Subfield subcodes of projective Reed-Muller codes

TL;DR

This work studies subfield subcodes of projective Reed-Muller codes over projective spaces, deriving explicit bases for the subfield subcodes on $\mathbb{P}^2$ and their duals and a dimension formula. It develops a toolkit based on trace maps, homogenization, cyclotomic sets, and universal Gröbner bases for $I(\mathbb{P}^m)$ to extend results to $\mathbb{P}^m$ and to provide practical bases in the plane case, with a concrete basis for both primal and dual codes and bounds on minimum distance. The paper introduces detailed constructions (via $\mathcal{T}_a$, $\mathcal{T}^h_a$, and $\,D_i$ sets) that enable basis extraction and dimension computation, and demonstrates through extensive examples that the resulting long codes over small fields can have competitive or best-known parameters, including many exceeding the Gilbert-Varshamov bound. Overall, it provides a comprehensive framework for subfield subcodes of projective Reed-Muller codes, with explicit results for $\mathbb{P}^2$ and a scalable approach to $\mathbb{P}^m$.

Abstract

Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we generalize the necessary tools to deal with this case as well: we obtain a universal Gröbner basis for the vanishing ideal of the set of standard representatives of the projective space and we show how to reduce any monomial with respect to this Gröbner basis. With respect to the parameters of these codes, by considering subfield subcodes of projective Reed-Muller codes we obtain long linear codes with good parameters over a small finite field.

Theorems & Definitions (73)

• Example 2.1
• Example 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Lemma 2.6
• proof
• Theorem 2.7
• Theorem 3.1
• Theorem 3.2