How to Find Good Finite-Length Codes: From Art Towards Science

We explain how to optimize finite-length LDPC codes for transmission over the binary erasure channel. Our approach relies on an analytic approximation of the erasure probability. This is in turn based on a finite-length scaling result to model large scale erasures and a union bound involving minimal stopping sets to take into account small error events. We show that the performances of optimized ensembles as observed in simulations are well described by our approximation. Although we only address the case of transmission over the binary erasure channel, our method should be applicable to a more general setting.Comment: 13 pages, 13 eps figures, enhanced version of an invited paperat the 4th International Symposium on Turbo Codes and Related Topics, Munich, Germany, 200

Spatially Coupled Turbo Codes

In this paper, we introduce the concept of spatially coupled turbo codes (SC-TCs), as the turbo codes counterpart of spatially coupled low-density parity-check codes. We describe spatial coupling for both Berrou et al. and Benedetto et al. parallel and serially concatenated codes. For the binary erasure channel, we derive the exact density evolution (DE) equations of SC-TCs by using the method proposed by Kurkoski et al. to compute the decoding erasure probability of convolutional encoders. Using DE, we then analyze the asymptotic behavior of SC-TCs. We observe that the belief propagation (BP) threshold of SC-TCs improves with respect to that of the uncoupled ensemble and approaches its maximum a posteriori threshold. This phenomenon is especially significant for serially concatenated codes, whose uncoupled ensemble suffers from a poor BP threshold.Comment: in Proc. 8th International Symposium on Turbo Codes & Iterative Information Processing 2014, Bremen, Germany, August 2014. To appear. (The PCC ensemble is changed with respect to the one in the previous version of the paper. However, it gives identical thresholds

A Unified Ensemble of Concatenated Convolutional Codes

We introduce a unified ensemble for turbo-like codes (TCs) that contains the four main classes of TCs: parallel concatenated codes, serially concatenated codes, hybrid concatenated codes, and braided convolutional codes. We show 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. We also derive the density evolution (DE) equations for this ensemble over the binary erasure channel. The thresholds obtained from the DE indicate that the TC ensembles from the unified ensemble have similar asymptotic behavior to the original TC ensembles

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

Density Evolution for Deterministic Generalized Product Codes with Higher-Order Modulation

Generalized product codes (GPCs) are extensions of product codes (PCs) where coded bits are protected by two component codes but not necessarily arranged in a rectangular array. It has recently been shown that there exists a large class of deterministic GPCs (including, e.g., irregular PCs, half-product codes, staircase codes, and certain braided codes) for which the asymptotic performance under iterative bounded-distance decoding over the binary erasure channel (BEC) can be rigorously characterized in terms of a density evolution analysis. In this paper, the analysis is extended to the case where transmission takes place over parallel BECs with different erasure probabilities. We use this model to predict the code performance in a coded modulation setup with higher-order signal constellations. We also discuss the design of the bit mapper that determines the allocation of the coded bits to the modulation bits of the signal constellation.Comment: invited and accepted paper for the special session "Recent Advances in Coding for Higher Order Modulation" at the International Symposium on Turbo Codes & Iterative Information Processing, Brest, France, 201

Design Of Fountain Codes With Error Control

This thesis is focused on providing unequal error protection (uep) to two disjoint sources which are communicating to a comdestination via a comrelay by using distributed lt codes over a binary erasure channel (bec), and designing fountain codes with error control property by integrating lt codes with turbo codes over a binary input additive white gaussian noise (bi-awgn) channel. A simple yet efficient technique for decomposing the rsd into two entirely different degree distributions is developed and presented in this thesis. These two distributions are used to encode data symbols at the sources and the encoded symbols from the sources are selectively xored at the relay based on a suitable relay operation before the combined codeword is transmitted to the destination. By doing so, it is shown that the uep can be provided to these sources. The performance of lt codes over the awgn channel is well studied and presented in this thesis which indicates that these codes have weak error correction ability over the channel. But, errors introduced into individual symbols during the transmission of information over noisy channels need correction by some error correcting codes. Since it is found that lt codes alone are weak at correcting those errors, lt codes are integrated with turbo codes which are good error correcting codes. Therefore, the source data (symbols) are at first turbo encoded and then lt encoded and transmitted over the awgn channel. When the corrupted encoded symbols are received at receiver, lt decoding is conducted folloby turbo decoding. The overall performance of the integrated system is studied and presented in this thesis, which suggests that the errors left after lt decoding can be corrected to some extent by turbo decoder