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

TL;DR

This paper introduces a Metzner-Kapturowski-like decoding algorithm for high-order interleaved sum-rank-metric codes, using an error code perspective to improve understanding, analyze success probability, and demonstrate versatility and efficiency.

Contribution

It presents a novel decoding algorithm based on error codes, applicable to any linear constituent code, with high success probability even beyond the unique decoding radius.

Findings

Decodes errors up to d-2 sum-rank weight with large interleaving order.

Provides an efficient method to estimate success probability for random errors.

Maintains high success probability beyond the unique decoding radius.

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.