Codes from Zero-divisors and Units in Group Rings

A new construction method for codes using encodings from group rings is presented. They consist primarily of two types, zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; e.g. cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in ideals.
It is proved that a group ring is isomorphic to a certain well-defined ring of matrices, and thus every group ring element has an associated matrix. This new result allows the use of matrix algebra as needed in the study and production of codes, enabling the creation of standard generator and check matrices.
Group rings are a fruitful source of units and zero-divisors from which new codes result. Many code properties may be expressed in terms of the group ring. These facilitate the presentation of example LDPC and self-dual codes. Example codes over the integers and from group rings over groups such as the dihedral group are also described. In certain cases, techniques for determining minimum distance are presented. An initial framework for convolutional-type codes from group ring encodings is also described.