On codes over R_{k,m} and constructions for new binary self-dual codes

TL;DR

This paper explores codes over a specific ring extension of binary fields, introduces a Gray map preserving key properties, and constructs numerous new binary self-dual codes, including extremal and previously unknown codes.

Contribution

It introduces a new Gray map for codes over R_{k,m} and uses it to construct many new binary self-dual codes, including extremal and previously unknown codes.

Findings

Constructed 175 new Type I binary self-dual codes of parameters [72,36,12]

Constructed 105 new Type II binary self-dual codes of parameters [72,36,12]

Developed a distance and duality preserving Gray map for R_{k,m}

Abstract

In this work, we study codes over the ring R_{k,m}=F_2[u,v]/<u^{k},v^{m},uv-vu>, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from R_{k,m} to F_2^{km} together with a Lee weight. After proving the MacWilliams identities for codes over R_{k,m} for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over R_{k,m}. In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72,36,12] and 105 new Type II binary self-dual codes of parameter [72,36,12].