Contributions to the theory and practice of group codes

Group codes, introduced by Forney and Trott (1), are a general algebraic approach to the study of convolutional-type error correcting codes. A Euclidean-space (ES) code is the implementation of a group code with a modulation for use on a communication channel. All group codes are assumed to be "complete" and "finitely-controllable" in this dissertation. It has been speculated in the literature that group codes over non-abelian groups may give ES codes with better performances than those using group codes over abelian groups. The main result of this dissertation is than many finite non-abelian groups are shown to support only group codes whose ES codes have the same performance as those using abelian group codes. Also a practical technique of using time-varying modulation called Rotating Euclidean-space Codes is proposed for channels with intersymbol interference.;A pair of group codes will be defined in this thesis to be "conformant" if there exists a bijection between the codes based on a componentwise group bijection. An encoder for the conformant abelian group code followed by the inverse bijection is a surrogate encoder for the non-abelian group code. Forney and Trott introduced the definition of "dynamic-equivalence" between group codes. Dynamic-equivalence implies conformance. Several sufficient conditions are given for non-abelian groups to have the property that support only group codes that are dynamically-equivalent (hence conformant) to abelian group codes. These sufficient conditions are satisfied by many familiar non-abelian groups, by at least every group with order 23 or less, and by every finite group of orthonormal transformations in 3-dimensional vector-space.;Several incomplete inductive conditions are derived for additional non-abelian groups to have the property that they support only group codes that are dynamically-equivalent to abelian group codes. A special type of group, which is a smallest-case exception to the inductive conditions, is shown to have the abelian-conformance property despite supporting group codes that are not dynamically-equivalent to abelian group codes. Groups will have the abelian-conformance property when they are shown to support only group codes, which are dynamically-equivalent to group codes over groups known to have the abelian-conformance property.