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

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