Abstract
In this paper minimum distance properties of multiple-serial turbo codes, obtained by coupling an outer code with a cascade of m rate-1 recursive convolutional encoders through uniform random interleavers, are studied. The parameters that make the ensemble asymptotically good are identified. In particular, it is shown that, if m = 2 and the free distance of the outer encoder d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sup> ≥ 3, or if m ≥ 3 and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sup> ≥ 2, then the minimum distance scales linearly in the interleaver length with high probability. Through the analysis of the asymptotic spectral functions, a lower bound for the asymptotic growth rate coefficient is provided. Finally, under a weak algebraic condition on the outer encoder, it is proved that the sequence of normalized minimum distances of these concatenated coding schemes converges to the Gilbert-Varshamov (GV) distance when m goes to infinity.
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