Strict Half-Singleton Bound, Strict Direct Upper Bound for Linear Insertion-Deletion Codes and Optimal Codes

Let ${\mathcal C}$ be an $[n, k]$ linear code over the finite field ${\mathbb F}_{q}$ . Let $d_{I}({\mathcal C})$ denote its insertion-deletion (insdel for short) distance, which characterizes the insdel error-correcting capability of ${\mathcal C}$ . To determine the insdel distances of linear codes is a very challenging problem. In this paper we propose a strict half-Singleton upper bound $d_{I}({\mathcal C}) \leq 2(n-2k+1)$ if ${\mathcal C}$ does not contain the codeword with all 1s, which generalizes the half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li, and a stronger direct upper bound $d_{I}({\mathcal C}) \leq 2(d_{H}({\mathcal C})-t)$ under a weak condition, where $t\geq 1$ is a positive integer determined by the generator matrix and $d_{H}({\mathcal C})$ denotes the Hamming distance of ${\mathcal C}$ . A sufficient condition for a linear code attaining the strict half-Singleton bound is given. We prove that the code length of an optimal binary linear insdel code with respect to the (strict) half-Singleton bound is about twice its dimension and conjecture that optimal binary linear insdel codes have exact parameters $[{2k, k, 4}]$ or $[{2k+1, k, 4}]$ with respect to the half-Singleton bound or the strict half-Singleton bound, respectively. Moreover, interestingly explicit optimal linear insdel codes attaining the (strict) half-Singleton bound, with the code length being independent of the finite field size, are given.