Bounds and Constructions for Insertion and Deletion Codes

Abstract

Insertion and deletion (insdel for short) codes have recently attracted a lot of attention due to their applications in many interesting fields such as DNA storage, DNA analysis, race-track memory error correction and language processing. The present paper mainly studies limits and constructions of insdel codes. The paper can be divided into two parts. The first part focuses on various bounds, while the second part concentrates on constructions of insdel codes.Although the insdel-metric Singleton bound has been derived before, it is still unknown if there are any nontrivial codes achieving this bound. Our first result shows that any nontrivial insdel codes do not achieve the insdel-metric Singleton bound. The second bound shows that every [n, k] Reed-Solomon code has insdel distance upper bounded by 2n − 4k + 4 and it is known in literature that an [n, k] Reed-Solomon code can have insdel distance 2n − 4k + 4 as long as the field size is sufficiently large. The third bound shows a trade-off between insdel distance and code alphabet size for codes achieving the Hamming-metric Singleton bound. In the second part of the paper, we first provide a non-explicit construction of nonlinear codes that can approach the insdel-metric Singleton bound arbitrarily when the code alphabet size is sufficiently large. The second construction gives two-dimensional Reed-Solomon codes of length n and insdel distance 2n − 4 with field size q = O(n 5 ).The non-explicit construction of insdel codes is based on constant-weight L 1 -codes that are introduced in this paper. We first establish a relation between constantweight L 1 -codes and insdel codes. Based on this relation, we construct constantweight L 1 -codes with reasonable parameters and subsequently give insdel codes approaching the insdel-metric Singleton bound. Via automorphism group of rational function field, we provide a necessary and sufficient condition under which a twodimensional Reed-Solomon code of length n has insdel distance 2n − 4. Based on this criterion, we present a construction of q-ary two-dimensional Reed-Solomon codes of length n and insdel distance 2n − 4 with q = O(n 5 ). Though this is worse than the current best field size, we provide a new angle to look into the problem.

A Lower Bound on the List-Decodability of Insdel Codes

“…Here, ρ (δ,L) (1 − τ D ) is a piecewise linear function with L linear pieces where it coincides with the unique decoding bound τ I < δ − τ D when τ D 1 − L+1 L−1 (1 − δ). Such result provides a lower bound on the insdellist-decodability of a code given the values of δ, τ I , τ D and L. As illustrated in Section VI, Theorem IV.1 provides an insdel-list-decodability results for various constructions of codes, including some Reed-Solomon codes [31], [32], [33], [34], [35], Varshamov-Tenengolts (VT for short) codes [14], [15], [17] and Helberg codes [36], [37]. This provides such constructions with a stronger property on their insdel-list-decodability, as can be observed in Theorems VI.1, VI.2 and VI.3 respectively.…”

“…This implies that for such choice of the evaluation points, the minimum Levenshtein distance is at least 2n − 4k + 4. Although there have been numerous works on the insdel-correcting capabilities of Reed-Solomon codes (see, for example [31], [32], [33], [34], [35]), there have not been any study on its list-decoding capability. It is easy to see from Theorem IV.1 that the upper bound for t I increases as the minimum Levenshtein distance increases.…”