Paper 2026/2007

Sharp Minimum-Distance Tails for BAA Codes

Abstract

Fast randomized linear codes used in pseudorandom correlation generators and code-based proof systems require quantitative guarantees on the probability of sampling a code with insufficient distance. For chosen-block Block--Accumulate--Accumulate (BAA) codes, prior work provides finite-length upper bounds, sparse failure mechanisms, and permutation obstructions that persist under every nonzero coordinate scaling. We determine the global minimum-distance lower tail across field sizes. For a fixed-length constituent of minimum distance $d\ge2$, sufficiently small fixed $\delta>0$, and $Q=q-1$, we prove, uniformly over finite fields and constituent coefficients, \[ \PP[d_{\min}\le\floor{\delta N}] =\Theta\!\left(N^{1-d} +N^{1-\ceil{d/2}}Q^{-\floor{d/2}}\right). \] A support-dependent weighted-gap estimate controls every active-block count and every cancellation layer. It identifies linear field growth as the exact order needed for the $N^{1-d}$ law. Certified extensions reach distance $0.30$ for $[16,8,9]_q$ MDS blocks and $0.60$ for $[16,4,13]_q$ MDS blocks over all $q\ge2^{20}$; binary instances reach their stated useful thresholds. For the two MDS families, when $Q/N\to\infty$, we further prove that the known marked one-block coefficient is the leading constant of the actual failure probability. Grouping full-block gap events and bounding their dependence gives near-exact two-sided finite certificates at the published large-field benchmarks. These show that the existing numerical guarantees are essentially optimal for the unchanged ensemble. \keywords{Random linear codes \and BAA codes \and Minimum distance \and Pseudorandom correlation generators \and Sampling failure}