An Error-Code Perspective on Metzner--Kapturowski-like Decoders

TL;DR

This work introduces an error‑code perspective on Metzner–Kapturowski–like decoding for high‑order interleaved sum‑rank‑metric codes, enabling an intuitive interpretation of decoding as operating on the error code. It presents a two‑stage, linear‑algebraic decoder that first recovers the error support and then erases accordingly, achieving correction up to $t\le d-2$ in general and under additional conditions up to $t\le n-k-1$, with a complexity of $O(\max\{n^3, n^2 s\})$ over $\mathbb{F}_{q^m}$ that is independent of constituent code structure. The paper analyzes probabilistic decoding for uniform random errors, derives bounds and a DP‑based method to compute the success probability, and demonstrates high success probabilities beyond the unique decoding radius as parameters grow. The results provide practical insights for designing and security‑assessing code‑based cryptosystems using interleaved sum‑rank codes and unify MK‑style decoding across Hamming, rank, and sum‑rank metrics, while also outlining open problems such as horizontal interleaving adaptations.

Abstract

In this paper we consider a Metzner-Kapturowski-like decoding algorithm for high-order interleaved sum-rank-metric codes, offering a novel perspective on the decoding process through the concept of an error code. The error code, defined as the linear code spanned by the vectors forming the error matrix, provides a more intuitive understanding of the decoder's functionality and new insights. The proposed algorithm can correct errors of sum-rank weight up to $d-2$, where $d$ is the minimum distance of the constituent code, given a sufficiently large interleaving order. The decoder's versatility is highlighted by its applicability to any linear constituent code, including unstructured or random codes. The computational complexity is $O(\max\{n^3, n^2 s\})$ operations over $\mathbb{F}_{q^m}$, where $n$ is the code length and $s$ is the interleaving order. We further explore the success probability of the decoder for random errors, providing an efficient algorithm to compute an upper bound on this probability. Additionally, we derive bounds and approximations for the success probability when the error weight exceeds the unique decoding radius, showing that the decoder maintains a high success probability in this regime. Our findings suggest that this decoder could be a valuable tool for the design and security analysis of code-based cryptosystems using interleaved sum-rank-metric codes. The new insights into the decoding process and the high success probability of the algorithm even beyond the unique decoding radius underscore its potential to contribute to various coding-related applications.

Figures (9)

• Figure 1: Logarithmic failure probability vs. $s-t$ for different values of $\ell$ with $q$, $m$, $n$ and $t$.
• Figure 2: Illustration of the decoding regions for Algorithm \ref{['alg:decode_high_order_int_sum_rank']} if full-rank condition is satisfied.
• Figure 3: Relationships between parameters, conditions and decoding success for uniform errors.
• Figure 4: Success probability vs error weight $t$ for $q=2$, $m=2$, $n=24$, $k=8$, $\eta=1$, and $\ell=24$ with interleaving order $s = t$.
• Figure 5: Success probability vs error weight $t$ for $q=2$, $m=2$, $n=24$, $k=8$, $\eta=2$, and $\ell=12$ with interleaving order $s = t$.

Theorems & Definitions (17)

• Definition 1: Interleaved Sum-Rank-Metric Code
• Lemma 1: Column-Erasure Decoder puchingerGenericDecodingSumRank2022
• Remark 1
• Lemma 2
• Theorem 1
• Remark 2
• Remark 3
• Theorem 2
• Theorem 3
• Lemma 3