Universal Maximum Likelihood (List) Decoding via Fast Vector-Matrix Multiplication
Hoang Ly, Emina Soljanin
TL;DR
This work tackles the fundamental challenge of maximum-likelihood decoding by introducing a universal, code-agnostic framework that reexpresses codeword likelihoods as inner products and represents the entire codebook as a single binary matrix. ML decoding then reduces to a single vector–matrix multiplication followed by a max operation, enabling an overall worst-case complexity of $\mathcal{O}(q^{k})$ and a space cost of $\mathcal{O}(q^{k+1} n)$ when accelerated with the Mailman algorithm. The approach applies to linear and nonlinear codes, hard and soft decision channels, erasures, and even ISI channels, and extends to list decoding with only logarithmic growth in complexity with the list size. This time-space tradeoff enables low-latency, high-reliability scenarios (e.g., URLLC) by decoupling decoding latency from block length and is supported by theoretical arguments about near-optimality under existing complexity conjectures. Overall, the paper provides a principled, broadly applicable method that significantly reduces ML decoding time at the cost of increased memory, with practical relevance for next-generation communication systems.
Abstract
Maximum-likelihood (ML) decoding for arbitrary block codes remains fundamentally hard, with worst-case time complexity-measured by the total number of multiplications-being no better than straightforward exhaustive search, which requires $q^{k} n$ operations for an $[n,k]_q$ code. This paper introduces a simple, code-agnostic framework that reduces the worst-case complexity by a factor of $n$, down to $q^{k}$ operations, a highly desirable reduction in practice. The result holds for both linear and nonlinear block codes over general memoryless channels and under both hard-decision and soft-decision decoding. It naturally extends to intersymbol-interference (ISI) channels and ML list decoding with only a negligible increase in complexity. Our core insight is that, upon receipt of each sequence at the receiver, the conditional probability of that sequence for each codeword in the codebook (i.e., the \emph{likelihood}) can be expressed as the inner product of two carefully constructed vectors -- the first depending on the received sequence, and the second on that codeword itself. As a result, evaluating the likelihoods for all codewords in the codebook reduces to a single vector-matrix multiplication, and ML decoding (MLD) becomes the simple task of picking the maximum entry in the resulting vector. The only non-trivial cost lies in the vector-matrix product. However, our matrix construction allows the use of the Mailman algorithm to reduce this cost. This time reduction is achieved at the cost of high space complexity, requiring $\mathcal{O}(q^{k+1} n)$ space to store the pre-computed codebook matrix.