New binary coding results by circulants

Abstract

The Introduction contains a new circulant echelon canonical form for the perfect(23,12)Golay code and some tentative conclusions are suggested. Section I gives an account of the properties of circulant matricesA, and a number of lemmas that make it possible to determine the minimum weight of codes generated by the rows of a matrix of the form|E|A|. In Section II, it is shown that many quadratic residue codes are almost of this form. The following new minimum weight results are obtained: For the(79, 40)code,w = 15; (103, 52), w = 19; (151, 76), w = 19; (89, 45), w = 17and for(113, 57), w = 15. In Section III, high-quality (noncyclic) group codes are constructed by means of circulants. In some cases a definite improvement is obtained on the best previously known Bose-Chaudhuri-Hocquenghem cyclic codes (including the(31, 16)code). Methods of coding and decoding circulant codes are not discussed.