Skew cyclic codes through Drinfeld modules

Abstract

We construct skew cyclic codes in the rank metric from the torsion of the supersingular Drinfeld module $\phi_T=t+\tau^n$ 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 $\mathfrak{p}\neq\mathfrak{p}_0$ of degree $d$ and every divisor $m$ of $n$, the torsion $\phi[\mathfrak{p}]$ 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 $\phi[\mathfrak{p}]$, and a rank-BCH bound whose designed distance is the true distance.