RESEARCH PROGRAM TO EXTEND THE THEORY OF WEIGHT DISTRIBUTION AND RELATED PROBLEMS FOR CYCLIC ERROR-CORRECTING CODES.
Summary scientific rept. for Jun 62-May 63,
SYLVANIA ELECTRIC PRODUCTS INC WALTHAM MASS
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In Section I of the report methods of induced representations were applied to questions of weight distribution in certain quadratic residue codes. This section is partly an exposition of work of Gleason and Prange. If the prime p denotes the block length of this cyclic p,p12 code over GFq for certain q, the coordinate indices are identified with the finite points on the projective line over GFp. A new coordinate was then introduced for the infinite point, so that the code is replaced by a p1, p12 code. The main result due to Gleason and Prange was that this new code is invariant under the permutation of coordinates with sign changes, in general induced by the projective unimodular group acting on the coordinate indices. This property allows one to deduce some results on the congruence of the weights modulo various m, especially when q 2. In Section II a class of cylcic codes over GF2 of block length 3p, where p is prime and 2 has even multiplicative order h modulo p, were defined. These codes have the property that all odd weights in them are at least p. Whether the even weights are at least of the order of p is not known but the smallest even weight found in any of these codes so far is at least p - 1.