On QC and GQC algebraic geometry codes

TL;DR

This paper introduces new constructions of quasi-cyclic and generalized quasi-cyclic codes derived from algebraic curves, expanding beyond elliptic curves to include hyperelliptic, norm-trace, and Hermitian curves, enabling more flexible code parameters.

Contribution

It presents a novel method for constructing QC and GQC codes from a broader class of algebraic curves, with explicit parameter formulas based on automorphism groups.

Findings

Codes with flexible co-index achieved

Explicit formulas for code parameters derived

Applicable to a wide range of algebraic curves

Abstract

We present new constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) codes from algebraic curves. Unlike previous approaches based on elliptic curves, our method applies to curves that are Kummer extensions of the rational function field, including hyperelliptic, norm-trace, and Hermitian curves. This allows QC codes with flexible co-index. Explicit parameter formulas are derived using known automorphism-group classifications.