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L-STEP MAJORITY LOGIC DECODING,

Robert Tien-Wen Chien, Spencer W. Ng

Published Jun 1, 1970• 0 citations
OpenAlex

Abstract

In the search for multiple-error-correcting codes which can be easily and rapidly decoded, some coding theorists have turned their attentions to threshold (majority logic) decoding. Much work has been done in this area recently, and quite a few classes of new codes suitable for threshold decoding have been developed. The main desirable feature of majority logic decoding is that it can be very simply implemented. Furthermore majority decoding can also automatically correct many more patterns of errors other than those guaranteed by the decoding schemes without any additional equipment. This paper is a survey of the L-step majority decoding.

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IEEE Transactions on Information Theory•Jan 1, 1973•2 citations
In the search for multiple-error-correcting codes which can be easily and rapidly decoded, some coding theorists have turned their attentions to threshold (majority logic) decoding. Much work has been done in this area recently, and quite a few classes of new codes suitable for threshold decoding have been developed. The main desirable feature of majority logic decoding is that it can be very simply implemented. Furthermore majority decoding can also automatically correct many more patterns of errors other than those guaranteed by the decoding schemes without any additional equipment. This paper is a survey of the L-step majority decoding.
The attractiveness of majority-logic decoding is its simple implementation. Several classes of majority-logic decodable block codes have been discovered for the past two decades. In this paper, a method of constructing a new class of majority-logic decodable block codes is presented. Each code in this class is formed by combining majority-logic decodable codes of shorter lengths. A procedure for orthogonalizing codes of this class is formulated. For each code, a lower bound on the number of correctable errors with majority-logic decoding is obtained. An upper bound on the number of orthogonalization steps for decoding each code is derived. Several majority-logic decodable codes that have more information digits than the Reed-Muller codes of the same length and the same minimum distance are found. Some results presented in this paper are extensions of the results of Lin and Weldon [11] and Gore [12] on the majority-logic decoding of direct product codes.
The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability.
In a recent article, a technique was proposed to step up the majority logic decoding of variation set low density parity check codes. This is helpful as majority logic decoding can be implemented serially with trouble-free hardware but requires a huge decoding time. The method detect whether a word has errors in the initial iterations of majority logic decoding, and at what time there are no errors the decoding split ends without completing the rest of the iterations. While the majority words in a memory will be error-free, the average decoding time is significantly reduced. The outcome obtained proves that the technique is also effective for EG-LDPC codes. General simulation results are given to precisely estimate the prospect of error detection for different code sizes and numbers of errors.