Faulty Successive Cancellation Decoding of Polar Codes for the Binary Erasure Channel

In this paper, faulty successive cancellation decoding of polar codes for the binary erasure channel is studied. To this end, a simple erasure-based fault model is introduced to represent errors in the decoder and it is shown that, under this model, polarization does not happen, meaning that fully reliable communication is not possible at any rate. Furthermore, a lower bound on the frame error rate of polar codes under faulty SC decoding is provided, which is then used, along with a well-known upper bound, in order to choose a blocklength that minimizes the erasure probability under faulty decoding. Finally, an unequal error protection scheme that can re-enable asymptotically erasure-free transmission at a small rate loss and by protecting only a constant fraction of the decoder is proposed. The same scheme is also shown to significantly improve the finite-length performance of the faulty successive cancellation decoder by protecting as little as 1.5% of the decoder.Comment: Accepted for publications in the IEEE Transactions on Communication

Broadcast Coded Slotted ALOHA: A Finite Frame Length Analysis

We propose an uncoordinated medium access control (MAC) protocol, called all-to-all broadcast coded slotted ALOHA (B-CSA) for reliable all-to-all broadcast with strict latency constraints. In B-CSA, each user acts as both transmitter and receiver in a half-duplex mode. The half-duplex mode gives rise to a double unequal error protection (DUEP) phenomenon: the more a user repeats its packet, the higher the probability that this packet is decoded by other users, but the lower the probability for this user to decode packets from others. We analyze the performance of B-CSA over the packet erasure channel for a finite frame length. In particular, we provide a general analysis of stopping sets for B-CSA and derive an analytical approximation of the performance in the error floor (EF) region, which captures the DUEP feature of B-CSA. Simulation results reveal that the proposed approximation predicts very well the performance of B-CSA in the EF region. Finally, we consider the application of B-CSA to vehicular communications and compare its performance with that of carrier sense multiple access (CSMA), the current MAC protocol in vehicular networks. The results show that B-CSA is able to support a much larger number of users than CSMA with the same reliability.Comment: arXiv admin note: text overlap with arXiv:1501.0338

A Rate-Distortion Exponent Approach to Multiple Decoding Attempts for Reed-Solomon Codes

Algorithms based on multiple decoding attempts of Reed-Solomon (RS) codes have recently attracted new attention. Choosing decoding candidates based on rate-distortion (R-D) theory, as proposed previously by the authors, currently provides the best performance-versus-complexity trade-off. In this paper, an analysis based on the rate-distortion exponent (RDE) is used to directly minimize the exponential decay rate of the error probability. This enables rigorous bounds on the error probability for finite-length RS codes and leads to modest performance gains. As a byproduct, a numerical method is derived that computes the rate-distortion exponent for independent non-identical sources. Analytical results are given for errors/erasures decoding.Comment: accepted for presentation at 2010 IEEE International Symposium on Information Theory (ISIT 2010), Austin TX, US

On Multiple Decoding Attempts for Reed-Solomon Codes: A Rate-Distortion Approach

One popular approach to soft-decision decoding of Reed-Solomon (RS) codes is based on using multiple trials of a simple RS decoding algorithm in combination with erasing or flipping a set of symbols or bits in each trial. This paper presents a framework based on rate-distortion (RD) theory to analyze these multiple-decoding algorithms. By defining an appropriate distortion measure between an error pattern and an erasure pattern, the successful decoding condition, for a single errors-and-erasures decoding trial, becomes equivalent to distortion being less than a fixed threshold. Finding the best set of erasure patterns also turns into a covering problem which can be solved asymptotically by rate-distortion theory. Thus, the proposed approach can be used to understand the asymptotic performance-versus-complexity trade-off of multiple errors-and-erasures decoding of RS codes. This initial result is also extended a few directions. The rate-distortion exponent (RDE) is computed to give more precise results for moderate blocklengths. Multiple trials of algebraic soft-decision (ASD) decoding are analyzed using this framework. Analytical and numerical computations of the RD and RDE functions are also presented. Finally, simulation results show that sets of erasure patterns designed using the proposed methods outperform other algorithms with the same number of decoding trials.Comment: to appear in the IEEE Transactions on Information Theory (Special Issue on Facets of Coding Theory: from Algorithms to Networks

A Decoding Algorithm for LDPC Codes Over Erasure Channels with Sporadic Errors

none4An efficient decoding algorithm for low-density parity-check (LDPC) codes on erasure channels with sporadic errors (i.e., binary error-and-erasure channels with error probability much smaller than the erasure probability) is proposed and its performance analyzed. A general single-error multiple-erasure (SEME) decoding algorithm is first described, which may be in principle used with any binary linear block code. The algorithm is optimum whenever the non-erased part of the received word is affected by at most one error, and is capable of performing error detection of multiple errors. An upper bound on the average block error probability under SEME decoding is derived for the linear random code ensemble. The bound is tight and easy to implement. The algorithm is then adapted to LDPC codes, resulting in a simple modification to a previously proposed efficient maximum likelihood LDPC erasure decoder which exploits the parity-check matrix sparseness. Numerical results reveal that LDPC codes under efficient SEME decoding can closely approach the average performance of random codes.noneG. Liva; E. Paolini; B. Matuz; M. ChianiG. Liva; E. Paolini; B. Matuz; M. Chian