Paper 2026/2712

Concrete Minimum Distance Bounds of Accumulate Codes over any Fixed Finite Field

Abstract

We present a new concrete unified distance analysis of three accumulate-type error-correcting codes over any finite field, specifically the Repeat-Accumulate (RA), Repeat-Accumulate-Accumulate (RAA), and Expand-Accumulate (EA) codes. For message length $k$, codeword length $n$, finite field size $q>2$, and target relative minimum distance $\delta$, we give new upper bounds on the probability that each of the above codes do not have minimum distance at least $\delta$. Our bounds are concrete and explicit: they do not rely on $n$ or $q$ being sufficiently large. We present (multiple) closed-form upper bounds on the failure probability for each code, and complement these bounds with asymptotic analyses that (nearly) match prior-known asymptotic bounds. Our technical core is a new fine-grained union bound for accumulate codes, which may be of independent interest, that examines inputs to the accumulator from a purely distributional perspective, deriving properties that are necessary for an input vector to accumulate to a targeted output weight, and analyzing the probability the input distribution achieves these properties. This fine-grained union bound generalizes the input-output weight enumerator approach used by many prior works. Our concrete analysis shows that (1) the RA code has concretely small distance (a maximum of $\delta\leq 2.4979\times10^{-8}$ for $k=2^{40}$ and $q=2^{32}$, except with probability $2^{-60}$), and (2) the RAA and EA codes achieve somewhat large concrete distance for smaller field sizes $q \leq 2^{32}$. Concretely, for rate $k/n = 1/4$, $k\in\{2^{40},2^{50}\}$, and $q=2^{8}$, the RAA code has distance $\delta=1/8$ except with $2^{-60}$ probability, whereas the rate $1/2$ and $1/4$ EA code for the same $k$ and $q=2^{4}$, respectively, has distance $\delta\leq 0.0689756$ and $\delta \leq0.232955$ except with probability $2^{-60}$. For fields of size $q\geq 2^{32}$, under our concrete bounds, all codes experience a sharp drop-off in attainable minimum distance, as our analysis targets (small) fixed $q$.