The $b$-weight distribution for MDS codes

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.