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

TL;DR

This work develops linear exact repair schemes for erasures in decreasing monomial-Cartesian codes (DM-CC) and their high-rate augmentations, augmented Reed-Muller (ARM) and augmented Cartesian (ACar) codes. It provides one- and two-erasure repair schemes built on trace-based duals, with one-erasure schemes requiring a locality condition and two-erasure schemes for augmented codes free of erasure-position restrictions, yielding bandwidths that can compare favorably to RS/Hermitian in certain regimes. The paper analyzes asymptotics as the extension degree $t$ grows, showing that repair bandwidth per symbol vanishes while rates can approach high values (up to or near 1 for some augmented codes), highlighting the practical potential for high-rate, repair-efficient storage systems. By expanding the toolbox beyond GW-based RS repairs, the work offers flexible, high-rate alternatives when conventional GW applicability is limited, with clear parameter regimes where ARM/ACar outperform traditional families in either bandwidth or rate. Key results are formalized through explicit bandwidth formulas, dual-structure lemmas, and asymptotic rate analyses, all expressed within the trace-function framework $\mathrm{Tr}:K\to\mathbb{F}_q$.

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.

Figures (5)

• Figure 1: The code $\textrm{ACar1}(S_1 \times S_2, (2,2))$ in Example \ref{['21.01.01']} with $K=\mathbb{F}_{17}$ is generated by the vectors $\textrm{ev}_{S_1 \times S_2} (\text{$M$}),$ where $M$ is a monomial whose exponent is a point in (a). The dual code $\textrm{ACar1}(S_1 \times S_2, (2,2))^{\perp}$ is generated by the vectors $\operatorname{Res}_{S_1 \times S_2} (\text{$M$}),$ where $M$ is a monomial whose exponent is a point in (b).
• Figure 2: The $\textrm{ARM1}(K^2, 2)$ code in Example \ref{['21.05.16']} with $K=\mathbb{F}_7$ is generated by the vectors $\textrm{ev}_{K^2} (\text{$M$}),$ where $M$ is a monomial whose exponent corresponds to a point in (a). $\textrm{ARM1}(K^2, 2)^{\perp}$ is generated by the vectors $\textrm{ev}_{K^2} (\text{$M$}),$ where $M$ is a monomial whose exponent corresponds to a point in (b).
• Figure 3: The code $\textrm{ACar2}(S_1 \times S_2, (2, 5))$ in Example \ref{['21.05.24']} with $K=\mathbb{F}_{17}$ is generated by the vectors $\textrm{ev}_{S_1 \times S_2} (\text{$M$}),$ where $M$ is a monomial whose exponent is a point in (a). The dual $\textrm{ACar2}(S_1 \times S_2, (2, 5))^{\perp}$ is generated by the vectors $\operatorname{Res}_{S_1 \times S_2} (\text{$M$}),$ where $M$ is a monomial whose exponent is a point in (b).
• Figure 4: The code $\textrm{ARM2}(K^2, 3)$ in Example \ref{['21.01.03']} with $K=\mathbb{F}_7$ is generated by the vectors $\textrm{ev}_{K^2} (\text{$M$}),$ where $M$ is a monomial whose exponent corresponds to a point in (a). The dual $\textrm{ARM2}(K^2, 3)^{\perp}$ is generated by the vectors $\textrm{ev}_{K^2} (\text{$M$}),$ where $M$ is a monomial whose exponent corresponds to a point in (b).
• Figure 5: Rate versus the repair bandwidth of the repair schemes of $\textrm{RM}(\mathbb{F}_{5^4}^3, k)$, $\textrm{ARM1}(\mathbb{F}_{5^4}^3, k)$, and $\textrm{ARM2}(\mathbb{F}_{5^4}^3, k),$ for all values of $k$ where the repair schemes developed in GW and Corollary \ref{['21.01.07']} can be applied.

Theorems & Definitions (32)

• Remark 2.1
• Remark 2.2
• Example 3.1
• Example 3.2
• Proposition 3.3
• proof
• Example 3.4
• Example 3.5
• Proposition 3.6
• proof