Application of circulant matrices to the construction and decoding of linear codes

An r × r matrix A = [aij] over a field F is called circulant if aij = a0, ( j−i) mod r . An [n = 2r, k = r] linear code over F = GF(q) is called double-circulant if it is generated by a matrix of the form [I A], where A is an r × r circulant matrix. In this work we first employ the Fourier transform technique to analyze and construct several families of double-circulant codes. The minimum distance of the resulting codes is lower-bounded by 2√⎯r and can be decoded easily employing the standard BCH decoding algorithm or the majority-logic decoder of Reed-Muller codes. Second, we present a decoding procedure for Reed-Solomon codes, based on a representation of the parity-check matrix by circulant blocks. The decoding procedure inherits both the (relatively low) time complexity of the Berlekamp-Massey algorithm, and the hardware simplicity characteristic of Blahut’s algorithm. The proposed decoding procedure makes use of the encoding circuit together with a reduced version of Blahut’s decoder. This work was presented in part at the Beijing International Workshop on Information Theory, July 1988, and in part at the IEEE International Symposium on Information Theory, San Diego, California, January 1990.