New binary coding results by circulants

The Introduction contains a new circulant echelon canonical form for the perfect <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(23,12)</tex> Golay code and some tentative conclusions are suggested. Section I gives an account of the properties of circulant matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> , 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|E|A|</tex> . 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(79, 40)</tex> code, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w = 15; (103, 52), w = 19; (151, 76), w = 19; (89, 45), w = 17</tex> and for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(113, 57), w = 15</tex> . 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(31, 16)</tex> code). Methods of coding and decoding circulant codes are not discussed.