On QC and GQC algebraic geometry codes

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.

💡 Research Summary

This paper introduces a novel framework for constructing quasi‑cyclic (QC) and generalized quasi‑cyclic (GQC) algebraic‑geometry (AG) codes by exploiting the rich automorphism groups of Kummer extensions of the rational function field. While earlier works on QC AG codes have largely been confined to elliptic curves—whose automorphism groups have order at most six, limiting the achievable co‑index—the authors broaden the scope to curves defined by equations of the form
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