The extended codes of a family of reversible MDS cyclic codes

TL;DR

This paper investigates the extended codes of a family of reversible MDS cyclic codes, constructing new MDS and NMDS codes with applications in cryptography, data storage, and finite geometry, and analyzing their weight distributions.

Contribution

It introduces new extended MDS and NMDS codes derived from classical reversible MDS cyclic codes, expanding the code families and their applications.

Findings

Two new families of MDS codes are constructed.

Several NMDS codes with practical applications are obtained.

The weight distributions of some extended codes are explicitly determined.

Abstract

A linear code with parameters $[n, k, n-k+1]$ is called a maximum distance separable (MDS for short) code. A linear code with parameters $[n, k, n-k]$ is said to be almost maximum distance separable (AMDS for short). A linear code is said to be near maximum distance separable (NMDS for short) if both the code and its dual are AMDS. MDS codes are very important in both theory and practice. There is a classical construction of a $[q+1, 2u-1, q-2u+3]$ MDS code for each $u$ with $1 \leq u \leq \lfloor\frac{q+1}2\rfloor$, which is a reversible and cyclic code. The objective of this paper is to study the extended codes of this family of MDS codes. Two families of MDS codes and several families of NMDS codes are obtained. The NMDS codes have applications in finite geometry, cryptography and distributed and cloud data storage systems. The weight distributions of some of the extended codes are determined.