MDS and AMDS symbol-pair codes 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 codes and providing explicit classifications for certain lengths and generator polynomial degrees.

Contribution

It introduces novel MDS and AMDS symbol-pair codes derived from repeated-root cyclic codes, including classifications for specific lengths and polynomial degrees.

Findings

Constructed two classes of MDS symbol-pair codes from repeated-root cyclic codes.

Derived all MDS and AMDS symbol-pair codes with length 3p and degree of generator polynomials ≤ 10.

Provided solutions to certain equations over finite fields to determine code parameters.

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. Based on repeated-root cyclic codes, we construct two classes of MDS symbol-pair codes for more general generator polynomials and also give a new class of almost MDS (AMDS) symbol-pair codes with the length $lp$. In addition, we derive all MDS and AMDS symbol-pair codes with length $3p$, when the degree of the generator polynomials is no more than 10. The main results are obtained by determining the solutions of certain equations over finite fields.