AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

TL;DR

This paper constructs algebraic geometric codes and quantum codes from cyclic covers of Suzuki and Ree curves, providing explicit models and analyzing their algebraic properties.

Contribution

It introduces new AG codes and quantum codes from specific cyclic covers of Suzuki and Ree curves, with detailed analysis of their algebraic structures.

Findings

Codes derived from these curves have symmetric Weierstrass semigroups.

Explicit plane models for the curves are provided.

The constructed codes have potential applications in quantum error correction.

Abstract

We investigate several types of linear codes constructed from two families $\tilde{\mathcal S}_q$ and $\tilde{\mathcal R}_q$ of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup $H(P)$ at an $\mathbb{F}_{q}$-rational point $P$ is shown to be symmetric.