Strict Half-Singleton Bound, Strict Direct Upper Bound for Linear Insertion-Deletion Codes and Optimal Codes
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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(𝒞) ≤ 2d_H(𝒞) is valid for any insdel code. In this paper, for a linear insdel code 𝒞 we propose a strict half-Singleton upper bound d_I(𝒞) ≤ 2(n-2k+1) if 𝒞 does not contain the codeword with all 1s, and a stronger direct upper bound d_I(𝒞) ≤ 2(d_H(𝒞)-t) under a weak condition, where t≥ 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.
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