Cyclic Codes. Research Program to Extend the Theory of Weight Distribution and Related Problems for Cyclic Error-Correcting Codes and Constructive Coding Theory

摘要

Each perfect code on q symbols which corrects e errors 'contains' a tactical configuration of type (q-1) to the e th power; (e+1)-(2e+1)-n, where n is the block length. The H-Golay codes also 'contain' in the case q=2 a closed k-th order Steiner system, and for q > 2 an analogous new configuration. An error in the literature is pointed out. A lower bound on the number of inequivalent Steiner triple systems is established. Work of Lloyd and Golay on perfect codes is recast. Other necessary conditions for perfect codes are also derived. We give an updated account of previous work on weights in quadratic-residue codes, and improve on the square-root bound on the minimum distance. The H-Golay code of type (n, n-k) over GF(q) is cyclic if and only if the gcd (k, q-1) is 1. Minimum weights in several cyclic codes are determined. The possible isomorphism of the combinatorial designa defined by certain difference sets with identical parameters is studied. At most two of the designs considered are isomorphic. (Author)