Skew cyclic codes through Drinfeld modules

Abstract

We construct skew cyclic codes in the rank metric from the torsion of the supersingular Drinfeld module ${\varphi }_T=t+{\tau }^n$ over ${\mathbb{F}}_{q^r}$, where $t$ is a root of a monic irreducible ${\mathfrak{p}}_0\in {\mathbb{F}}_q[T]$ of degree $r$ coprime to $n$. For every prime $\mathfrak{p}\ne {\mathfrak{p}}_0$ of degree $d$ and every divisor $m$ of $n$, the torsion $\varphi [\mathfrak{p}]$ is a free module over the subring $R_0=\kappa [{\pi }^m]$ generated by a power of the Frobenius $\pi ={\tau }^r$, $\kappa ={\mathbb{F}}_q[T]/\mathfrak{p}$, and the codes are the left ideals of the centralizer $R\cong {\mathrm{Mat}}_m(R_0)$ of $R_0$. We prove a lattice anti-isomorphism between codes and $R_0$-submodules of $\varphi [\mathfrak{p}]$, and a rank-BCH bound whose designed distance is the true distance.