Self-orthogonal and self-dual codes from maximal curves
Puyin Wang, Jinquan Luo
TL;DR
This work addresses the challenge of characterizing duals of algebraic-geometric codes by constructing explicit self-orthogonal and self-dual codes from maximal curves. Two main pathways are developed: (i) Euclidean self-orthogonal/self-dual codes from the curve y^q+y=x^m with n=mq^2−mq+q and (ii) codes from Hermitian curves, using both multiplicative and additive group structures to realize Euclidean and Hermitian self-orthogonality, with selective cases achieving self-duality and enabling quantum constructions via the Hermitian method. The resulting [n,k,d]_{q^2} codes have k+d close to n, delivering strong error-correcting performance relative to length, and yield quantum codes with large minimum distance through standard Hermitian construction. The methods leverage Riemann-Roch spaces and differential descriptions to tightly control dimensions and distances, offering a framework potentially extendable to other maximal or Kummer-type extensions. Overall, the paper advances practical and theoretically well-founded routes to long self-orthogonal/self-dual AG codes and their quantum counterparts.
Abstract
In the field of algebraic geometric codes (AG codes), the characterization of dual codes has long been a challenging problem which relies on differentials. In this paper, we provide some descriptions for certain differentials utilizing algebraic structure of finite fields and geometric properties of algebraic curves. Moreover, we construct self-orthogonal and self-dual codes with parameters $[n, k, d]_{q^2}$ satisfying $k + d$ is close to $n$. Additionally, quantum codes with large minimum distance are also constructed.