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

Insertion-deletion codes (insdel codes for short) are used for correcting synchronization errors in communications, and in other many interesting fields such as DNA storage, date analysis, race-track memory error correction and language processing, and have recently gained a lot of attention. To determine the insdel distances of linear codes is a very challenging problem. The half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li is a basic upper bound on the insertion-deletion error-correcting capabilities of linear codes. On the other hand the natural direct upper bound $d_I(\mathcal{C})\le 2d_H(\mathcal{C})$ is valid for any insdel code. In this paper, for a linear insdel code $\mathcal{C}$ we propose a strict half-Singleton upper bound $d_I(\mathcal{C})\le 2(n-2k+1)$ if $\mathcal C$doesnotcontainthecodewordwithall1s,andastrongerdirectupperbound$d_I(\mathcal C) \leq 2(d_H(\mathcal C)-t)$underaweakcondition,where$t\geq 1$ is a positive integer determined by the generator matrix. We also give optimal linear insdel codes attaining our strict half-Singleton bound and direct upper bound, and show that the code length of optimal binary linear insdel codes with respect to the (strict) half-Singleton bound is about twice the dimension. 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.