Affiliation:
1. Department of Mathematics Fatih University 34500 Buyukcekmece Istanbul Turkey
Abstract
Abstract
In this work, we study codes over the ring 𝓡
k, m
= 𝔽2[u, v]/ 〈uk
, vm
, 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
k
m
$\mathbb{F}_2^{km}$
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].
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