Combinatorial analysis of the minimum distance of turbo codes

Abstract

In this paper, new upper bounds on the maximum attainable minimum Hamming distance of Turbo codes with arbitrary-including the best-interleavers are established using a combinatorial approach. These upper bounds depend on the interleaver length, on the code rate and on the scramblers employed in the encoder. Examples of the new bounds for particular Turbo codes are given and discussed. The new bounds are tighter than all existing ones and prove that the minimum Hamming distance of Turbo codes cannot asymptotically grow at a rate more than the third root of the codeword length.

A Codeword Weight Lower Bound for a Class of Tail-Biting Turbo Codes

“…where u' is a weight-I input sequence with the nonzero element located at coordinate i. scrbfL (uij) is lower-bounded by a i +H13 or a (L Tij) + 13 [4], where T, is the period of the convolutional code A. Partition rule Systematic recursive convolutional code used in a CTC is equivalent to an IIR scrambler whose period has a great impact on the distance property of the associated CTC. A finite weight codeword can be generated by a weight-k input sequence, k > 2.…”

“…Most CTC interleaver designs [6], [3] take this class of error events into account, trying to maximize the minimum weight of these error events. Breiling [4] suggested a novel partition strategy to derive upper bounds for the weight-2 error events. Although the upper bound is not as tight as more general upper bounds [4], [5] which consider other error events as well, weight-2 error event remains an important design concern.…”

“…Weiss et al [2] proposed a product code (without the check-on-check part) whose column and row vectors are tail-biting encoded convolutional codewords and derived some distance properties. The codeword associated with a weight-2 input sequence was called a weight-2 error event by Breiling [4] for an obvious reason. Most CTC interleaver designs [6], [3] take this class of error events into account, trying to maximize the minimum weight of these error events.…”

“…By analyzing the effects of selected particular system parameters on this general bound we obtain some useful design guidelines. We use a simplified partition rule presented in [4] and apply a regular permutation function to derive the bound. We also examine some special cases and evaluate distance lower bounds of the weight-2 error events for different block lengths.…”

Distance measurement method for double binary turbo codes and a new interleaver design for DVB-RCS

“…Even for very large random interleavers, d min can still be produced by an input sequence of any weight. The necessity of considering input-weights larger than two has been confirmed analytically in [6]. Thus d min (2) gives a loose upper bound on d min .…”