Generalized minimum distance decoding schemes are found for majority logic decodable codes. These decoding schemes are parallel to the classical majority logic decoding schemes and thus can be easily implemented.
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 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.
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.