Linear Codes over Z_4+uZ_4: MacWilliams identities, projections, and formally self-dual codes

TL;DR

This paper explores linear codes over the ring Z_4+uZ_4, establishing weight enumerator identities, analyzing projections, and constructing formally self-dual codes with practical examples.

Contribution

It introduces new MacWilliams identities, studies projections to related rings, and provides constructions for formally self-dual codes over Z_4+uZ_4.

Findings

Proved MacWilliams identities for various weight enumerators.

Analyzed projections to Z_4 and F_2+uF_2 rings.

Constructed new classes of formally self-dual codes.

Abstract

Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and self-dual codes over Z_4+uZ_4 are studied in connection with these projections. Finally three constructions are given for formally self-dual codes over Z_4+uZ_4 and their Z_4-images together with some good examples of formally self-dual Z_4-codes obtained through these constructions.