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.