Erasures repair for decreasing monomial-Cartesian and augmented Reed-Muller codes of high rate

TL;DR

This paper introduces new repair schemes for decreasing monomial-Cartesian and augmented Reed-Muller codes, enabling efficient erasure correction with lower bandwidth and no position restrictions, especially at high rates.

Contribution

It develops novel repair schemes for augmented codes that outperform traditional codes in bandwidth and flexibility, with detailed analysis of asymptotic behavior.

Findings

Augmented Reed-Muller and Cartesian codes can repair one or two erasures efficiently.

The repair schemes for augmented codes have no restrictions on erasure positions.

Augmented codes can achieve lower bandwidth compared to Reed-Solomon and Hermitian codes.

Abstract

In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes DM-CC, a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.