Three results on twisted $G-$codes and skew twisted $G-$codes
Alvaro Otero Sanchez
TL;DR
This work addresses when twisted skew $G$-codes are code-checkable, extending prior results from untwisted group algebras to the twisted setting. It develops a framework based on crossed systems and Frobenius-duality to obtain three main contributions: (i) a code-checkability criterion for twisted group algebras when $G$ is $p$-nilpotent with a cyclic Sylow $p$-subgroup, (ii) a generalization showing that every twisted $G$-code with $\dim_K(C)\le 3$ is permutation-equivalent to an abelian group code, and (iii) a dimension–distance bound $|G| \le d(C)\dim C$ for nonzero twisted codes, with a precise equality condition. These results extend established group-algebra results to twisted contexts, providing structural insights and concrete criteria for code construction in twisted group rings.
Abstract
In this paper we solve an open question formulated in the original paper of twisted skew group codes regarding when a twisted skew group code is checkable. Also, we prove that all ideals of dimension 3 over a twisted group algebra are abelian group codes, generalising another previous result over group algebras. Finally, we prove a bound on the dimension and distance of a twisted group code, as well as when such bound is reached.