On Reed-Muller subcodes, Grassmannian partitions and sum-free functions

2026-05-21

AI Summary

This work investigates the existence of $k$-sum-free functions and their intrinsic connections to Reed–Muller codes and colorings of Grassmann graphs. By establishing an equivalence between $k$-sum-free functions and high-dimensional subcodes of the Reed–Muller code $\mathrm{RM}(n-k,n)$, the study introduces a new family of subcodes that avoid all minimum-weight codewords of the original code. Leveraging tools from algebraic coding theory, affine subspace analysis over finite fields, and the structural properties of Grassmann graphs, the authors derive the first nontrivial lower bound on the parameter $m$, construct subcodes whose minimum distance reaches 1.5 times that of the parent code, and consequently improve the known upper bound on the chromatic number of Grassmann graphs, thereby enabling an optimized partitioning of constant-dimension subspace codes on Grassmann manifolds.

Abstract

A function $F:\mathbb{F}_{2}^{n}\to \mathbb{F}_{2}^{m}$ is called $k$th-order sum-free if the sum of its values over any $k$-dimensional affine subspace of $\mathbb{F}_2^n$ is non-zero. Carlet recently introduced this notion and constructed such functions for every $2\le k\le n$. We prove that, for $2\le k\le n-2$ and $m \leq n$, the existence of a (non-degenerate) $\mathbb{F}_{2}^{m}$-valued $k$th-order sum-free function on $\mathbb{F}_{2}^{n}$ is equivalent to the existence of a codimension $m$ linear subcode of the Reed-Muller code $\mathrm{RM}(n-k,n)$ with minimum distance $3\cdot 2^{k-1}$. In particular, this yields a new family of Reed-Muller subcodes that avoid all minimum weight codewords of $\mathrm{RM}(n-k,n)$, and thus have minimum distance $3/2$ times that of $\mathrm{RM}(n-k,n)$. We also derive new necessary conditions for the existence of $k$th-order sum-free functions and present the first nontrivial lower bound on $m$. Finally, we observe that $k$th-order sum-free functions lead to a partition of the Grassmannian of all $k$-dimensional (linear) subspaces of $\mathbb{F}_2^n$ into constant-dimension subspace codes. Under the assumption that functions exist that are $k$th-order sum-free for multiple values of $k$, we obtain an improved partitioning result and a stronger upper bound on the chromatic number of the Grassmann graphs.

Problem

• sum-free functions
• Reed-Muller codes
• Grassmannian partitions
• constant-dimension subspace codes
• Grassmann graphs

Innovation

• sum-free functions
• Reed-Muller subcodes
• minimum distance
• Grassmannian partitions
• constant-dimension subspace codes