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

This document reports on work done in the period July 1964-April 1965 on error-correcting codes and questions arising in the study of them. In a major section, devoted to follow-up and additions to the 1964 Report, even lower weights in quadratic-residue codes were found in continuing the study using cyclotomic numbers. The proof that all but a finite number of cyclic codes are optimal has been greatly simplified, and by the application of a result available even in the 1964 Report. In addition, the question of the action of the projective unimodular group on the extended quadratic-residue codes is treated. A determination of the weight distribution in optimal codes is made, and a proof is given of the fact that, except for the obvious exceptions, optimal codes are never perfect. One section relates the known perfect codes (other than Hamming codes) and the Mathieu groups; it will be published in the Archive der Mathematik. Another section contains general results relating perfect codes to Steiner systems, complete Steiner systems, and tactical configurations. (Author)