Coxeter codes: Extending the Reed-Muller family

TL;DR

This work introduces Coxeter codes, a broad binary linear family obtained by replacing the RM domain with finite Coxeter groups $W$, preserving key RM-like properties such as nestedness, duality $\C_W(r)^ op = \C_W(m-r-1)$, and a multiplication rule $\C_W(r_1)\odot\C_W(r_2) \subseteq \C_W(r_1+r_2)$. Code dimensions are governed by $\dim \C_W(r) = \sum_{i=0}^r \genfrac{\langle}{\rangle}{0pt}{}{W}{i}$, and the asymptotic rate for irreducible families follows a Gaussian limit with mean $\tfrac{m}{2}$ and variance $\tfrac{m}{12}$, yielding a sharp phase transition around $r\approx m/2$. The paper provides explicit parameter formulas for infinite families, notably types $A_m$ and $I_2(n)^{\mu}$, and proposes a distance conjecture $\text{dist}(\C_W(r))=\min_{|J|=m-r}|\langle J\rangle|$ together with a universal lower bound $\ge 2^{m-r}$, with proven cases for large $r$. It extends to quantum codes via CSS constructions, defining quantum Coxeter codes $\QC_W(q,r)$ with $n=|W|$, $k=\sum_{i=q+1}^r\langle W\rangle_i$, and $d=2^{\min(q+1,m-r)}$, and analyzes the dihedral family $I_2(n)^{\mu}$ in detail, including explicit examples such as $[[216,88,8]]$ and $[[1296,454,16]]$, while outlining decoding and potential generalizations. Overall, the work establishes a rich algebraic-combinatorial framework linking RM codes, Coxeter group theory, and quantum error-correcting codes, with concrete parameter results and several promising directions for future improvements and applications in quantum information and coding theory.

Abstract

Binary Reed-Muller (RM) codes are defined via evaluations of Boolean-valued functions on $\mathbb{Z}_2^m$. We introduce a class of binary linear codes that generalizes the RM family by replacing the domain $\mathbb{Z}_2^m$ with an arbitrary finite Coxeter group. Like RM codes, this class is closed under duality, forms a nested code sequence, satisfies a multiplication property, and has asymptotic rate determined by a Gaussian distribution. Coxeter codes also give rise to a family of quantum codes for which transversal diagonal $Z$ rotations can perform non-trivial logic.

Figures (5)

• Figure 1: A useful way to visualize a Coxeter system $(W,S)$ is a Cayley graph, $(V,E)$, where $V=W$ and $(w,w')\in E$ iff there is a generator $s\in S$ such that $w'=ws$. The figure shows the Cayley graph of the 4-letter symmetric group, $A_3$, with generators given by adjacent transpositions. The shaded vertex represents the identity element. The polytope obtained by embedding this graph in $\mathbb{R}^3$ is called a permutohedron.
• Figure 2: The code $\C_{{A_3}}({1})$ is generated by indicators of faces of the Cayley graph of $A_3$. The bit assignment shown in the figure represents the codeword in $\C_{{A_3}}({1})$ generated by the indicators of the colored hexagonal and square faces. The same codeword is equivalently generated by the indicators of the three solid red edges, indicative of the containment $\C_{{A_3}}({1})\subseteq \C_{{A_3}}({2})$.
• Figure 3: This figure shows extensions (blue) and reverse extensions (red) of the elements $w_1$ and $w_2$ in $A_3$. The identity element is shown as the shaded vertex of the graph.
• Figure 4: Cayley graphs for Cartesian products of two dihedral groups: (a) $I_2(3)$--- note that $I_2(3)\cong A_2$, the symmetric group on 3 letters--- and (b) $I_2(4)$. The Coxeter system $I_2(4)\cong B_2$, the hyperoctahedral group, or signed symmetric group, on 3 letters.
• Figure 5: Cayley graph for the symmetric groups $A_4$

Theorems & Definitions (65)

• Theorem 1.1: barg2024geometric, Fact II.3
• Definition 1.2
• Definition 1.3
• Definition 1.4: Coxeter codes
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1