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

Abstract

We study the maximum length of $q$-ary codes as a function of alphabet size, code size, and Singleton defect. For an $(n, M, d)_q$ code with dimension $\kappa = \log_q M \ge 2$ and Singleton defect $s = n - \lceil\kappa\rceil + 1 - d$, we establish a \emph{maximal-arc-type bound}. For $M = q^k$, we call codes with $n = (s+1)(q+1) + k - 2$ \emph{length-maximal}, and show such codes are necessarily symbol-uniform, have pairwise distances confined to $\{d\} \cup \{n-k+3, \ldots, n\}$, and satisfy the divisibility condition $(s+2) \mid q(q+1)$. An equivalent form yields an improved Singleton-type inequality extending a result of Guerrini, Meneghetti, and Sala for binary systematic codes. When $s \ge 2q$, the bound tightens to $n \le s(q+1)+k-1$; more finely, when $\alpha q \le s < (\alpha+1)q$ for integer $\alpha \ge 2$, it tightens to $n \le (s+2-\alpha)(q+1)+\alpha+k-3$, improving on the main bound by $(\alpha-1)q$. We identify several conditions under which nonlinear codes satisfy the Griesmer bound, including: $d \le q^2$; $s \le q-1$; $s \ge \beta q$ with $d \le \beta q^2$; and a parametric family of binary conditions. We also show that near-length-maximal $A^1$MDS codes of length $k+2q-1$ cannot exist for $k \ge 5$ when $q=2$, nor for $k \ge 7$ when $q=3$. For codes of non-integer dimension $\kappa \in (k, k+1)$, an analogous bound holds but is never attained. This forces the corresponding Singleton-type inequality one unit tighter than the integer-dimension case. For rational non-integer $\kappa$, our bounds specialise to a length bound for additive codes of fractional dimension, complementing recent geometric results on additive codes. Throughout, the results parallel the theory of maximal arcs. Whether length-maximal nonlinear codes can exist for parameter ranges within which no linear length-maximal codes exist is the principal open problem raised by this work.