Paper 2026/1075

Sharp Minimum-Distance Lower Tails for RAA Codes

Abstract

Repeat--accumulate--accumulate (RAA) codes are sparse random linear codes with linear-time encoders. Although RAA ensembles can have linear minimum distance with high probability, this does not determine how often a randomly sampled code falls below a prescribed relative-distance threshold. We determine this lower-tail probability and identify the rare mechanisms that dominate it. Let \(n\) be the message length, \(r\ge4\) a fixed repetition factor, and \(N=rn\) the block length. For the randomly scaled ensemble \[ G=R\Pi_1V_1A\Pi_2V_2A, \] where \(R\) is repetition, \(A\) is the prefix-sum accumulator, \(\Pi_1,\Pi_2\) are independent uniform interleavers, and \(V_1,V_2\) have independent uniform nonzero diagonal entries in \(\mathbb F_q\), we prove that, for every fixed \(0<\delta\le1/2\) and \(\gamma>0\), \[ q-1\ge\gamma N \quad\Longrightarrow\quad \mathbb P[d_{\min}(G)\le\delta N] =\Theta_{r,\delta,\gamma}(N^{1-r}). \] For the unscaled ensemble \[ G_0=R\Pi_1A\Pi_2A, \] if \(\operatorname{char}(\mathbb F_q)>r\) and \(0<\delta\le1/10\), then \[ \mathbb P[d_{\min}(G_0)\le\delta N] =\Theta_{r,\delta}(N^{2-r}). \] Thus random scaling improves the failure probability by one power of \(N\). We also prove \[ \mathbb P[d_{\min}(G)\le\delta N] =\Omega_{r,\delta}\!\left( N^{1-\lceil r/2\rceil}(q-1)^{-\lfloor r/2\rfloor} \right), \] showing that linear field growth is necessary, in asymptotic order, for the constant-factor \(N^{1-r}\) law.

Note: Major revision and new title. The scaled RAA result is strengthened from a polylog-tight bound for r >= 9 and q - 1 >= (eN)^2 to the sharp Theta_{r,delta,gamma}(N^{1-r}) law for every r >= 4 under the order-optimal linear condition q - 1 >= gamma N. The revision also proves the sharp Theta_{r,delta}(N^{2-r}) tail for the unscaled ensemble when char(F_q) > r and delta <= 1/10, using a new cancellation-trace and shared-rank proof framework.