Application of Circulant Matrices

A k x k matrix A = [aU lover a field F is called circulant if aij = a (j-i) mod k' A [2k ,k l linear code over F = GF (q) is called double-circulant if it is generated by a matrix of the fonn [I A l, where A is a circulant matrix. In this work we ftrst employ the Fourier transform techJ nique to analyze and construct se:veral families of double-circulant codes. The minimum distance of the resulting codes is lower-bounded by 2"k and can be decOded easily employing the standard BCH 'de~oding 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 blPcks. The decoding' proCedure inherits both the (relatively low) tim~ 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 Infonnation Theory, July 1988. T ec hn io n C om pu te r Sc ie nc e D ep ar tm en t T eh ni ca l R ep or t C S0 53 5 19 89