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IEEE Transactions on Communications

Information-bit, information-symbol, and decoded-symbol error rates for linear block codes

Don J. Torrieri

Published May 1, 1988• 45 citations
Publisher

Abstract

There are two types of bounded-distance decoders for linear block codes: erasing decoders that discard uncorrectable received words, and reproducing decoders that reproduce uncorrectable received words. Exact expressions for the information-symbol and decoded-symbol error rates are derived for both types. Necessary and sufficient conditions are derived for the quality of the information-symbol and decoded symbol error rates, It is formally proved that these two error rates are equal for cyclic codes with either erasing or reproducing decoders. For reproducing decoders, two approximations to the information-bit error rate and their applicability are examined.>

DOI
10.1109/26.1477

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There are two types of bounded-distance decoders for linear block codes: erasing decoders that discard uncorrectable received words, and reproducing decoders that reproduce uncorrectable received words. Exact expressions for the information-symbol and decoded-symbol error rates are derived for both types. Necessary and sufficient conditions are derived for the quality of the information-symbol and decoded symbol error rates, It is formally proved that these two error rates are equal for cyclic codes with either erasing or reproducing decoders. For reproducing decoders, two approximations to the information-bit error rate and their applicability are examined.>
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Decoding rules to minimize the probability of information bit errors in binary linear block codes are investigated. A novel decoding rule is proposed and compared to a recently proposed rule (see M. Elia and G. Prati, 1985) and to the standard rule, with respect to both the information bit error rate and the information word error rate. It is concluded that the proposed rule not only can be implemented much more efficiently that the standard rule, but also is superior with respect to either error rate. Some particular codes have been presented to show that the authors' rule can coincide or can differ from Elia and Prati's rule when applied to a given code.>
This dissertation investigates the complexity of minimum-distance decoding (MDD) algorithms for general binary linear block codes, and also investigates the use of such codes in constructing error correcting/detecting codes that are effective against both random and unidirectional errors. A new MDD algorithm is proposed, and its complexity is analyzed and compared to that of other such algorithms. The question of whether all good codes satisfy certain properties required for efficient implementation of the algorithm is also considered. The performance of decoding algorithms, with the information bit error rate as the criterion instead of the block error rate, is also studied. New decoding rules which can yield lower information bit error rates than the MDD rule are proposed and evaluated. A new class of binary codes is developed which corrects random errors and detects all unidirectional errors. These codes are obtained by appending a suffix to each codeword of a linear block code. To evaluate the efficiency of the new codes, a thorough comparison with the best previously obtained codes is made.
The performance of optimum linear block codes has been examined by computer simluation, and the error distribution at the decoder output, the block-error probability and the bit-error probability have been determined for the codes when they are used on a binary symmetric channel with random errors. A table showing the distribution of errors in the decoder output is presented, and it is shown how the table can be used to determine exact values for the bit-error probability and the block-error probability. The exact values of bit-error probability computed from the table are compared with values obtained using a known theoretical approximation to the bit-error probability, and some conclusions are drawn as to the general usefulness of the theoretical approximation. Also, block-error probabilities are computed from the information presented in the table and these values are compared with the estimates of block-error probability obtained from minimumdistance considerations.
Linear block codes over noisy channels causing both erasures and errors can be decoded by deleting the erased symbols and decoding the resulting vector with respect to a punctured code and then retrieving the erased symbols. This can be accomplished using separating parity-check matrices. For a given maximum number of correctable erasures, such matrices yield parity-check equations that do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. Separating parity-check matrices typically have redundant rows. An upper bound on the minimum number of rows in separating parity-check matrices, which is called the separating redundancy, is derived which proves that the separating redundancy tends to behave linearly as a function of the code length.
It has been claimed that the performance of a linear block code under iterative decoding on the binary erasure channel is determined by the stopping distance, i.e., the size of the smallest non-empty stopping set in the associated Tanner graph. Indeed, this is true from the perspective of code word retrieval. However, we show that with respect to the retrieval of just the information bits within the code word, the stopping distance may not be the main performance indicator since the smallest non-empty stopping sets might not hit the information set. We present expressions for decoding failure probabilities for code/information words/bits under optimal or iterative decoding and demonstrate the importance of the choice of the set of bits carrying the information for a fixed code with a fixed decoder