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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.

In this article, some row-cyclic error-correcting codes are shown to be ideals in group rings in which the underlying group is metacyclic. For a given underlying group, several nonequivalent codes with this structure may be generated. Each is related to a cyclic code generated in response, to the metrics associated with the underlying metacyclic group. Such codes in the same group ring are isomorphic as vector spaces but may vary greatly in weight distributions and so are nonequivalent. If the associated cyclic code is irreducible, examining the structure of its isomorphic finite field yields all nonequivalent codes with the desired structure. Several such codes have been found to have minimum distances equalling those of the best known linear codes of the same length and dimension.

This paper is an exposition of two methods of formulating a lower bound for the minimum distance of a code which is an ideal in an abelian group ring. The first, a generalization of the cyclic BCH (Bose-Chaudhuri-Hoquenghem) bound, was proposed by Camion [2]. The second method, presented by Jensen [4], allows the application of the BCH bound or any of its improvements by viewing an abelian code as a direct sum of concatenations of cyclic codes. This second method avoids the mathematical analysis required for a direct generalization of a cyclic bound to the abelian case. It can produce a lower bound that improves the generalized BCH bound. We present simple algorithms for 1) deriving the generalized BCH bound for an abelian code 2) determining direct sum decompositions of an abelian code to concatenated codes and 3) deriving a bound on an abelian code, viewed as a direct sum of concatenated codes, by applying the cyclic BCH bound to the inner and outer code of each concatenation. Finally, we point out the applicability of these methods to codes that are not ideals in abelian group rings.