Non-binary Two-Deletion Correcting Codes and Burst-Deletion Correcting Codes

TL;DR

This paper introduces new systematic q-ary two-deletion and burst-deletion correcting codes with improved redundancy and near-linear encoding complexity, advancing error correction capabilities for sequence transmission.

Contribution

The paper presents novel constructions of systematic q-ary two-deletion and burst-deletion correcting codes with reduced redundancy and efficient encoding algorithms.

Findings

Redundancy for two-deletion codes is $5 ext{log} n + O( ext{log} q ext{log} ext{log} n)$.

Binary burst-deletion codes achieve redundancy $ ext{log} n + 9 ext{log} ext{log} n + ext{constant}$, improving previous results.

q-ary burst-deletion codes have redundancy $ ext{log} n + (8 ext{log} q + 9) ext{log} ext{log} n + o( ext{log} q ext{log} ext{log} n)$.

Abstract

In this paper, we construct systematic $q$-ary two-deletion correcting codes and burst-deletion correcting codes, where $q\geq 2$ is an even integer. For two-deletion codes, our construction has redundancy $5\log n+O(\log q\log\log n)$ and has encoding complexity near-linear in $n$, where $n$ is the length of the message sequences. For burst-deletion codes, we first present a construction of binary codes with redundancy $\log n+9\log\log n+\gamma_t+o(\log\log n)$ bits $(\gamma_t$ is a constant that depends only on $t)$ and capable of correcting a burst of at most $t$ deletions, which improves the Lenz-Polyanskii Construction (ISIT 2020). Then we give a construction of $q$-ary codes with redundancy $\log n+(8\log q+9)\log\log n+o(\log q\log\log n)+\gamma_t$ bits and capable of correcting a burst of at most $t$ deletions.