MDS and AMDS symbol-pair codes are constructed from repeated-root codes

TL;DR

This paper constructs new classes of MDS and AMDS symbol-pair codes from repeated-root cyclic codes, expanding the known code families and their error-correcting capabilities for specific lengths.

Contribution

It introduces general generator polynomials for MDS symbol-pair codes and derives all such codes of length 3p with degree constraints, also providing new AMDS code classes.

Findings

All MDS symbol-pair codes of length 3p with generator degree ≤ 10 are characterized.

Two new classes of AMDS symbol-pair codes with lengths lp or 4p are constructed.

Solutions to certain equations over finite fields are used to determine code properties.

Abstract

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against the pair errors in symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. In this paper, we construct more general generator polynomials for two classes of MDS symbol-pair codes with code length $lp$. Based on repeated-root cyclic codes, we derive all MDS symbol-pair codes of length $3p$, when the degree of the generator polynomials is no more than 10. We also give two new classes of (almost maximal distance separable) AMDS symbol-pair codes with the length $lp$ or $4p$ by virtue of repeated-root cyclic codes. For length $3p$, we derive all AMDS symbol-pair codes, when the degree of the generator polynomials is less than 10. The main results are obtained by determining the solutions of certain equations over finite fields.