &{}\quad \text { if } s< s_0\le s+t, \\ 0 &{}\quad \text { if } s_0\le s. \end{array}\right. \end{aligned}$$
and with the additional property that the maximal point multiplicity does not exceed \(\sigma +s\). In fact, the maximal point multiplicity will be exactly \(\sigma +s\) if \(\mathcal {K}\) has a 0-point.

Problem C

Find the maximal value of t such that for all d given by (2) there exists a minihyper in \({{\mathrm{PG}}}(k-1,q)\) with parameters
$$\begin{aligned} (\sigma v_k+\lambda _{k-2}v_{k-1}+\lambda _1v_2+\lambda _0v_1-t, \sigma v_{k-1}+\lambda _{k-2}v_{k-2}+\lambda _1v_1-t), \end{aligned}$$
with maximal point multiplicity at most \(\sigma +s\).

This formulation though very useful is a bit complicated and we shall explain it by an example.

Example 1

For a fixed prime power q, take \(k=4\) and \(d=2q^3-4q^2\). Then in the above notation \(s=2\), \(\lambda _2=4\), \(\lambda _0=\lambda _1=0\). The existence of a Griesmer code is equivalent to the existence of a \((4v_3,4v_2)\)-minihyper with a maximal point multiplicity \(s=2\). Since such a minihyper is necessarily a sum of four planes (cf. [19]) there is necessarily a point of multiplicity 3 and there exists no 4-dimensional Griesmer code with minimum distance \(d=2q^3-4q^2\) for all q. An \([n,4,d]_q\) code with \(n=g_q(4,d)+t\) exists iff there exists a minihyper for one of the parameter sets in the row indexed by t in the table below

t

\(s_{0}\)

   

2

3

4

\(\ldots \)

0

\((4v_3,4v_2)\)

   

1

\((4v_3-1,4v_2-1)\)

\((v_4+4v_3-1,v_3+4v_2-1)\)

  

2

\((4v_3-2,4v_2-2)\)

\((v_4+4v_3-2,v_3+4v_2-2)\)

\((2v_4+4v_3-2,2v_3+4v_2-2)\)

 

3

\((4v_3-3,4v_2-3)\)

\((v_4+4v_3-3,v_3+4v_2-3)\)

\((2v_4+4v_3-3,2v_3+4v_2-3)\)

\(\ldots \)

For the parameters in the column indexed by \(s_0\) the maximal point multiplicity is bounded by \(s_0\). Thus the minihypers with parameters in the first column should have a maximal point multiplicity two, in the second column – maximal point multiplicity three and so on. Let us take the sum of four planes in \({{\mathrm{PG}}}(3,q)\) that have a common point P, but no three of them with a common line. Reduce the multiplicity of P by two (see Theorem 4). We get a \((4v_3-2,4v_2-2)\)-minihyper and hence we can get a code with \(t=2\). Let us note that in the case of \(q=5\) a \([189,4,150]_5\)-code is known to exist, whence \(n_5(4,150)=g_5(4,150)+1\).

Example 2

If we take \(d=q^3-4q^2\) then \(s=1\). Then the maximal point multiplicities in the table above are reduced by one. So the minihypers with parameters in the first column must be projective, in the second column must have maximal multiplicity two and so on. The construction from the previous example produces codes with a very large t since we have to reduce the multiplicities of all points P with \(\mathcal {K}(P)>1\). We can use another approach and take the sum of 4q lines contained in a spread. This gives a \((4qv_2,4qv_1)\)-minihyper. Of course, we have to be sure that the spread contains 4q lines, but this is indeed true for all \(q\ge 4\). Now using \(v_i=qv_{i-1}+1\) we get that this minihyper has parameters \((4v_3-4,4v_2-4)\), i.e. for \(k=4\), \(d=q^3-4q^2\) we have \(t\le 4\) for all q. This is not the best possible construction for all q. It is known that for \(q=5\) we can construct a code with \(d=25\) and \(t=2\). In fact this is the only value for d in the case \(q=5\), \(k=4\), for which \(t=2\); for all other d’s one has \(t\le 1\).

3 A general construction

We start this section with a generalization of Example 1 which is in some way an analogue of the construction by Belov, Logachev and Sandimirov. The proof of the theorem is obvious.

Theorem 4

Let \(d=sq^{k-1}-\lambda _{k-2}q^{k-2}-\cdots -\lambda _1q-\lambda _0\), and let the multiset \(\mathcal {F}\) be a minihyper in \({{\mathrm{PG}}}(k-1,q)\) with parameters
$$\begin{aligned} (\sigma v_k+\lambda _{k-2}v_{k-1}+\cdots +\lambda _0v_1-\tau _1, \sigma v_{k-1}+\lambda _{k-2}v_{k-2}+\cdots +\lambda _1v_1-\tau _1). \end{aligned}$$
Define the multiset \(\mathcal {F}'\) by
$$\begin{aligned} \mathcal {F}'(x)=\left\{ \begin{array}{ll} \mathcal {F}(x) &{} \text { if } \mathcal {F}(x)\le \sigma +s, \\ \sigma +s &{} \text { if } \mathcal {F}(x)> \sigma +s. \end{array}\right. \end{aligned}$$
Let \(N=|\mathcal {F}|\) and \(N'=|\mathcal {F}'|\). If \(\mathcal {F}-\mathcal {F}'\) is an \((N-N',\tau _2)\)-arc then there exists a \((g_q(k,d)+t,w_q(k,d)+t)\)-arc in \({{\mathrm{PG}}}(k-1,q)\), or, equivalently, a code with parameters \([g_q(k,d)+t,k,d]_q\), with \(t=\tau _1+\tau _2\).

Proof

The minihyper \(\mathcal {F}'\) has parameters
$$\begin{aligned}&(\sigma v_k+\lambda _{k-2}v_{k-1}+\cdots +\lambda _0v_1-\tau _1-(N-N'), \sigma v_{k-1}\\&\quad +\,\lambda _{k-2}v_{k-2}+\cdots +\lambda _1v_1-\tau _1-\tau _2) \end{aligned}$$
since by the construction the multiplicity of each hyperplane is reduced by at most \(\tau _2\). Moreover, the multiplicity of each point is at most \(\sigma +s\). By increasing the multiplicities of some suitable points we can produce a minihyper \(\mathcal {F}''\) with parameters
$$\begin{aligned} (\sigma v_k+\lambda _{k-2}v_{k-1}+\cdots +\lambda _0v_1-(\tau _1+\tau _2), \sigma v_{k-1}+\lambda _{k-2}v_{k-2}+\cdots +\lambda _1v_1-(\tau _1+\tau _2)) \end{aligned}$$
and with maximal point multiplicity \(\sigma +s\). Now if \(\mathcal {P}\) denotes the set of points of \({{\mathrm{PG}}}(k-1,q)\), the multiset \((\sigma +s)\chi _{\mathcal {P}}-\mathcal {F}''\) is a \((g_q(k,d)+t,w_q(k,d)+t)\)-arc in \({{\mathrm{PG}}}(k-1,q)\).\(\square \)
Now let t and k be fixed integers and let q be a prime power. Define the set of integers
$$\begin{aligned} D_q^{(t)}(k):=\{d\in \mathbb {Z}^+\mid n_q(k,d)=g_q(k,d)+t\}. \end{aligned}$$

Lemma 3

If \(d\in D_q^{(t)}(k)\) then \(d-1\in D_q^{(t')}(k)\) where
$$\begin{aligned} t'\le t+\sum _{j=1}^{k-2}\left( \left\lceil \frac{d}{q^j}\right\rceil -\left\lceil \frac{d-1}{q^j}\right\rceil \right) . \end{aligned}$$

Proof

By the condition of the lemma there exists a code with parameters \([g_q(k,d)+t,k,d]_q\). Using a suitable puncturing we can construct a linear \([g_q(k,d)+t-1,k,d-1]_q\) code. Hence \(n_q(k,d-1)\le g_q(k,d)+t-1\).

Let d be written in the form (2) and denote by i the smallest index for which
$$\begin{aligned} \lambda _0=\cdots =\lambda _{i-1}=q-1, \lambda _i<q-1. \end{aligned}$$
Then \(d-1=q^{k-1}-\lambda _{k-2}q^{k-2}-\cdots -(\lambda _i+1)q^i\) and from (3) we get
$$\begin{aligned} g_q(k,d)-g_q(k,d-1)=i+1. \end{aligned}$$
(8)
On the other hand we have
$$\begin{aligned} \left\lceil \frac{d}{q^j}\right\rceil =\left\{ \begin{array}{ll} \left\lceil \frac{d-1}{q^j}\right\rceil +1 &{} \text { for } j=1,\ldots ,i, \\ \left\lceil \frac{d-1}{q^j}\right\rceil &{} \text { for } j=i+1,\ldots ,k-2, \end{array}\right. \end{aligned}$$
whence
$$\begin{aligned} \sum _{j=1}^{k-2}\left( \left\lceil \frac{d}{q^j}\right\rceil -\left\lceil \frac{d-1}{q^j}\right\rceil \right) =i. \end{aligned}$$
(9)
Now if \(d-1\in D_q^{(t')}(k)\) we get from (8) and (9) that
$$\begin{aligned} t'\le & {} g_q(k,d) - g_q(k,d-1) + t -1 \\= & {} t + \sum _{j=1}^{k-2}\left( \left\lceil \frac{d}{q^j}\right\rceil -\left\lceil \frac{d-1}{q^j}\right\rceil \right) . \end{aligned}$$
\(\square \)

Corollary 2

If \(d\in D_q^{(t)}(k)\) and \(d'<d\) then \(d'\in D_q^{(t')}(k)\) where
$$\begin{aligned} t'\le t+\sum _{j=1}^{k-2}\left( \left\lceil \frac{d}{q^j}\right\rceil -\left\lceil \frac{d'}{q^j}\right\rceil \right) . \end{aligned}$$
In particular, if \(k=3\)
$$\begin{aligned} t'\le t+ \left\lceil \frac{d}{q}\right\rceil -\left\lceil \frac{d'}{q}\right\rceil . \end{aligned}$$
Now we are going to prove that for even dimensions k the function \(t_q(k)\) is bounded roughly by \(q^{k/2}\). This estimate looks very rough, but it is the best one available. Let d be given by (2) and set \(k=2l\). If for every such d there exists a projective minihyper in \({{\mathrm{PG}}}(k-1,q)\) with parameters
$$\begin{aligned} (\lambda _{k-2}v_{k-1}+\cdots +\lambda _1v_2-t_0, \lambda _{k-2}v_{k-2}+\cdots +\lambda _1v_1-t_0) \end{aligned}$$
then \(t_q(k)\le t_0\).

Theorem 5

If \(k=2l\) then
$$\begin{aligned} t_q(k)\le 2\frac{q^l-1}{q-1}-(2l+q-1). \end{aligned}$$

Proof

Let d be given by (2). Since \(k=2l\) there exists a partition \(\mathcal {S}\) of the geometry \({{\mathrm{PG}}}(k-1,q)\) in subspaces of dimension \(l-1\), called also an \((l-1)\)-spread in \({{\mathrm{PG}}}(k-1,q)\). Taking subspaces from \(\mathcal {S}\) we can construct a minihyper in \({{\mathrm{PG}}}(k-1,q)\) with parameters
$$\begin{aligned} \left( (\lambda _{k-2}q^{k/2-1}+\cdots +\lambda _{k/2-1})v_{k/2}, (\lambda _{k-2}q^{k/2-1}+\cdots +\lambda _{k/2-1})v_{k/2-1}\right) . \end{aligned}$$
Now using \(v_{i+1}=qv_i+1\)
$$\begin{aligned} \left( \sum _{i=0}^{k/2-1} \lambda _{k/2+i-1}q^i\right) v_{k/2}= & {} (\lambda _{k-2}q^{k/2-2}+\cdots +\lambda _{k/2})v_{k/2+1} \\&+\, \lambda _{k/2-1}v_{k/2}-(\lambda _{k-2}q^{k/2-2}+\cdots +\lambda _{k-2}) \\= & {} (\lambda _{k-2}q^{k/2-3}+\cdots +\lambda _{k/2+1})v_{k/2+1}+\lambda _{k/2}v_{k/2+1} \\&+\, \lambda _{k/2-1}v_{k/2}-(\lambda _{k-2}q^{k/2-2}+\cdots +\lambda _{k/2}) \\&+\, (\lambda _{k-2}q^{k/2-3}+\cdots +\lambda _{k/2+1}) \\= & {} \lambda _{k-2}v_{k-1}+\cdots +\lambda _{k/2-1}v_{k/2}-\sum _{i=k/2}^{k-2}\lambda _iv_{i-k/2+1} \\= & {} \sum _{i=1}^{k-2} \lambda _iv_{i+1}-\sum _{i=k/2}^{k-2}\lambda _iv_{i-k/2+1}-\sum _{i=1}^{k/2-2}\lambda _iv_{i+1} \end{aligned}$$
Note that \(\lambda _{k-2}v_{k-1}+\cdots +\lambda _{k/2-1}v_{k/2}<q^{k/2}+1\), i.e. there exists the required number of \((l-1)\)-subspaces in the spread. Thus the constructed minihyper has parameters (NW), where
$$\begin{aligned} N= & {} \lambda _{k-2}v_{k-1}+\cdots +\lambda _1v_2- t_0 \\ W= & {} \lambda _{k-2}v_{k-2}+\cdots +\lambda _1v_1-t_0, \end{aligned}$$
where
$$\begin{aligned} t_0=\sum _{i=k/2}^{k-2}\lambda _iv_{i-k/2+1}-\sum _{i=1}^{k/2-2}\lambda _iv_{i+1} . \end{aligned}$$
Now since \(\lambda _i\le q-1\) for all i we have
$$\begin{aligned} t_0\le & {} \sum _{i=k/2}^{k-2}(q-1)v_{i-k/2+1}-\sum _{i=1}^{k/2-2}(q-1)v_{i+1} \\= & {} \sum _{i=1}^{k/2-1}(q^i-1)-\sum _{i=2}^{k/2-1}(q^i-1) \\= & {} 2\cdot \sum _{i=1}^{k/2-1}(q^i-1)-(q-1) \\= & {} 2\frac{q^l-1}{q-1}-(2l+q-1). \end{aligned}$$
\(\square \)

In the special case of \(k=4\), we get the following corollary.

Corollary 3

$$\begin{aligned} t_q(4)\le q-1. \end{aligned}$$
\(\square \)

For codes of odd dimension the idea of Theorem 5, i.e. taking subspaces of dimension k / 2 from a k / 2-spread, does not give a good estimate. A better bound can be obtained from the obvious \(t_q(k-1)\le t_q(k)\).

4 The case \(k=3\)

The construction of good three-dimensional codes over \(\mathbb {F}_q\) is equivalent to the construction of good arcs in the projective planes \({{\mathrm{PG}}}(2,q)\). The sizes of the best arcs are known for planes of order \(q\le 9\) and reasonable bounds are available for planes of order \(q\le 29\) [1, 14]. In [1] the following question was asked which is closely related to our Problem B:

For a fixed \(n-d\), is there always a three-dimensional linear [n, 3, d] code meeting the Griesmer bound (or at least close to the Griesmer bound, may be constant or \(\log (q)\) away)?

The answer to the first part is obviously “no” . For instance, if \(w=n-d=q+2\) the optimal arcs have size \(q^2+q+2\) and are not associated with Griesmer codes [2]. The second part states that \(\log _2q\) is an upper bound on \(t_q(3)\). The existing tables show that \(t_q(3)\le 2\) for all \(q\le 29\) and we do not know an instance of w for which the optimal (nw)-arc gives a code whose length exceeds the Griesmer bound by 2. Thus we can reformulate the second part of Ball’s question in the following way:

Does there exist a constant C with the following property. For a fixed \(w=n-d\le q^2-q\) is there always a three-dimensional linear code with minimum distance d whose length exceeds the Griesmer bound by at most C.

Though \(\log _2q\) seems to be a rough estimate on \(t_q(3)\) there is no straightforward proof of this statement for q odd. In what follows we shall prove that for even q this upper bound on t follows from the existence of maximal arcs.

Let us note that we can restrict ourself to codes with \(d\le q^2\) (cf. Lemma 1). So we can write \(d=q^2-\lambda _1q-\lambda _0\) whence
$$\begin{aligned} g_q(3,d)=v_3-\lambda _1v_2-\lambda _0, w_q(3,d)=v_2-\lambda _1\le q+1. \end{aligned}$$
Since there exist trivial examples of Griesmer arcs with \(w=q\) and \(q+1\) (\({{\mathrm{AG}}}(2,q)\) and \({{\mathrm{PG}}}(2,q)\)), we shall consider values for w that satisfy \(w\le q-1\). We start with an useful lemma which relates the Griesmer bound with the trivial upper bound on the size of an (nw)-arc in \({{\mathrm{PG}}}(2,q)\).

Lemma 4

Let \(\mathcal {K}\) be an (nw)-arc in \({{\mathrm{PG}}}(2,q)\) with \(w\le q-1\). If \(n=(w-1)q+w-\alpha \) and \(d=n-w\), then \(n=t+g_q(3,d)\) where \(t=\lfloor \alpha /q\rfloor \).

Proof

Since \(d=(w-1)q-\alpha \) and \(w\le q-1\), we have \(\left\lceil d/q^2\right\rceil =1\) and
$$\begin{aligned} g_q(3,d)= & {} d+\left\lceil \frac{d}{q}\right\rceil +\left\lceil \frac{d}{q^2}\right\rceil \\= & {} (w-1)q-\alpha +(w-1)-\lfloor \frac{\alpha }{q}\rfloor +1 \\= & {} n-\lfloor \frac{\alpha }{q}\rfloor . \end{aligned}$$
\(\square \)

Now we observe that the sum of r maximal arcs for q even gives a linear code which deviates by \(r-1\) from the Griesmer bound.

Lemma 5

Let \(q=2^h\) and let \(\mathcal {K}_i\), \(i=1,\ldots ,r\), be maximal arcs. Define the arc \(\mathcal {K}=\sum _{i=1}^r \mathcal {K}_i\). If the code \(C_{\mathcal {K}}\) associated with \(\mathcal {K}\) has parameters \([n,3,d]_q\) then \(n=g_q(3,d)+ (r-1)\).

Proof

Let \(\mathcal {K}_i\) have parameters \(((2^{a_i}-1)q+2^{a_i},2^{a_i})\), \(a_i\in \{1,\ldots ,h-1\}\), \(i=1,\ldots ,r\). The arc \(\mathcal {K}=\sum _{i=1}^r\mathcal {K}_i\) has parameters
$$\begin{aligned} (N,W)=((2^{a_1}+\cdots +2^{a_r}-r)q+2^{a_1}+\cdots +2^{a_r}, 2^{a_1}+\cdots +2^{a_r}). \end{aligned}$$
We can choose the arcs \(\mathcal {K}_i\) in such way that W has exactly this value. Now the cardinality N can be expressed as
$$\begin{aligned} N= & {} (2^{a_1}+\cdots +2^{a_r}-1)q+(2^{a_1}+\cdots +2^{a_r})-(r-1)q \\= & {} (W-1)q+W - (r-1)q. \end{aligned}$$
Now the result is obtained from Lemma 4 with \(\alpha =(r-1)q\).\(\square \)

Lemma 6

Let \(q=2^h\). Every integer \(m\le q\) can be represented in the form \(m=2^{a_1}+\cdots +2^{a_r}-r\) for some \(a_i\in \{1,\cdots ,h-1\}\) and some integer \(r\le h\).

Proof

Induction on h. For small values of h such representations are easily found. Assume the lemma is true for \(q=2^h\) and let \(q'=2^{h+1}\). We have to show that such representation exists only for the integers \(m=q+\beta \) with \(\beta =1,\ldots ,q-1\), since \(q'=2^{h+1}+2^1-2\). By the induction hypothesis we have \(\beta +1=2^{a_1}+\cdots +2^{a_r}-r\) with \(a_i\in \{1,\ldots ,h-1\}\), \(r\le h\). Now
$$\begin{aligned} q+\beta =(2^h-1)+(\beta +1) = (2^h+2^{a_1}+\cdots +2^{a_r})-(r+1), \end{aligned}$$
which is the desired representation.\(\square \)

Theorem 6

If \(q=2^h\) then \(t_q(3)\le \log _2q-1\).

Proof

By Corollary 2, we need to consider only the d’s with \(d\le q^2-2q\). Let us write d in the form \(d=q^2-\lambda q=(q-\lambda )q=jq\), \(j=1,\ldots ,q-2\). By Lemma 6 every \(j\in \{2,\ldots ,q-2\}\) can be represented as \(j=2^{a_1}+\cdots +2^{a_r}-r\) for some \(r\le h=\log _2 q\). By Lemma 5 there exists an (nw)-arc with \(n=jq+j+r\) and \(w=j+r\) which is the sum of r maximal arcs in \({{\mathrm{PG}}}(2,q)\). Now by Lemma 4 the length of the code associated with this arc exceeds by \(r-1=\log _2q-1\) the Griesmer bound \(g_q(3,d)\).\(\square \)

Theorem 7

If q is a square then \(t_q(3)\le \sqrt{q}-1\).

Proof

By a result of Hill and Mason [13] (cf. also [14]) for q square there exists an (nw)-arc with \(n=(w-1)q+w-\sqrt{q}(q-w+1)\). This makes sense for \(n-w>0\), i.e. \(w\ge \sqrt{q}+\frac{1}{\sqrt{q}+1}\). Hence we consider values for w bounded by
$$\begin{aligned} \sqrt{q}+1\le w\le q-1. \end{aligned}$$
Set \(w=\sqrt{q}+\epsilon \), where \(\epsilon \in \{1,\ldots ,q-\sqrt{q}-1\}\). These arcs are associated with codes of minimum distance \(d_{\epsilon }=\epsilon q-(\epsilon -1)\sqrt{q}\). It can be checked that for \(\epsilon =1\), i.e. \(d=q\), \(n=g_q(3,d)+(\sqrt{q}-1)\). For \(\epsilon \ge 1\), \(n=g_q(3,d_{\epsilon })+t\), where \(t\le \sqrt{q}+2\). Now the result follows from Corollary 2.\(\square \)

Hamada [10] made the conjecture that \(t_q(3)=1\). This is equivalent to saying (cf. Lemma 4) that for every w there exists an (nw)-arc in \({{\mathrm{PG}}}(2,q)\) with \(n\ge (w-2)q+w\) or that the length of an optimal 3-dimensional code deviates by at most one from the Griesmer bound. Though this conjecture looks very optimistic, at present it has neither been proved nor disproved. A weaker version of this conjecture would be that \(t_q(3)\le C\), where C is a constant not depending on q. In a private conversation, Maruta made the more general conjecture that \(t_q(k)\le k-2\) for all q.

Notes

Acknowledgements

This result was partially supported by the Research Scientific Fund of Sofia University “St. Kl. Ohridski” under Contract No. 80-10-55/19.04.2017.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sofia UniversitySofiaBulgaria
  2. 2.New Bulgarian UniversitySofiaBulgaria
  3. 3.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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