New tailbiting encoders

In B tailbiting trellis encoder, the starting state of the encoder is set to the state at which the machine will be at the end of the encoded frame. By this means the probability of decoding error at the end of the frame can be kept low without the addition of termination bits and the consequent rate loss. We report on an extensive search for short and moderatelength convoluticinal encoders for tailbiting trellis representations of bllock codes at rates 1/4, 1/3, 1/2, and 2/3. The short tailbiting representations found are typically as good as the best known block codes

Approximate Linear Time ML Decoding on Tail-Biting Trellises in Two Rounds

A linear time approximate maximum likelihood decoding algorithm on tail-biting trellises is prsented, that requires exactly two rounds on the trellis. This is an adaptation of an algorithm proposed earlier with the advantage that it reduces the time complexity from O(mlogm) to O(m) where m is the number of nodes in the tail-biting trellis. A necessary condition for the output of the algorithm to differ from the output of the ideal ML decoder is reduced and simulation results on an AWGN channel using tail-biting rrellises for two rate 1/2 convoluational codes with memory 4 and 6 respectively are reporte

Codes on Graphs and More

Modern communication systems strive to achieve reliable and efficient information transmission and storage with affordable complexity. Hence, efficient low-complexity channel codes providing low probabilities for erroneous receptions are needed. Interpreting codes as graphs and graphs as codes opens new perspectives for constructing such channel codes. Low-density parity-check (LDPC) codes are one of the most recent examples of codes defined on graphs, providing a better bit error probability than other block codes, given the same decoding complexity. After an introduction to coding theory, different graphical representations for channel codes are reviewed. Based on ideas from graph theory, new algorithms are introduced to iteratively search for LDPC block codes with large girth and to determine their minimum distance. In particular, new LDPC block codes of different rates and with girth up to 24 are presented. Woven convolutional codes are introduced as a generalization of graph-based codes and an asymptotic bound on their free distance, namely, the Costello lower bound, is proven. Moreover, promising examples of woven convolutional codes are given, including a rate 5/20 code with overall constraint length 67 and free distance 120. The remaining part of this dissertation focuses on basic properties of convolutional codes. First, a recurrent equation to determine a closed form expression of the exact decoding bit error probability for convolutional codes is presented. The obtained closed form expression is evaluated for various realizations of encoders, including rate 1/2 and 2/3 encoders, of as many as 16 states. Moreover, MacWilliams-type identities are revisited and a recursion for sequences of spectra of truncated as well as tailbitten convolutional codes and their duals is derived. Finally, the dissertation is concluded with exhaustive searches for convolutional codes of various rates with either optimum free distance or optimum distance profile, extending previously published results

Threshold Saturation for Spatially Coupled Turbo-like Codes over the Binary Erasure Channel

In this paper we prove threshold saturation for spatially coupled turbo codes (SC-TCs) and braided convolutional codes (BCCs) over the binary erasure channel. We introduce a compact graph representation for the ensembles of SC-TC and BCC codes which simplifies their description and the analysis of the message passing decoding. We demonstrate that by few assumptions in the ensembles of these codes, it is possible to rewrite their vector recursions in a form which places these ensembles under the category of scalar admissible systems. This allows us to define potential functions and prove threshold saturation using the proof technique introduced by Yedla et al..Comment: 5 pages, 3figure

Spatially Coupled Turbo-Like Codes

The focus of this thesis is on proposing and analyzing a powerful class of codes on graphs---with trellis constraints---that can simultaneously approach capacity and achieve very low error floor. In particular, we propose the concept of spatial coupling for turbo-like code (SC-TC) ensembles and investigate the impact of coupling on the performance of these codes. The main elements of this study can be summarized by the following four major topics. First, we considered the spatial coupling of parallel concatenated codes (PCCs), serially concatenated codes (SCCs), and hybrid concatenated codes (HCCs).We also proposed two extensions of braided convolutional codes (BCCs) to higher coupling memories. Second, we investigated the impact of coupling on the asymptotic behavior of the proposed ensembles in term of the decoding thresholds. For that, we derived the exact density evolution (DE) equations of the proposed SC-TC ensembles over the binary erasure channel. Using the DE equations, we found the thresholds of the coupled and uncoupled ensembles under belief propagation (BP) decoding for a wide range of rates. We also computed the maximum a-posteriori (MAP) thresholds of the underlying uncoupled ensembles. Our numerical results confirm that TCs have excellent MAP thresholds, and for a large enough coupling memory, the BP threshold of an SC-TC ensemble improves to the MAP threshold of the underlying TC ensemble. This phenomenon is called threshold saturation and we proved its occurrence for SC-TCs by use of a proof technique based on the potential function of the ensembles.Third, we investigated and discussed the performance of SC-TCs in the finite length regime. We proved that under certain conditions the minimum distance of an SC-TCs is either larger or equal to that of its underlying uncoupled ensemble. Based on this fact, we performed a weight enumerator (WE) analysis for the underlying uncoupled ensembles to investigate the error floor performance of the SC-TC ensembles. We computed bounds on the error rate performance and minimum distance of the TC ensembles. These bounds indicate very low error floor for SCC, HCC, and BCC ensembles, and show that for HCC, and BCC ensembles, the minimum distance grows linearly with the input block length.The results from the DE and WE analysis demonstrate that the performance of TCs benefits from spatial coupling in both waterfall and error floor regions. While uncoupled TC ensembles with close-to-capacity performance exhibit a high error floor, our results show that SC-TCs can simultaneously approach capacity and achieve very low error floor.Fourth, we proposed a unified ensemble of TCs that includes all the considered TC classes. We showed that for each of the original classes of TCs, it is possible to find an equivalent ensemble by proper selection of the design parameters in the unified ensemble. This unified ensemble not only helps us to understand the connections and trade-offs between the TC ensembles but also can be considered as a bridge between TCs and generalized low-density parity check codes