It is shown that a majority-logic decoding algorithm proposed by Lin and Weldon for the product of anL-step and a one-step orthogonalizable code is incomplete whenLis greater than unity. An improvement is presented to overcome this disadvantage in the binary case.
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.
In this correspondence additional results on majority-logic decoding of product codes are presented. If one of the component codes is decodable using Rudolph's nonorthogonal algorithm, and the other using Reed-Massey'sL-step orthogonalization, then the resulting product code is majority-logic decodable inLsteps, where nonorthogonal checks are used in the first step and orthogonal checks in the remaining steps. If the component codes are both decodable using Rudolph's algorithm, the product code is majority-logic decodable using the same procedure.
A method of using reliability information in one-step majority-logic decoders is presented. The idea is, basically, that the received binary digits are corrected in an order such that the least reliable digit is first corrected. With this method the error correction capability of the decoder is extended for large signal-to-noise ratios (SNR's). Different decoding algorithms are used when the number of orthogonal parity-check sums are even and odd, respectively. Computer simulations are presented for some short codes with binary antipodal signals on the additive white Gaussian noise channel.