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Proceedings of the IEEE

Soft decoding performance of extremal self-dual codes

V.K. Bhargava

Published Jan 1, 1983• 0 citations
Publisher

Abstract

Performance of soft decoded extremal self-dual codes of lengths 8 to 72 are obtained over the Gaussian channel. The results indicate that for decoded bit-error rates below 10-3, which is the main region of interest for coding application, substantial coding gains can be obtained by using extremal self-dual codes.
DOI
10.1109/proc.1983.12545

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The authors present and compare the performance of maximum likelihood (ML) soft decoding algorithms for binary linear self-dual (SD) codes. The codes considered have a minimum distance of 4, but due to their different weight distributions they differ in their coding gain. Remarkably, there is a code, denoted F/sub 24/, which has a decoder nearly 4 times faster than the fastest published decoder (6) for the (24,12,8) Golay code. Nontheless, these two codes have almost equal coding gains. Simulation results are given to compare between the codes. An analytical discussion of the results appears in an Appendix.>
There exist almost complete convolutional encodings of self-dual codes, i.e., block codes of rate 1/2 with weights w, w = 0 mod 4. The codes are of length 8m with the convolutional portion of length 8m-2 and the nonsystematic information of length 4m-1. The last two bits are parity checks on the two (4m-1) length parity sequences. The final information bit complements one of the extended parity sequences of length 4m. Solomon and van Tilborg have developed algorithms to generate these for the Quadratic Residue (QR) Codes of lengths 48 and beyond. For these codes and reasonable constraint lengths, there are sequential decodings for both hard and soft decisions. There are also possible Viterbi-type decodings that may be simple, as in a convolutional encoding/decoding of the extended Golay Code. In addition, the previously found constraint length K = 9 for the QR (48, 24;12) Code is lowered here to K = 8.
The decoding error probability of a code CCC measures thequality of performance when CCC is used for error correction in data transmission.In this note we compare different types of codes with regard to the decoding errorprobability.
A new decoder is proposed to decode the (23, 12, 7) binary Golay-code up to five errors. It is based on the algorithm that can correct up to four errors for the (24, 12, 8) extended Golay-code proposed by Lin et al., thereby achieving the soft decoding in the real sense for the Golay-code. For a weight-2 or weight-3 error pattern decoded by the hard decoder for correcting up to three errors, one can find the corresponding 21 weight-4 or weight-5 error patterns and choose the one with the maximum emblematic probability value, which is defined as the product of individual bit-error probabilities corresponding to the non-zero locations of the error pattern as the ultimate choice. Finally, simulation results of this decoder over additive white Gaussian noise (AWGN) channels show that the proposed method provides 0.9 dB coding gain than that of Lin et al.'s algorithm at bit-error rate of 10−5.