A novel analytical approach to performance evaluation of soft-decoding algorithms for binary linear block codes based on probabilistic iterative error correction is presented. A convergence condition establishing the critical noise rate below which the expected bit-error probability tends to zero is theoretically derived. It explains the capability of iterative probabilistic decoding of binary linear block codes with sparse parity-check matrices to correct, with probability close to one, error patterns with the number of errors (far) beyond half the code minimum distance. Systematic experiments conducted on truncated simplex codes seem to agree well with the convergence condition. The method may also be interesting for the theoretical analysis of the so-called turbo codes.
A novel analytical approach to performance evaluation of soft-decoding algorithms for binary linear block codes based on probabilistic iterative error correction is presented. A convergence condition establishing the critical noise rate below which the expected bit-error probability tends to zero is theoretically derived. It explains the capability of iterative probabilistic decoding of binary linear block codes with sparse parity-check matrices to correct, with probability close to one, error patterns with the number of errors (far) beyond half the code minimum distance. Systematic experiments conducted on truncated simplex codes seem to agree well with the convergence condition. The method may also be interesting for the theoretical analysis of the so-called turbo codes.
Iterative decoding methods have gained interest, initiated by the results of the so-called "turbo" codes. The theoretical description of this decoding, however, seems to be difficult. Therefore, we study the iterative decoding of block codes. First, we discuss the iterative decoding algorithms developed by Gallager (1962), Battail et al. (1979), and Hagenauer et al. (1996). Based on their results, we propose a decoding algorithm which only uses parity check vectors of minimum weight. We give the relation of this iterative decoding to one-step majority-logic decoding, and interpret it as gradient optimization. It is shown that the used parity check set defines the region where the iterative decoding decides on a particular codeword. We make plausible that, in almost all cases, the iterative decoding converges to a codeword after some iterations. We derive a computationally efficient implementation using the minimal trellis representing the used parity check set. Simulations illustrate that our algorithm gives results close to soft decision maximum likelihood (SDML) decoding for many code classes like BCH codes. Reed-Muller codes, quadratic residue codes, double circulant codes, and cyclic finite geometry codes. We also present simulation results for product codes and parallel concatenated codes based on block codes.
Today the error correcting codes are present in all the telecom standards, in particular the low density parity check (LDPC) codes. The choice of a good code for a given network is essentially linked to the decoding performance obtained by the bit error rate (BER) curves. This approach requires a significant simulation time proportional to the length of the code, to overcome this problem Exit chart was introduced, as a fast technique to predict the performance of a particular class of codes called Turbo codes. In this paper, we success to apply Exit chart to analyze convergence behavior of iterative threshold decoding of one step majority logic decodable (OSMLD) codes. The iterative decoding process uses a soft-input soft-output threshold decoding algorithm as component decoder. Simulation results for iterative decoding of simple and concatenated codes transmitted over a Gaussian channel have shown that the thresholds obtained are a good indicator of the Bit Error Rate (BER) curves.
Asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes N and the number of iterations I go to infinity. Three classes of codes are considered. These are Gallager's regular low-density parity-check (LDPC) codes, Tanner's generalized LDPC (GLDPC) codes, and the turbo codes due to Berrou et al. It is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability P/sub b/, but also the block (frame) error probability P/sub B/, goes to zero as N and I go to infinity.
Binary cyclic codes achieve good error correction performance and allow the implementation of very simpleencoder and decoder circuits. Among them, BCH codesrepresent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that softdecision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on “extended parity-check matrices”, and show that such method is not suitable for high rates or long codes. We propose a new approach, based on “reduced parity-check matrices” and “spread parity-check matrices”, that can achieve better correction performance in many practical cases, without increasing the complexity.
For iterative soft-decision decoding algorithms for binary linear block codes, error performance and computational complexity are very important factors.In this thesis, suÆcient conditions of the optimality of decoded codeword and a suÆcient condition which is used for ruling-out some useless iterations of bounded distance decoding which reduces the computational complexity of iterative soft-decision decoding algorithms are shown.New iterative soft-decision decoding algorithms which uses the suÆcient conditions of the optimality of decoded codeword to achievegood error performance with small computational complexity are also shown.First, the suÆcient conditions are derived based on: (1) partial knowledge of the distance pro le of the code, and (2) previously generated candidate codewords.When the numberof previously generated candidate codewords is greater than 1, the conditions presented here are less stringent than those derived by Taipale and Pursley [4] and Kaneko et.al.[5].The condition based on three previously generated candidate codewords is the rst such s u Æ c i e n t condition that has ever been derived.It is shown by computer simulation that these suÆcient condions reduce the computational complexity of iterative decoding algorithms without degradation of error performance.Second, a suÆcient condition to rule-out some useless test error patterns is derived.In an iterative decoding algorithm, such as Chase Type-II decoding algorithm[2] and its improved versions, candidate codewords for a received vector are generated
Binary cyclic codes achieve good error correction performance and allow the implementation of very simple encoder and decoder circuits. Among them, BCH codes represent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that soft-decision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on "Extended Parity-Check Matrices", and show that such method is not suitable for high rates or long codes. We propose a new method, based on "Spread Parity-Check Matrices", that has the same complexity, but can achieve better correction performance in many practical cases.