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

TL;DR

This paper establishes new upper bounds for linear insertion-deletion codes' error correction capabilities, introduces optimal codes attaining these bounds, and explores their properties in relation to code length and field size.

Contribution

It proposes a strict half-Singleton bound and a stronger direct upper bound for linear insdel codes, and constructs optimal codes that meet these bounds.

Findings

New strict half-Singleton upper bound for linear insdel codes.

Stronger direct upper bound under weak conditions.

Explicit optimal codes achieving the bounds with length independent of field size.

Abstract

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 $d_I(\mathcal C) \leq 2d_H(\mathcal C)$ is valid for any insdel code. In this paper, for a linear insdel code $\mathcal C$ we propose a strict half-Singleton upper bound $d_I(\mathcal C) \leq 2(n-2k+1)$ if $\mathcal C$ does not contain the codeword with all 1s, 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. 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.