Metacyclic error-correcting codes

Abstract

This paper examines error-correcting codes which are ideals in group rings where the underlying group is metacyclic and non-abelian. Such a group G(M,N,R) is the extension of a finite cyclic group $\doubz\sb{\rm M}$ by a finite cyclic group $\doubz\sb{\rm N}$ and has a presentation of the form (S, T: S$\sp{\rm M}$ = 1, T$\sp{\rm N}$ = 1, T $\cdot$ S = S$\sp{\rm R}$ $\cdot$ T) where gcd (M, R) = 1, R$\sp{\rm N}$ $\equiv$ 1 mod M, R $\not=$ 1. Only group rings are semi-simple, i.e., where the characteristic of the field does not divide the order of the group, were examined in depth. In all cases, the field of the group ring is of characteristic two and MN is odd. Algebraic analysis of the structure of the group ring yielded a unique direct sum decomposition of FG(M,N,R) to minimal two-sided ideals (central codes). In every case, such codes were found to be combinatorially equivalent to abelian codes and of minimum distance that was not particularly desirable. Certain minimal central codes decompose to a direct sum of N minimal left ideals (left codes). The decomposition is not unique. Using cosets of associated cyclic codes, all non-equivalent minimal left codes were determined. An extension of the BCH bound that may used with these codes was formulated. Metacyclic codes were compared and contrasted with several classes of known codes: abelian, concatenated, and quasi-cyclic codes. In general, 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. The well-defined algebraic structure of these codes should allow encoding and decoding with simple circuitry.