New binary coding results by circulants

The Introduction contains a new circulant echelon canonical form for the perfect Golay code and some tentative conclusions are suggested. Section I gives an account of the properties of circulant matrices , 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 . 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 code, and for . 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 code). Methods of coding and decoding circulant codes are not discussed.