Decoding the (23, 12, 7) Binary Golay Code

Abstract

In this paper, an algebraic decoding method is proposed for the (23, 12, 7) binary Golay code. By applying a technique developed in [7], the computation complexity of this decoder is much less compared with the conventional step-by-step decoding algorithm, since the variable calculations and the operations of multiplication can be reduced significantly. I. INTRODUCTIONThe (23, 12, 7) binary Golay code is a unique multipleerror-correcting perfect code. It was described by Golay [1], and several applications of the code are listed in Reference 2. The algebraic method such as Kasami's error-trapping ecoder and systematic-search decoder [3], can be applied to decode this Golay code. Massey presented another algebraic decoding method, the stepby-step decoding algorithm, for general BCH codes [4]. Based on the step-by-step decoding algorithm and some results of [5], wei and wei [6] proposed a fast step-bystep algebraic decoding algorithm for the (23, 12, 7) binary Golay codes. However, it has not tried to reduce the computation complexity.In this paper, a modified step-by-step decoding algorithm for the (23, 12, 7) binary Golay is presented. By using some results of [7], we show that the computation complexity of the decoder [6] can be further simplified. The novel decoder significantly reduces the calculations of variables (i.e., the variables T 3 , T 9 , and F of [6]) compared with the algorithm in [6].