Suzuki-invariant codes from the Suzuki curve

Abstract

In this paper we consider the Suzuki curve $y^q + y = x^{q_0}(x^q + x)$ over the field with $q = 2^{2m+1}$ elements. The automorphism group of this curve is known to be the Suzuki group $Sz(q)$ with $q^2(q-1)(q^2+1)$ elements. We construct AG codes over $\mathbb{F}_{q^4}$ from a $Sz(q)$-invariant divisor $D$, giving an explicit basis for the Riemann-Roch space $L(\ell D)$ for $0 < \ell \leq q^2-1$. These codes then have the full Suzuki group $Sz(q)$ as their automorphism group. These families of codes have very good parameters and are explicitly constructed with information rate close to one. The dual codes of these families are of the same kind if $2g-1 \leq \ell \leq q^2-1$.