Generalized Hermitian Codes over GF(2^r)

TL;DR

This paper generalizes Hermitian codes over GF(2^r), computes their semigroups, constructs new codes, improves distance estimates, and presents a record-breaking code over GF(8).

Contribution

It introduces a new class of generalized Hermitian codes, analyzes their algebraic properties, and demonstrates improved code parameters with explicit formulas.

Findings

Computed Weierstrass semigroup with three generators for q=2, r≥3.

Constructed generalized Hermitian codes with improved Feng-Rao distances.

Presented a new record [32,16,≥12] code over GF(8).

Abstract

In this paper we studied generalization of Hermitian function field proposed by A.Garcia and H.Stichtenoth. We calculated a Weierstrass semigroup of the point at infinity for the case q=2, r>=3. It turned out that unlike Hermitian case, we have already three generators for the semigroup. We then applied this result to codes, constructed on generalized Hermitian function fields. Further, we applied results of C.Kirfel and R.Pellikaan to estimating a Feng-Rao designed distance for GH-codes, which improved on Goppa designed distance. Next, we studied the question of codes dual to GH-codes. We identified that the duals are also GH-codes and gave an explicit formula. We concluded with some computational results. In particular, a new record-giving [32,16,>=12]-code over GF(8) was presented.