Subfield-Subcodes of Generalized Toric codes

TL;DR

This work develops a concrete framework for subfield-subcodes of Generalized Toric codes by leveraging trace maps and cyclotomic cosets to characterize polynomials in $R$ that evaluate to $\mathbb{F}_p$. It provides explicit bases and a dimension formula $\dim D_U=\sum_{I_{\underline{b}}\subseteq U} n_{\underline{b}}$, and shows that the dual $D_U^{\perp}$ corresponds to the subfield-subcode $D_{\hat U}$, enabling practical decoding strategies. The approach yields constructive constructions of $D_U$ with new, often best-known, parameters for fixed length and dimension, validated by extensive Magma computations. The results offer a scalable pathway to high-performance subfield-subcodes of GT codes with explicit generators and decoding routes, broadening the toolbox for algebraic-geometric coding in applications.

Abstract

We study subfield-subcodes of Generalized Toric (GT) codes over $\mathbb{F}_{p^s}$. These are the multidimensional analogues of BCH codes, which may be seen as subfield-subcodes of generalized Reed-Solomon codes. We identify polynomial generators for subfield-subcodes of GT codes which allows us to determine the dimensions and obtain bounds for the minimum distance. We give several examples of binary and ternary subfield-subcodes of GT codes that are the best known codes of a given dimension and length.

Theorems & Definitions (20)

• Proposition 1.1
• Proposition 1.3
• Definition 2.1
• Theorem 2.2
• Proposition 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4