Decoding rank metric Reed-Muller codes

TL;DR

This work addresses decoding rank-metric Reed–Muller codes RM_theta(r, n) over arbitrary Abelian Galois extensions L/K by exploiting G-Dickson matrices. The authors develop a polynomial-time decoder that reconstructs the error theta-polynomial via a majority-voting scheme on the entries of the G-Dickson matrix, enabling correct decoding up to half the minimum distance. The method extends Gabidulin-style decoding from cyclic G to general Abelian G, and provides detailed complexity analyses, including a refinement that achieves essentially a single Gaussian elimination with O~(N^4) base-field operations. The approach broadens the class of efficiently decodable rank-metric codes over infinite fields and offers a framework potentially adaptable to other skew-group-algebra codes. This has implications for cryptography, network coding, and data-storage applications where robust rank-metric decoding over extensions is desirable.

Abstract

In this article, we investigate the decoding of the rank metric Reed--Muller codes introduced by Augot, Couvreur, Lavauzelle and Neri in 2021. These codes are defined from Abelian Galois extensions extending the construction of Gabidulin codes over arbitrary cyclic Galois extensions. We propose a polynomial time algorithm that rests on the structure of Dickson matrices, works on any such code and corrects any error of rank up to half the minimum distance.

Figures (6)

• Figure 1: The sequence of minors that permit to recover the unknown coefficients. At each step, unknown coefficients are in black font.
• Figure 2: Description of the algorithm in the general case
• Figure 3: $\textrm{G}$-Dickson matrix of $F = \sum_{i,j=0}^{2} f_i^j \theta_1^{i} \theta^{j}$. Compared to the $\textrm{G}$-Dickson matrix for $\textrm{G}$ cyclic, which is $q$-circulant, for $\textrm{G} \cong \mathbb{Z}/{n\mathbb{Z}} \times \mathbb{Z}/{n\mathbb{Z}}$, it is a block circulant matrix. The colors depict that the $9 \times 9$ matrix can be seen as a $3 \times 3$ block matrix where each block is a sort of circulant matrix and the blocks appear in a circular manner (up to applications of elements of the Galois group). If we ignore the applications of the Galois group elements, then the coefficients of $F$ are exactly in a block circulant form as defined in Tr73.
• Figure 4: Illustration of positions of the unknown coefficients.
• Figure 5: Comparison with ACLN in the case $\textrm{G} = \mathbb{Z} /n \mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ when $r<n$ and $n \rightarrow \infty$. The $x$-axis represents $\rho = \frac{r}{n}$ and the $y$--axis the relative decoding radius $\frac{t}{n^2}$.

Theorems & Definitions (65)

• Remark 2.1
• Remark 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7: ACLN
• Remark 2.8
• Theorem 2.9: ACLN