Tensor Reed-Muller Codes: Achieving Capacity with Quasilinear Decoding Time

2026-01-22

AI Summary

This work addresses the construction of efficient error-correcting codes that approach Shannon capacity while supporting quasi-linear time decoding. By leveraging a tensor product framework, the authors introduce two new families of tensorized Reed–Muller codes based on multivariate low-degree polynomial evaluations, which achieve capacity at any fixed rate. The key innovation lies in the first demonstration of reliable quasi-linear time decoding for such codes, circumventing the conventional requirement that component subcodes be polynomial-time decodable. Specifically, for $t = 3$, decoding runs in $O(n \log \log n)$ time with failure probability $n^{-\omega(\log n)}$; for $t \geq 4$, decoding complexity is $O(n \log n)$ and the error probability decays exponentially as $2^{-n^{1/2 - 1/(2(t-2)) - o(1)}}$.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Define the codewords of the Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;r_2,m_2;\dots;r_t,m_t)$ to be the evaluation vectors of all multivariate polynomials in the variables $\left\{x_{ij}\right\}_{i=1,\dots,t}^{j=1,\dots m_i}$ with degree at most $r_i$ in the variables $x_{i1},x_{i2},\dots,x_{im_i}$. The generator matrix of $\mathsf{TRM}(r_1,m_1;\dots;r_t,m_t)$ is thus the tensor product of the generator matrices of the Reed-Muller codes $\mathsf{RM}(r_1,m_1),\dots, \mathsf{RM}(r_t,m_t)$. We show that for any constant rate $R$ below capacity, one can construct a Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;\dotsc;r_t,m_t)$ of rate $R$ that is decodable in quasilinear time. For any blocklength $n$, we provide two constructions of such codes: 1) Our first construction (with $t=3$) has error probability $n^{-\omega(\log n)}$ and decoding time $O(n\log\log n)$. 2) Our second construction, for any $t\geq 4$, has error probability $2^{-n^{\frac{1}{2}-\frac{1}{2(t-2)}-o(1)}}$ and decoding time $O(n\log n)$. One of our main tools is a polynomial-time algorithm for decoding an arbitrary tensor code $C=C_1\otimes\dotsc\otimes C_t$ from $\frac{d_{\min}(C)}{2\max\{d_{\min}(C_1),\dotsc,d_{\min}(C_t) \}}-1$ adversarial errors. Crucially, this algorithm does not require the codes $C_1,\dotsc,C_t$ to themselves be decodable in polynomial time.

Problem

• Tensor Reed-Muller codes
• capacity
• quasilinear decoding
• error correction
• tensor codes

Innovation

• Tensor Reed-Muller codes
• quasilinear decoding
• tensor product codes
• adversarial error correction
• capacity-achieving codes