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Abstract

An improved and extended Reed-Solomon-like method for providing a redundancy of m.gtoreq.3 is disclosed. A general expression of the codes is described, as well as a systematic criterion for proving correctness and finding decoding algorithms for values of m.gtoreq.3. Examples of codes are given for m=3, 4, 5, based on primitive elements of a finite field of dimension N where N is 8, 16 or 32. A Horner's method and accumulator apparatus are described for XOR-efficient evaluation of polynomials with variable vector coefficients and constant sparse square matrix abscissa. A power balancing technique is described to further improve the XOR efficiency of the algorithms. XOR-efficient decoding methods are also described. A tower coordinate technique to efficiently carry out finite field multiplication or inversion for large dimension N forms a basis for one decoding method. Another decoding method uses a stored one-dimensional table of powers of .alpha. and Schur expressions to efficiently calculate the inverse of the square submatrices of the encoding matrix.

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