Metacyclic error-correcting codes

Abstract

Error-correcting codes which are ideals in group rings where the underlying group is metacyclic and non-abelian are examined. Such a groupG(M, N,R) is the extension of a finite cyclic group ℤ_M by a finite cyclic group ℤ _N and has a presentation of the form (S, T:S ^M =1,T ^N =1, T· S=S ^R ·T) where gcd(M, R)=1, R ^N =1 modM, R ≠ 1. Group rings that are semi-simple, i.e., where the characteristic of the field does not divide the order of the group, are considered. In all cases, the field of the group ring is of characteristic 2, and the order ofG is odd.

Algebraic analysis of the structure of the group ring yields a unique direct sum decomposition ofFG(M, N, R) to minimal two-sided ideals (central codes). In every case, such codes are found to be combinatorically equivalent to abelian codes and of minimum distance that is not particularly desirable. Certain minimal central codes decompose to a direct sum ofN minimal left ideals (left codes). This direct sum is not unique. A technique to vary the decomposition is described. p]Metacyclic codes that are one-sided ideals were found to display higher minimum distances than abelian codes of comparable length and dimension. In several cases, codes were found which have minimum distances equal to that of the best known linear block codes of the same length and dimension.