Suzuki-invariant codes from the Suzuki curve

TL;DR

This paper constructs algebraic geometry codes from the Suzuki curve that are invariant under the Suzuki group, achieving high information rates and explicit bases, with automorphism groups matching the Suzuki group.

Contribution

It introduces a method to construct Suzuki-invariant algebraic geometry codes with explicit bases and automorphism groups equal to the Suzuki group.

Findings

Codes have very good parameters.

Codes have automorphism group equal to Suzuki group.

Explicit basis for Riemann-Roch spaces provided.

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 $2g-1 \leq \ell \leq q^2-1$.