Systematic Single-Deletion Multiple-Substitution Correcting Codes

TL;DR

This paper introduces a new family of systematic codes capable of correcting one deletion and multiple substitutions with reduced redundancy and polynomial encoding/decoding complexity, improving on previous bounds.

Contribution

It presents a novel construction of systematic single-deletion s-substitution correcting codes with asymptotical redundancy bounds and efficient polynomial encoding and decoding algorithms.

Findings

Redundancy at most (3s+4) log n + o(log n)

Encoding complexity is O(n^{s+3})

Decoding complexity is O(n^{s+2})

Abstract

Recent work by Smagloy et al. (ISIT 2020) shows that the redundancy of a single-deletion $s$-substitution correcting code is asymptotically at least $(s+1)\log n+o(\log n)$, where $n$ is the length of the codes. They also provide a construction of single-deletion and single-substitution codes with redundancy $6\log n+8$. In this paper, we propose a family of systematic single-deletion $s$-substitution correcting codes of length $n$ with asymptotical redundancy at most $(3s+4)\log n+o(\log n)$ and polynomial encoding/decoding complexity, where $s\geq 2$ is a constant. Specifically, the encoding and decoding complexity of the proposed codes are $O(n^{s+3})$ and $O(n^{s+2})$, respectively.