Length-Maximal Codes with Given Singleton Defect: Structure and Bounds

Abstract

Let $\mathcal C$ be an $(n,q^k,d)_q$ code of integer dimension $k\ge2$ and Singleton defect $s=n-k+1-d$. Repeated shortening followed by the Plotkin bound gives \[ n\le (s+1)(q+1)+k-2. \] We study the equality case, calling a code attaining this bound \emph{length-maximal}. Equality provides rigid structure: the code is symbol-uniform under every successive shortening, is an orthogonal array of strength at least $k-1$, and has pairwise distances in $\{d\}\cup\{n-k+3,\ldots,n\}$. For $k\ge3$ it also satisfies $s\le q-1$ and $(s+2)\mid q(q+1)$. In dimension two, length-maximal codes are equivalent to resolvable $2$-$(q^2,q,s+1)$ multidesigns. The (inner) distance distribution, and hence the Hamming weight enumerator after any codeword is normalised to zero, is determined by the parameters. In dimension four this gives three cases: apart from the MDS case, only $s=q-2$ and $s=q-1$ remain possible; in the latter case the alphabet size is constrained by $(q+2)\mid36$, leaving six values of $q$. For $q\ge4$, every length-maximal code of dimension at least five is MDS, and requires $q$ to be a multiple of $36$. The non-MDS cases are the dual and extended ternary Golay codes, of dimensions five and six. Consequently, for $s\ge1$ one has $k\le4$ unless $q=3$, where $k\le6$. By shortening the codes before applying Plotkin, we provide sharper large-defect bounds and several ranges in which nonlinear codes satisfy the Griesmer bound.