On Codes from Split Metacyclic Groups
Kirill Vedenev
TL;DR
This work addresses codes arising from non-abelian split metacyclic groups by deriving an explicit Wedderburn-like decomposition of the finite split metacyclic group algebra $\mathbb{F}_q G_{n,m,r}$ under $\gcd(q,n)=1$. Leveraging this decomposition, the authors develop a systematic theory of metacyclic codes, proving that they are generalized concatenated codes with cyclic inner codes and skew quasi-cyclic outer codes, and providing minimum-distance bounds via a modular, componentwise analysis. They also study induced codes (extending and restricting codes along subgroups) and derive structural and distance results, including a product-type bound for intersections of induced codes. Finally, the paper demonstrates the feasibility of a partial key-recovery attack on McEliece-type cryptosystems based on metacyclic codes, highlighting both cryptanalytic opportunities and practical considerations for code-based cryptography. Overall, the results offer a cohesive algebraic framework for constructing and analyzing metacyclic codes with potential benefits and caveats for cryptographic applications.
Abstract
The paper presents a comprehensive study of group codes from non-abelian split metacyclic group algebras. We derive an explicit Wedderburn-like decomposition of finite split metacyclic group algebras over fields with characteristic coprime to the group order. Utilizing this decomposition, we develop a systematic theory of metacyclic codes, providing their algebraic description and proving that they can be viewed as generalized concatenated codes with cyclic inner codes and skew quasi-cyclic outer codes. We establish bounds on the minimum distance of metacyclic codes and investigate the class of induced codes. Furthermore, we show the feasibility of constructing a partial key-recovery attack against certain McEliece-type cryptosystems based on metacyclic codes by exploiting their generalized concatenated structure.