Sharp Minimum-Distance Lower Tails for RAA Codes

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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 th