The $b$-weight distribution for MDS codes

TL;DR

This paper extends the understanding of the weight distribution of MDS codes within the new $b$-symbol coding framework, connecting it to the properties of shortened MDS codes and generalizing previous results.

Contribution

It establishes a general method to determine the $b$-weight distribution of MDS codes using solutions to equations and properties of shortened codes, broadening prior work.

Findings

Derived the $b$-weight distribution for MDS codes.

Connected $b$-weight distribution to the structure of shortened codes.

Generalized previous results by Ma and Luo.

Abstract

For a positive integer $b\ge2$, the $b$-symbol code is a new coding framework proposed to combat $b$-errors in $b$-symbol read channels. Especially, the $2$-symbol code is called a symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $b$-symbol code. Recently, for any MDS code $\mathcal{C}$, Ma and Luo determined the symbol-pair weight distribution of $\mathcal{C}$. In this paper, by calculating the number of solutions for some equations and utilizing some shortened codes of $\mathcal{C}$, we give the connection between the $b$-weight distribution and the number of codewords in shortened codes of $\mathcal{C}$ with special shape. Furthermore, note that shortened codes of $\mathcal{C}$ are also MDS codes, the number of these codewords with special shape are also determined by the shorten method. From the above calculation, the $b$-weight distribution of $\mathcal{C}$ is determined. Our result generalies the corresonding result of Ma and Luo.