Multilayer crisscross error and erasure correction

TL;DR

This paper introduces the multi-cover metric for multilayer crisscross errors in matrices, establishes bounds, explores code properties, and provides constructions and decoding algorithms for maximum multi-cover distance codes.

Contribution

It defines the multi-cover metric, derives bounds like the Singleton bound, studies duality properties, and constructs codes with efficient decoding algorithms.

Findings

Introduced the multi-cover metric for multilayer crisscross errors.

Derived bounds including the Singleton bound for these codes.

Provided constructions and decoding algorithms for maximum multi-cover distance codes.

Abstract

In this work, multilayer crisscross error and erasures are considered, which affect entire rows and columns in the matrices of a list of matrices. To measure such errors and erasures, the multi-cover metric is introduced. Several bounds are derived, including a Singleton bound, and maximum multi-cover distance (MMCD) codes are defined as those attaining it. Duality, puncturing and shortening of linear MMCD codes are studied. It is shown that the dual of a linear MMCD code is not necessarily MMCD, and those satisfying this duality condition are defined as dually MMCD codes. Finally, some constructions of codes in the multi-cover metric are given, including dually MMCD codes, together with efficient decoding algorithms for them.