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IEEE International Symposium on Information Theory (ISIT)

An algorithm for calculating the exact bit error probability of a binary linear code over the binary symmetric channel

T. Wadayama

Published Jun 25, 2003• 0 citations
Publisher

Abstract

An efficient algorithm for calculating the i-th bit error probability of a binary linear code over the binary symmetric channel (BSC) are presented. It is proved that the exact i-th bit error probability of maximum likelihood (ML) decoding, bounded distance decoding, symbolwise maximum a posteriori probability (MAP) decoding and symbolwise MAP decoding with erasure can be obtained with time complexity O(n2/sup n-k/), where n, k denote length and dimension of the target code.

DOI
10.1109/isit.2002.1023674

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An efficient algorithm for calculating the i-th bit error probability of a binary linear code over the binary symmetric channel (BSC) are presented. It is proved that the exact i-th bit error probability of maximum likelihood (ML) decoding, bounded distance decoding, symbolwise maximum a posteriori probability (MAP) decoding and symbolwise MAP decoding with erasure can be obtained with time complexity O(n2/sup n-k/), where n, k denote length and dimension of the target code.
An efficient algorithm for calculating the ith bit error probability of a binary linear code over the binary symmetric channel (BSC) is presented. It is proved that the exact ith bit error probability of maximum-likelihood (ML) decoding, bounded distance decoding, and symbol-wise maximum a posteriori probability (MAP) decoding can be obtained with time complexity O(n2/sup n-k/), where n and k denote the length and the dimension of the target code. The proposed methods are applicable to any binary linear code with redundancy up to nearly 25-30 bits with a typical personal computer.
A lower bound on the error rate of linear binary block codes(under maximum likelihood decoding) over BSC channels is proposed.According to the principle of the maximum likelihood(ML) decoding algorithm,the decoding error probability is firstly converted into the joint probability of the error events,and the judge rule of the redundant error events is deduced based on the optimization rule of the improved Dawson-Sankoff bound.Moreover,the calculation expression about lower bound of the error probability solely depends on the Hamming weight enumerator function of the code and the crossover probability of the channel.The simulation results applying to various LDPC codes show that the new lower bound outperforms those generic lower bounds and the sphere packing bound.Its computational complexity is also lower.
A coset representation of the bit error probabilities for binary linear block codes and arbitrary symmetric memoryless channels is derived. Such a representation leads to simple upper bounds on the bit error probabilities for code components expressed via 2n code spectra, where n is the code length, and to a "bounded distance decoding" algorithm. The performance of the algorithm is illustrated for the (23,12) Golay code and a particular channel having non-binary output alphabet where it attains a sub-optimum performance.
Maximum-likelihood (ML) decoding of linear block codes on a symmetric channel is studied. Exact ML decoding is known to be computationally difficult. We propose an algorithm that finds the exact solution to the ML decoding problem by performing a depth-first search on a tree. The tree is designed from the code generator matrix and pruned based on the statistics of the channel noise. The complexity of the algorithm is a random variable. We characterize the complexity by means of its first moment, which for binary symmetric channels we find in closed-form. The obtained results indicate that the expected complexity of the algorithm is low over a wide range of system parameters.