On the generalized Hamming weights of hyperbolic codes

TL;DR

This work investigates generalized Hamming weights of hyperbolic codes and their relation to Reed-Muller codes. It develops explicit containment criteria to identify the largest Reed-Muller code contained in a given hyperbolic code and the smallest Reed-Muller code that contains it, and proves that for hyperbolic codes the $r$-th generalized Hamming weight equals the $r$-th footprint: $\delta_r(\mathrm{Hyp}_q(d,m))=\mathrm{FB}_r(\mathrm{Hyp}_q(d,m))$, while also providing bounds in terms of the surrounding RM codes. The authors give concrete formulas to compute the minimal and maximal RM codes relative to a hyperbolic code, including special cases such as $m=2$, and discuss the sharpness of footprint-based bounds with numerical examples (e.g., $q=9$, $m=2$). These results deepen understanding of the dimension-distance tradeoffs for monomial evaluation codes and offer practical guidance for code construction with favorable GHW profiles.

Abstract

A hyperbolic code is an evaluation code that improves a Reed-Muller because the dimension increases while the minimum distance is not penalized. We give the necessary and sufficient conditions, based on the basic parameters of the Reed-Muller, to determine whether a Reed-Muller coincides with a hyperbolic code. Given a hyperbolic code, we find the largest Reed-Muller containing the hyperbolic code and the smallest Reed-Muller in the hyperbolic code. We then prove that similarly to Reed-Muller and Cartesian codes, the $r$-th generalized Hamming weight and the $r$-th footprint of the hyperbolic code coincide. Unlike Reed-Muller and Cartesian, determining the $r$-th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the $r$-th footprint of a hyperbolic code that, sometimes, are sharp.

Figures (7)

• Figure 1: (A) The lattice points under the red curve define $\mathrm{Hyp}_9(27,2)$. The lattice points under the blue curve define $\mathrm{RM}_9(7,2)$, the smallest Reed-Muller code that contains $\mathrm{Hyp}_9(27,2)$. (B) The lattice points under the red curve define $\mathrm{Hyp}_9(9,2)$. The lattice points under the blue curve define $\mathrm{RM}_9(12,2)$, the smallest Reed-Muller code that contains $\mathrm{Hyp}_9(9,2)$.
• Figure 2: (A) The lattice points under the red curve define $\mathrm{Hyp}_9(27,2)$. The lattice points under the black curve define $\mathrm{RM}_9(6,2)$, the largest Reed-Muller code in $\mathrm{Hyp}_9(27,2)$. (B) The lattice points under the red curve define $\mathrm{Hyp}_9(9,2)$. The lattice points under the black curve define $\mathrm{RM}_9(8,2)$, the largest Reed-Muller in $\mathrm{Hyp}_9(9,2)$.
• Figure 3: We observe that $\mathrm{RM}_9(6,2)\subseteq\mathrm{Hyp}_9(27,2)\subseteq \mathrm{RM}_9(7,2)$. The boxes represent the lattice points that help to compute the second GHWs. The number of lattice points inside: the red box is equal to $\delta_2(\mathrm{Hyp}_9(27,2))$, the black box is equal to $\delta_2(\mathrm{RM}_9(6,2))$, and the blue box is equal to $\delta_2(\mathrm{RM}_9(7,2))$.
• Figure 4: We observe that $\mathrm{RM}_9(8,2)\subseteq\mathrm{Hyp}_9(9,2)\subseteq \mathrm{RM}_9(12,2)$. The boxes represent the lattice points that help to compute the second GHWs. The number of lattice points inside: the red box is equal to $\delta_2(\mathrm{Hyp}_9(9,2))$, the black box is equal to $\delta_2(\mathrm{RM}_9(8,2))$, and the blue box is equal to $\delta_2(\mathrm{RM}_9(12,2))$.
• Figure 5: The number of lattice points inside of the blue box equals $\delta_1(\mathrm{Hyp}_9(27,2))$.

Theorems & Definitions (36)

• Proposition 2.1
• proof
• Theorem 2.2
• proof
• Example 2.3
• Corollary 2.4
• proof
• Proposition 3.1
• proof
• Example 3.2