The performance of serial turbo codes does not concentrate

Abstract—Minimum distances and maximum likelihood error probabilities of serial turbo codes with uniform interleaver are an-alyzed. It is shown that, for a fraction of interleavers approaching one as the block-length grows large, the minimum distance of se-rial turbo codes grows as a positive power of their block-length, while their error probability decreases exponentially fast in some positive power of their block-length, on sufficiently good memory-less channels. Such a typical code behavior contrasts the perfor-mance of the average serial turbo code, whose error probability is dominated by an asymptotically negligible fraction of poorly per-forming interleavers, and decays only as a negative power of the block-length. The analysis proposed in this paper relies on precise bounds of the minimum distance of the typical serial turbo code, whose scaling law is shown to depend both on the free distance of its outer constituent encoder, which determines the exponent of its sub-linear growth in the block-length, and on the effective free distance of its inner constituent encoder. The latter is defined as the smallest weight of codewords obtained when the input word of the inner encoder has weight two, and appears as a linear scaling factor for the minimum distance of the typical serial turbo code. Hence, despite the lack of concentration of the maximum likeli-hood error probability around its expected value, the main de-sign parameters suggested by the average-code analysis turn out to characterize also the performance of the typical serial turbo code. By showing for the first time that the typical serial turbo code’s minimum distance scales linearly in the effective free distance of the inner constituent encoder, the presented results generalize, and improve upon, the probabilistic bounds of Kahale and Urbanke, as well as the deterministic upper bound of Bazzi, Mahdian, and Spielman, where only the dependence on the outer encoder’s free distance was proved. Index Terms—Error probability, minimum distance, serially concatenated codes, turbo codes, typical code analysis. I