Recovering erasures by using MDS codes over extension alphabets

Abstract

A new family of ${\mathbb {F}}_{q}$-linear codes over ${\mathbb {F}}_{q}^{b}$ can be obtained replacing the elements in the large field ${\mathbb {F}}_{q^{b}}$ by elements in ${\mathbb {F}}_{q}[C]$, where C is the companion matrix of a primitive polynomial of degree b and coefficients in ${\mathbb {F}}_{q}$. In this work, we propose a decoding algorithm for this family of ${\mathbb {F}}_{q}$-linear codes over the erasure channel, based on solving linear systems over the field ${\mathbb {F}}_{q}$.