¹

1 Introduction and preliminaries↩︎

This section establishes the notation and background used throughout. We begin by recalling the definitions of linear codes and projective systems, then introduce the central parameters \(m^s(k,q)\), \(m^s_t(k,q)\), and \(\kappa(s,q)\), and survey existing bounds in the literature. Section 2 treats codes of small dimension, Section 3 reviews bounds from the theory of classical arcs, and Section 4 presents the main results. Section 5 concludes with open questions and conjectures.

For \(n\geq k\), a linear code \(C\) of length \(n\), dimension \(k\), and minimum distance \(d\), denoted an \([n,k,d]_q\) code, is a \(k\)-dimensional subspace of \(\mathbb{F}_q^n\) such that the minimum (Hamming) distance between any two elements (codewords) of \(C\) is \(d\); that is to say, there exist two codewords agreeing in \(n-d\) coordinates and no two codewords agree in as many as \(n-d+1\) coordinates. A linear code \(C \subset \mathbb{F}_q^n\) is degenerate if \(C \subseteq \mathbb{F}_q^{n-1} \subset \mathbb{F}_q^n\), where \(\mathbb{F}_q^{n-1}\) is the subspace of vectors with \(0\) at some fixed position, otherwise, we say that the code is non-degenerate. Unless otherwise stated, the codes considered here shall be non-degenerate. Given an \([n, k, d]_q\) code \(C\), the dual code \(C^\perp = \{ y \in \mathbb{F}_q^n \mid y\cdot x = 0 \text{ for all } x \in C \}\) is an \([n,k^\perp, d^\perp]_q\) code where \(k^\perp = n-k\).

Following Tsfasman and Vlǎduţ [1], we represent linear codes as projective systems — finite multisets of points in a projective space — thereby gaining a geometric vantage point from which to analyze code parameters. We now recall the necessary definitions.

The projective space of \(k\) dimensions over \(\mathbb{F}_q\) will be denoted by PG\((k, q)\). A projective subspace of dimension \(m\) of PG\((k-1,q)\) is called an \(m\)-flat; it is isomorphic to PG\((m,q)\) as an abstract projective space, though we retain the term flat to emphasize its embedding in the ambient space. We say that a set of \(m\) points of PG\((k-1, q)\) are independent if they are not contained in an \((m-2)\)-flat. More generally, the quotient of \(PG(k,q)\) by a \(d\)-flat \(\Omega\) is isomorphic to PG\((k-d-1,q)\). The flats of the quotient space are precisely the flats of PG\((k,q)\) containing \(\Omega\), (see e.g. [2]).

An \([n, k, d]_q\) code \(C\) is determined by an associated generator matrix, \(G\). Up to code equivalence, the columns of \(G\) can be considered as an \(n\)-multiset \(\mathcal{G}\) of points in \(PG(k-1,q)\), at most \(n-d\) per hyperplane (subspace of co-dimension \(1\)). Any hyperplane meeting \(\mathcal{G}\) in \(n-d\) points is said to be a secant of \(\mathcal{G}\). We may thus discuss codes in terms of equivalent objects known as projective systems (see e.g. [1]).

A projective \([n, k, d]_q\) system is a finite unordered family \(\mathcal{G}\) of points in \(\Sigma =\text{PG}(k-1,q)\) that does not lie entirely in a hyperplane of \(\Sigma\). Though generally, \(\mathcal{G}\) may be a multiset, we shall employ a slight abuse of notation, writing \(\mathcal{G} \subseteq \Sigma\), and \(|\mathcal{G}|\) to denote the cardinality of \(\mathcal{G}\) counting multiplicities. For a point \(P\), we denote by \(\mu(P)\) the multiplicity of \(P\) in \(\mathcal{G}\). The parameters \(n, k,\) and \(d\) are defined as follows: \[\label{eqn:32Proj32Sys32Code32parameters} n=|\mathcal{G}|, \quad k=\operatorname{dim} \Sigma +1, \quad n-d= \max \{|\mathcal{G}\cap H| \geq 1 \mid H \text{ is a hyperplane of } \Sigma\}.\tag{1}\]

Similarly, the set of parameters \([n,k^\perp,d^{\perp}]_q\) of a dual code \(\mathcal{G}^\perp\) can be characterized in terms of the projective system \(\mathcal{G}\). For a projective system \(\mathcal{G}\), \[\label{eqn:32Proj32Sys32dual32distance} d^{\perp}=\min \{|\mathcal{Q}|: \mathcal{Q} \subset \mathcal{G},|\mathcal{Q}|-\operatorname{dim} \operatorname{lin}\langle\mathcal{Q}\rangle=1\},\tag{2}\] where \(\langle\mathcal{Q}\rangle\) is the span of \(\mathcal{Q}\), and \(\operatorname{dim} \operatorname{lin}\langle\mathcal{Q}\rangle\) is its linear dimension (which is greater by 1 than the projective dimension of \(\langle\mathcal{Q}\rangle\) ). Equivalently, a set \(\mathcal{Q}\subset\mathcal{G}\) is dependent if \(|\mathcal{Q}|-\operatorname{dim}\operatorname{lin}\langle\mathcal{Q}\rangle=1\), i.e.the points of \(\mathcal{Q}\) fail to span a \(\mathrm{PG}(|\mathcal{Q}|-1,q)\); thus \(d^\perp\) is the minimum size of a dependent set of points of \(\mathcal{G}\). Note that degenerate codes are degenerate projective systems, in that they contain a point \(0\), corresponding to the \(0\)-dimensional vector subspace \(\langle 0 \rangle\). Though we generally avoid degenerate codes here, in such cases, the point \(0\) is said to have projective dimension \(-1\). Notice that according to Eqn. (2 ), a code \(\mathcal{G}\) is degenerate if and only if \(d^\perp=1\).

Our working definition of a (non-degenerate) linear code is as follows.

It follows immediately that an \([n,k,d ]_q\) code \(\mathcal{G}\) satisfies \(n-d\geq k-1\), which is the well known Singleton bound. The Singleton defect of an \([n,k,d]_q\) code is \(S(\mathcal{G})=n-k+1-d\). An \([n,k,d]_q\) code \(\mathcal{G}\) with \(S(\mathcal{G})=0\) is called a maximum distance separable (MDS) code. Codes of Singleton defect \(1\) are called Almost-MDS (AMDS) codes [3], and more generally, codes with \(S(\mathcal{G}) = s\) are denoted A\(^s\)MDS codes. The dual code of an MDS code is MDS, but generally speaking the dual of an A\(^s\)MDS code need not be A\(^s\)MDS. If \(S(\mathcal{G})=S(\mathcal{G}^\perp)=1\), \(\mathcal{G}\) is said to be dually-AMDS (or Near MDS, or NMDS). This notation extends naturally to dually A\(^s\)MDS (or N\(^s\)MDS) codes.

A linear code \(\mathcal{G}\) is projective if \(\mathcal{G}\) is a set, that is to say if \(P\in \mathcal{G}\) then \(\mu(P)=1\). A non-projective code is therefore equivalent to a code with one or more repeated coordinates. As we shall see, codes of sufficient length are necessarily projective.

Projective systems may be viewed as arcs or, sometimes as caps in PG\((k-1,q)\). An \((n,r)\)-arc \(\mathcal{K}\) in \(\Pi=PG(k-1,q)\), \(k\leq r+1\), is a collection of \(n\) points such that each hyperplane of \(\Pi\) is incident with at most \(r\) points in \(\mathcal{K}\), and some hyperplane is incident with \(r\) points of \(\mathcal{K}\). In the case where \(\mathcal{K}\) is a multi-set, we shall use the term multi-arc. An \((n,k-1)\)-arc in \(PG(k-1,q)\) is an \(n\)-arc. From the definitions, it is immediate that \((n,n-d)\)-arcs in \(PG(k-1,q)\) and projective \([n,k,d]_q\)-codes are equivalent objects, as are \((n,n-d)\)-multi-arcs in \(PG(k-1,q)\) and \([n,k,d]_q\)-codes. An \((n,r)\)-arc in \(PG(k-1,q)\) is said to be maximal if \(n=(r-k+2)(q+1)+k-2\). Such arcs are complete, in that they are not contained in an \((n+1,r)\)-arc. The (non)-existence question regarding maximal \((n,r)\)-arcs in PG\((2,q)\) has largely been answered (see e.g.[4] ). Here, for fixed \(q\), and \(s\) and with \(r=k+s-1\), we investigate the parameter \(\kappa(s,q)\), representing the maximum value \(k\) for which there exists a maximal \((n,r)\)-arc in \(PG(k-1,q)\).

An \(n\)-cap in PG\((k-1,q)\) is an \(n\)-set of points such that no three are collinear. It follows immediately that for \(k\ge 3\), a projective \([n,k,d]_q\) code, \(\mathcal{G}\) is an \(n\)-cap in PG\((k-1,q)\) if and only if \(d^\perp \ge 4\) (equivalently, \(S(C^\perp)\le k-3\)).

A fundamental problem in coding theory is that of determining the maximum length of a code with \(k\), \(q\), and \(S(\mathcal{G})\) fixed. To this end, we make the following definition.

Investigations into the bounds on \(m^s_t(k,q)\) initially focused on codes of small defect. In particular, the case \(s=0\) has received much attention by way of the Main Conjecture on MDS codes (see Section 3.1). The focus of the current work is on codes of defect \(s>0\). For \(s=1\), De Boer [3] and Dodunekov and Landjev [5], [6] provide the following.

Recall the notation \(m'(k,q):= m^1_1(k,q)\).

Dodunekov and Landjev furthermore showed that an \((n,k,d)_q\) AMDS code with \(n>q+k\) must in fact be NMDS. Written using our current notation, their result is as follows.

In [7], Tong discusses the case \(s=2\), establishing the following bound in the binary case.

Faldum and Willems [8] opened discussions to codes of arbitrary defect, providing bounds on \(m^s(k,q)\). In particular, they provided the following result which subsumes part [part:32thm32Dodunekov32P1] of Theorem 2, and Theorem 3.

If a code achieves the bound in Lemma 1 then it is said to be length-maximal. Faldum and Willems go on to show a code of sufficient length must be dual to an AMDS code.

In the sequel we shall establish bounds on \(m^{s}_t(k,q)\) which subsume and generalize the existing bounds. We are also interested in the existence question of length-maximal codes. Our work leads us to conjecture that such codes do not exist for code dimension \(k>4\).

Let us briefly describe code shortening, a technique that shall be employed in the sequel.

Shortening codes via quotient geometries↩︎

Code shortening is a common technique employed in coding theory (see e.g. constructions Y1–Y4, p. 592 in [9]). We shall employ shortening in some of our proofs. Within the framework of projective systems, shortening can be performed by way of quotients, described as follows.
Suppose \(\mathcal{G}\) is an \([n,k,d]_q\) A\(^s\)MDS code in \(\Sigma=\text{PG}(k-1,q)\), let \(\Lambda\) be an \(\ell\)-flat (i.e.an \(\ell\)-dimensional projective subspace) of codimension \(r\ge 2\), and let \(\alpha=|\Lambda \cap \mathcal{G}|\) (including multiplicities). Let \(\Sigma^*\) be the quotient geometry taken at \(\Lambda\), so that in particular, \(\Sigma^*\cong \text{PG}(r-1,q)\). Each point \(P\) of \(\Sigma^*\) is the image of an \((\ell+1)\)-flat, \(\Omega_P\), where \(\Lambda \subseteq \Omega_P\). If for each such point \(P\), we assign the multiplicity \(\mu(P) = |\Omega_P\cap \mathcal{G}|-\alpha\) then we obtain an \([n-\alpha, r, d^*]_q\) A\(^{s^*}\)MDS code \(\mathcal{G}^*\), where \(n-\alpha-d^*\le n-d-\alpha\), giving \(d^*\ge d\), and \[s^*=n-\alpha -r+1-d^* = n-\alpha +\ell-k+2-d^* \le n-\alpha +\ell-k+2-d =s-\alpha +\ell+1.\]

Consequently, we have the following.

Before delving into the main results of this paper, we explore bounds that follow from the basic theory, as well as those that follow from the established bounds on planar arcs.

For ease of reference, Table 1 collects the recurring notation. We follow the convention that Roman letters denote code- and arc-theoretic quantities, while Greek letters denote geometric objects (subspaces and flats) and point multiplicities; the one traditional exception is \(H\), reserved for a hyperplane.

2 Trivial codes and their bounds↩︎

We begin with 1-dimensional codes, where we temporarily widen the discussion to include degenerate codes. For any \(n,d,q\) one may construct the trivial (possibly degenerate) \([n,1,d]_q\) A\(^s\)MDS code by taking as \(\mathcal{G}\) a multiset from \(\{0,1\}\) (\(=\)PG\((0,q) \cup\{0\})\), where \(s=\mu(0)\), and \(d=\mu(1)\). By definition, \(\mathcal{G}\) will be MDS and non-degenerate if \(d=n\), and degenerate with \(s>0\) otherwise. The dual code \(\mathcal{G}^\perp\) is an \([n,n-1,d^\perp]_q\) code, which is MDS (\(d^\perp = 2\)) if \(d=n\), and is AMDS (\(d^\perp = 1\)) otherwise.

In the 2-dimensional case, for arbitrary \(n,d,q\), and \(s=n-d-1\), one may clearly select a multiset of points from PG\((1,q)\) with the maximum multiplicity of any point being \(n-d\). As such, one may construct an \([n,2,d]_q\)- A\(^s\)MDS code if and only if \(2+s\le n\le (s+1)(q+1)\). If \(\mathcal{G}\) is such a code with \(\mathcal{G}^\perp\) an \([n,n-2,d^\perp]_q\) code, then (Eqn. 2 ) \(d^\perp =3\) if \(s=0\), and \(d^\perp=2\) otherwise. For \(k>2q\), any \([k+2,2,d]_q\)-code necessarily satisfies \(d^\perp =2\) and is therefore dual to a \([k+2,k,2]_q\)-AMDS code. Since a \([k+2,2,d]_q\) code can only be AMDS if \(k+2\le 2(q+1)\), part [part:32thm32Dodunekov32P4] of Theorem 2 follows immediately, as does each of the following.

Let us denote by \(e_i\), \(1\le i \le k\), the projective point with homogeneous coordinates being the \(i\)’th standard basis vector in \(\mathbb{F}_q^k\). For fixed \(k,q,s\), the code \(\mathcal{G}\) comprising \(\{e_1,e_2,e_3,\ldots,e_k\}\), where \(\mu(e_1)=s+1\), and \(\mu(e_i)=1\) for \(2\le i \le k\), is an \([k+s,k,1]_q\) A\(^s\)MDS code, with \(d=1\), and \(d^\perp=2\). We thus have the following.

Codes of interest here are those which are non-degenerate and “long”—in the sense of being (near) length-maximal. If \(\mathcal{G}\) is an \([n,k,d]_q\) A\(^s\)MDS code with \(d^\perp=1\) then \(\mathcal{G}\) is degenerate, and if \(d=1\), then \(n=k+s\), so \(\mathcal{G}\) is far from length-maximal. Thus, having discussed codes of dimension \(2\) or less, in the sequel we shall generally limit discussion to cases where \(d,d^\perp>1\) and \(k>2\).

3 Some established code bounds based on arcs↩︎

We now explore the relationship between \((n,r)\)-arcs in projective planes and the bounds on the lengths of linear codes.

3.1 MDS Codes↩︎

In 1952, Bush [10] established that if \(k\ge q\) then \(m(k,q) = k+1\). For \(k<q\) there is a long-standing conjecture regarding linear MDS codes: every linear \([n,k,n-k+1]_q\) MDS code with \(1 < k < q\) satisfies \(n \le q + 1\), except when \(q\) is even and \(k = 3\) or \(k = q - 1\) in which cases \(n \le q + 2\). This conjecture is called the main conjecture on linear MDS codes. Though the main conjecture has been shown to hold in many cases, the topic is far from moribund. Below we list some of the cases where the main conjecture holds, but for a more complete summary see [11] and [12].

3.2 Bounds on 3-dimensional codes↩︎

The results of Barlotti (1956) [21], when paired with the later developments of Ball et al.[22], give bounds on \((n,r)\)-arcs in PG\((2,q)\) which provide the first three items in the following lemma. The fourth item is due to the construction of Denniston [23].

Further, we have the following.

4 Main Results↩︎

In this section, we present the primary findings, focusing on the bounds for the lengths of codes with non-zero Singleton defects. We first establish some elementary properties of \(m^s(k,q)\).

The bounds in Lemmas 5 and 6 may be extended to higher dimensions as follows.

4.1 Projective codes↩︎

Recall that a code \(\mathcal{G}\) is projective if \(\mathcal{G}\) is a set, i.e.every point has multiplicity \(1\). We now establish conditions under which codes are necessarily projective, and derive bounds on their lengths.

Note that according to equation (2 ), a (non-degenerate) code of dimension \(k>2\) is projective if and only if \(d^\perp\ne 2\).

We may also deduce the following bound on A\(^s\)MDS codes that are not dual to an AMDS code.

From Lemma 7, if \(t\ge 3\) and \(s>0\) then \(m^{s-1}(t,q)\le m^{s-1}(3,q)+t-3\). This bound may in turn be leveraged by bounds on \((n,s+1)\)-arcs in the plane, such as those in Lemmas 5, and 6. Specializing to the case \(s=1\): if \(t\ge 2\) then \(m(t,q)\le m(2,q)+t-2=q+t-1\). From part 1 of Lemma 2, the dual of any 2-dimensional non-MDS code is AMDS. Furthermore, if \(t\ge q\), then as observed in Section 3.1, \(m(t,q)=t+1\). We thus have the following corollary, the second part of which appears as Theorem 3.4 of [6].

4.2 Bounds on A\(^s\)MDS codes↩︎

It is noteworthy that \(S(C^\perp) =t > 1\) if and only if \(d^\perp < k\). Thus, for \(k = 3\), all projective codes are either MDS or dual to an AMDS code. For dimensions \(k > 3\), there exist projective codes that are neither MDS nor dual to an AMDS code. However, such codes are shown to necessarily be short, indicating that for dimensions three or greater, the longest codes are typically dual to AMDS codes. Ball and Hirschfeld observed in [4] that, in most cases, no examples exist of \((n, r)\)-arcs in \(\text{PG}(2, q)\) with a large \(n/q\) ratio, specifically \(n/q > r - 2\). The following demonstrates that for dimensions \(k > 3\), the longest codes that are neither AMDS nor dual to an AMDS code generally correspond (by shortening) to \((n, r)\)-arcs in \(\text{PG}(2, q)\) with \(n \leq (r - 2)q + r\).

Note that codes that achieve equality in part [part32132cor:32from32Barlotti] of Corollary 2 have no codewords of weight \(d+1\), whereas codes dual to those meeting the bound in part [part:32832thm32ub32ASMDS] of Theorem 16 have no codewords of weight \(d^\perp +1\).

Let \(C\) be an A\(^s\)MDS code, where \(k\ge3\), \(q>3\), and suppose \(C^\perp\) is AMDS. By part [part:32432thm32ub32ASMDS] of Theorem 16, if \(s=1\) then \(k\le 2q-2\), and if \(s>1,\) then by part [part:32132thm32ub32ASMDS], \(k\le q\). Thus, by part [part32132cor:32from32Barlotti] of Corollary 2 we obtain the following.

As an immediate consequence, we have the following bound for non-projective codes.

The next several corollaries of Lemma 9 highlight that both codes of large minimum distance, and those of large defect must be short.

We may improve upon the bound in Corollary 7 when \(a=1\), provided \(k\) and \(q\) are such that \(m(k-1,q)=q+1\) (as per the Main Conjecture on MDS Codes).

We may improve slightly if \(t>1\).

In [4], it is shown (for \(k=3\)) that if \(d\le q^2\), then an \([n,k,d]_q\) A\(^s\)MDS code \(\mathcal{G}\) meets the Griesmer bound if and only if \(n> s(q+1)+k-2\). The proof holds for \(k\ge 3\). Thus, with Corollary 9, Theorem 12, and part [part:32132thm32ub32ASMDS] of Theorem 16 we obtain the following.

The largest of the upper bounds on the length of linear \([n,k,d]_q\) codes presented here applies only to cases where \(t=1\): \[\label{eqn:32largest32bound} n\le (s+1)(q+1)+k-2.\tag{5}\] We are interested in determining existence conditions for codes meeting this bound. From Corollary 2, and Corollary 7 any code of dimension \(k\ge 4\) meeting the bound (5 ) satisfies either

1. \((s+2)|q\), \(q\) even, and \(2\le s \le q-4\), or

2. \(q-2 \le s\le q-1\).

There are clearly codes of dimension \(k=2\) meeting (5 ) for each \(s\ge 0\). When \(k=3\) there are optimal codes of type (b). For example, removing a single line from the plane \(PG(2,q)\) yields a \([q^2,3,q^2-q]_q\) code \(C\) with \(S(C)=q-2\), whereas the entire plane yields an \([q^2+q+1, 3, q^2]_q\) code \(C'\), with \(S(C')=q-1\). For each \(q>2\), there exists a \((q^2+1)\)-cap in PG\((3,q)\) with maximal plane intersection \(q+1\), and there exists a cap of size \(8\) in PG\((3,2)\) (see e.g. [27]). Thus for each \(q\ge 3\) there is a \([q^2+1,4,d]_q\) code \(C\) with \(S(C)=q-2\), and \(S(C^\perp)=1\), and in the binary case there is an \([8,4,4]_2\) NMDS code (\(s=q-1\)). For dimensions \(k\ge 5\) we are able to show that no codes of type (b) exist for \(q>3\).

With reference to Lemma 10, part [part:32132cor:32prime32divisors] of Corollary 14 gives the following.

Bounds on \(\kappa (s,q)\)↩︎

For fixed \(s,q\), let us denote by \(\kappa (s,q)\) the maximum \(k\) such that \(m^s(k,q)=(s+1)(q+1)+k-2\). Table 2 serves to set some of our results in context.

From the bounds in Corollary 13, and by computing the parameters in Lemmas 10 and 11 for some small values of \(s\) and \(q\), we may leverage the bounds in Table 2 to obtain the values of \(\kappa(s,q)\) in Table 3. Entries on the diagonal follow from Corollary 13, entries with an asterisk follow from Corollary 4, and remaining entries were verified numerically using Sage [30].

If \(\mathcal{G}\) is an \([n,k,d]_q\) AMDS code with \(n>q+k\), then by Theorem 3, \(\mathcal{G}\) is NMDS. In the next section, we address a gap in the literature by providing sufficient conditions on \(n\) and \(k\) under which an A\(^s\)MDS code must be N\(^s\)MDS, \(s\ge 1\).

4.3 More on N\(^s\)MDS codes↩︎

For \(q>3\), if an \([2q+3,2q,3]_q\) NMDS code were to exist, then the dual code would be an \([2q+3,3,2q]_q\) code, and thus, by part [part32232cor:32from32Barlotti] of Corollary 2, we arrive at the contradiction \(2q+3\le 2q+1\). By the discussion in Section 2, an \([2q+2,2,2q]_q\) code exists and its dual is an \([2q+2,2q,2]_q\) NMDS code.

If \(\mathcal{G}\) is an \([k+3,k,3]_q\) NMDS code, then \(\mathcal{G}^{\perp}\) is an \([k+3,3,k]_q\) NMDS code, whence \(k+3\le 2q+1\), so \(k\le 2q-2\). From the discussion in Section 2, the dual code of a \([k+2,2,k]_q\) code, \(1\le k\le 2q\), is an \([k+2,k,2]_q\) NMDS code. We thus have the following:

With reference to Theorem 16 we obtain the following.

The dual of an MDS code is also MDS, and if \(\mathcal{G}\) is an \([n,k,n-k]_q\) AMDS code with \(n>q+k\) then (Theorem 16, part [part:32132thm32ub32ASMDS]) \(\mathcal{G}^\perp\) is also AMDS, so \(\mathcal{G}\) is NMDS. For \(s>1\), Liao and Liao ([31], Theorem 3.6) provide lower bounds on \(n\) and \(k\) sufficient for an \([n,k,d]_q\) A\(^s\)MDS code to be N\(^s\)MDS, so too does Tong ([7], Theorem 4). However, in both works the authors require \(n>s(q+1)+k-1\), so according to part [part:32132thm32ub32ASMDS] of Theorem 16 their results hold vacuously. This observation identifies a gap in the existing literature, which we address by establishing feasible bounds within which such codes demonstrably exist. Towards this end, we first provide the following lemma.

The previous Lemma provides some easily obtained examples of N\(^2\)MDS binary codes.

We now present some bounds providing sufficiency conditions under which an A\(^s\)MDS code is necessarily an N\(^s\)MDS code.

We note that there are codes meeting the condition of the previous theorem, for example, both \([13,5,7]_4\), and \([14,5,8]_4\) A\(^2\)MDS codes exist (see e.g. [32]).

5 Conclusion, questions, and conjectures↩︎

Here, we have explored bounds on the lengths of codes with non-zero Singleton defects within the framework of projective systems. We introduced the parameters \(m^{s}(k,q)\), denoting the maximum length of a (non-degenerate) \([n,k,d]_q\) A\(^s\)MDS code, and \(m^{s}_t(k,q)\) denoting the maximum length of a (non-degenerate) \([n,k,d]_q\) A\(^s\)MDS code \(\mathcal{G}\) such that \(\mathcal{G}^\perp\) is an A\(^t\)MDS code. Tables 4, and 5 may help place our results in context.

Our results provide new insights that both extend and refine existing knowledge in the literature, with the bounds obtained through these methods often subsuming those previously established.

The findings presented highlight the effectiveness of interpreting linear codes as projective systems. This perspective not only allows us to derive tighter bounds on code lengths but also to gain insight into the structural properties of these codes. For example, we easily see that for fixed Singleton defect, dimension, and field, codes of reasonable length must meet the Griesmer bound, must be projective, and must be dual to an AMDS code.

The results and methods presented here have potential implications for the design and analysis of error-correcting codes, particularly in scenarios where maximizing code length is critical. Based on the findings and observations, we propose the following questions and conjectures for further investigation.

Question 1: Codes meeting the hypothesis of Theorem 20 form an extremely restricted family (especially in view of Corollary 3). Can the bounds be widened, or otherwise relaxed to include more codes?

Question 2: For which values of \(k,s,\) and \(q\) is the inequality in part [part:32232bounds32on32length32of32AsMDS] of Lemma 7 strict? For example, when \(s=0\) and \(k = 3\), the inequality is strict only if \(q\) is odd.

The dual of the perfect ternary Golay code is a projective two-weight \([11,5,6]_3\) code underlying the Berlekamp–van Lint–Seidel strongly regular graph \(\mathrm{SRG}(243,22,1,2)\). The \([11,5,6]_3\) code dual to the perfect ternary Golay code is a length-maximal two-weight code with weights \(6\) and \(9\). The self-dual extended ternary Golay code \([12,6,6]_3\) is a three-weight length-maximal code, with weights \(6\), \(9\) and \(12\). We suspect that, up to equivalence, these are the only examples of linear length maximal codes in dimensions \(k>4\).

References↩︎

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1. The first author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number 2019-04103]. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence 2019-04103].↩︎