Generalised triangular basis multipliers for the design of Reed-Solomon codecs

In this paper a generalised definition of a triangular basis for GF(2^m) is presented. This definition is more flexible than the traditional definition and allows a number of triangular bases to be determined to any given basis. The triangular basis to the polynomial basis can then be used which has the simplest basis transformation in modified Hasan-Bhargava multipliers. It is shown that when the defining irreducible polynomial for the field is a trinomial, the triangular basis is a permutation of the polynomial basis elements. Furthermore, if the defining irreducible polynomial is a pentanomial of a certain form, the triangular basis can be obtained from the polynomial basis with a reordering of the coefficients and two GF(2) additions. This new definition of triangularity allows for reduced complexity modified Hasan-Bhargava multipliers to be designed which in turn reduces the hardware complexity of Reed-Solomon codecs using these multipliers. Finally, a simple method of making large hardware savings in Reed-Solomon decoders utilising frequency domain decoding is presented