On the List-Decodability of Random (Linear) Sum-Rank Metric Codes
Yang Liu, Anna Baumeister, Antonia Wachter-Zeh
TL;DR
This work establishes a list-decoding capacity theorem for sum-rank metric codes, showing that random general sum-rank codes with rate up to the capacity $1-\kappa_b(\rho)$ are $(\rho, O(1/\epsilon))$-list-decodable with high probability. It extends prior approaches from the Hamming and rank metrics by introducing decomposable subspaces and a dimension lemma, then proves a limited-correlation property to enable polynomial list sizes for random $\mathbb{F}_q$-linear sum-rank codes. The results unify and generalize the known capacity phenomena to the sum-rank setting, with an initial exponential-list bound for linear codes that is improved to $O(1/\epsilon)$ via limited correlation. The findings have implications for multishot network coding and related algebraic coding theory, and open questions remain for $\mathbb{F}_{q^m}$-linear vector sum-rank codes. Overall, the paper fills a gap by showing that sum-rank codes can achieve near-optimal list-decodability with manageable list sizes in both random general and random linear models.
Abstract
In this paper, we establish the list-decoding capacity theorem for sum-rank metric codes. This theorem implies the list-decodability theorem for random general sum-rank metric codes: Any random general sum-rank metric code with a rate not exceeding the list-decoding capacity is $\left(ρ,O\left(1/ε\right)\right)$-list-decodable with high probability, where $ρ\in\left(0,1\right)$ represents the error fraction and $ε>0$ is referred to as the capacity gap. For random $\mathbb{F}_q$-linear sum-rank metric codes by using the same proof approach we demonstrate that any random $\mathbb{F}_q$-linear sum-rank metric code with a rate not exceeding the list-decoding capacity is $\left(ρ,\exp\left(O\left(1/ε\right)\right)\right)$-list-decodable with high probability, where the list size is exponential at this stage due to the high correlation among codewords in linear codes. To achieve an exponential improvement on the list size, we prove a limited correlation property between sum-rank metric balls and $\mathbb{F}_q$-subspaces. Ultimately, we establish the list-decodability theorem for random $\mathbb{F}_q$-linear sum-rank metric codes: Any random $\mathbb{F}_q$-linear sum-rank metric code with rate not exceeding the list decoding capacity is $\left(ρ, O\left(1/ε\right)\right)$-list-decodable with high probability. For the proof of the list-decodability theorem of random $\mathbb{F}_q$-linear sum-rank metric codes our proof idea is inspired by and aligns with that provided in the works \cite{Gur2010,Din2014,Gur2017} where the authors proved the list-decodability theorems for random $\mathbb{F}_q$-linear Hamming metric codes and random $\mathbb{F}_q$-linear rank metric codes, respectively.