Tensor Reed-Muller Codes: Achieving Capacity with Quasilinear Decoding Time
Emmanuel Abbe, Colin Sandon, Oscar Sprumont
TL;DR
This work addresses decodability of Reed-Muller based codes at constant rate and proposes Tensor Reed-Muller codes $\mathsf{TRM}(r_1,m_1;\dots; r_t,m_t)$, whose codewords are tensor products of RM evaluation vectors. It develops a general polynomial-time decoder for arbitrary tensor codes that tolerates adversarial errors up to $\lceil d_{\min}(C)/(2\max_i d_{\min}(C_i))\rceil-1$, and leverages it alongside RM-specific decoders to obtain efficient, capacity-achieving decoding for TRM under both adversarial and random noise. The paper provides two concrete TRM constructions: (i) $t=3$ with $O(n\log\log n)$ decoding time and $n^{-\omega(\log n)}$ failure, and (ii) $t\ge 4$ with $O(n\log n)$ time and $2^{-n^{\frac{1}{2}-\frac{1}{2(t-2)}-o(1)}}$ failure, achieving rates arbitrarily close to capacity $R<1-h(p)$. These results yield practical, capacity-achieving decoding for a structured family of tensorized RM codes across noise regimes, advancing efficient coding for constant-rate channels.
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 $n^{-ω(\log n)}$ 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.