Minimum distances and maximum likelihood error probabilities of serial turbo codes with uniform interleaver are analyzed. It is shown that, for a fraction of interleavers approaching one as the block-length grows large, the minimum distance of serial 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 memoryless channels. Such a typical code behavior contrasts the performance of the average serial turbo code, whose error probability is dominated by an asymptotically negligible fraction of poorly performing 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 likelihood error probability around its expected value, the main design 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.

A class of serial turbo codes admitting low-density parity-check (LDPC) representation is considered. Their parity matrix has a random and a structured part. Thanks to their turbo structure, these codes are linear-time encodable, while they can be decoded as LDPC codes. Previous works enlightened the role of the inner encoder in the error floor region and suggested the use of a non-binary iterative decoding algorithm. In this paper, a density-evolution analysis is developed giving insight into the performance of these codes in the waterfall region. The inner encoder is optimized in order to guarantee the best tradeoff between error floor and threshold.

In this paper we study serial turbo interconnections of Abelian group codes to be used on symmetric channels. Particular attention is devoted to AWGN channel with input restricted to m-PSK constellation with corresponding group structure $\mathbb{Z}_m$. We establish the exact asymptotic decay of the average symbol and word error probabilities when the interleaver length goes to infinity (interleaver gain). Moreover, we give a detailed characterization of the distance parameter characterizing the behavior for the signal-to-noise ratio going to infinity (effective free distance). Some of our results are new also in the binary context; in particular, the lower bound to the error probability decay.

We consider a family of codes which can be seen both as a special kind of serial turbo codes and as LDPC codes having a parity check matrix which is partly random and partly structured. These codes are linear-time encodable, thanks to the turbo structure, and can be decoded as LDPC codes. We provide an ensemble analysis for the waterfall region, on the line of classical results for serial turbo codes, and we find some design parameters.

In this paper we study the ML error probability of serially concatenated schemes averaged over different interleaver ensembles. We prove asymptotic results when the interleaver length goes to infinity: differently from the parallel case, the choice of the ensemble can change the decreasing speed of error probability

In this paper we present a theoretical analysis of serial concatenation schemes for transmission over AWGN channels employing a geometrically uniform constellation having Zopfmas a generating group. Our schemes possess a uniform error property both with respect to the word and to the symbol. This allows to prove exact convergence results on the probability of error. We then show that, as in the binary case, there is a natural concept of distance which, at the design level of the constituent encoders, should be maximized, in order to optimize performances