Erasures Repair for Decreasing Monomial-Cartesian and Augmented Reed-Muller Codes of High Rate

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.

Automated summary

In this work, linear exact repair schemes are proposed for one or two erasures in decreasing monomial-Cartesian codes. Families of augmented Reed-Muller codes and augmented Cartesian codes are used for the repair schemes. Unlike the repair scheme for two erasures in decreasing monomial-Cartesian codes, the repair scheme for two erasures in the augmented codes has no restrictions on the positions of the erasures. Examples are provided where the augmented codes have lower bandwidth compared to other codes.