RESEARCH TO DEVELOP THE ALGEBRAIC THEORY OF CODES

Abstract
A number of new combinatorial designs are found as direct applications of the theory of error-correcting codes. Results on the automorphism groups of Hadamard matrices are presented. A simple proof of the well-known MacWilliams relations, together with a generalization, is given. We have constructed the (24, 12) extended code of Golay over GF(2) in a particularly simple way. We prove that teach of the seven 6; 2-15-36 designs ((v,k,lambda) designs) arising from the difference sets of size 15 in the Abelian groups of order 36 has only the obvious group as automorphism group. Each of these designs gives a Hadamard matrix of order 36. We determine the covering radius for BCH codes of design distance 2 over GF(q) for all odd prime powers q. We give two extensions of the Peterson-Kasami-Lin result on necessary and sufficient conditions for an extension of a cyclic code to be invariant under the affine group. Explicit factorizations of (x(to the lth power)-1) over the appropriate quadratic-number field for l=7, 11, 13, and 23 are given.

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