The codes in some non-semisimple dihedral group algebras and their properties.
Kirill Vedenev, Vladimir Mikhailovich Deundyak · arXiv (Cornell University) · 2020
In this paper, the group codes, i.e., left ideals, in the non--semisimple dihedral group algebra $\mathbb{F}_qD_{2n}$, $gcd(q, n)=1$, are studied. In order to do this, we consider one generalization of Wedderburn decomposition of $\mathbb{F}_qD_{2n}$. In particular, for every dihedral code its dual code is algebraically described. In addition, the bases, generating and check matrices of the dihedral codes are explicitly constructed. Some connections with the theory of cyclic codes are established. Finally, for dihedral codes some results about code parameters are obtained and several illustrative examples are considered.