Minimum distance properties of multiple-serially concatenated codes

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 dof >= 3, or if m >= 3 and dof >= 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.