The nonexistence of code words of weight 16 in a projective plane of order 10
The well known unsolved problem of the existence of a projective plane of order 10 is being studied by means of the binary error correction code generated by the rows of the incidence matrix of the plane, assuming that the plane does exist. \textit{F. J. MacWilliams}, \textit{N. J. A. Sloane} and \textit{J. G. Thompson} [J. Comb. Theory, Ser. A 14, 66-78 (1973; Zbl 0251.05020)] found that the weight enumerator of this code is determined by the number of vectors of weights 12, 15 and 16. In later work, it was found that no code words of weights 12 and 15 exist in the code. The search for code words of weight 16 can be divided into 6 cases. \textit{J. L. Carter} [''On the existence of a projective plane of order ten'', Ph. D. Thesis, Univ. California, Berkeley (1974)] finished the search for 4 of these cases and part of the fifth. In the paper under review, the study of the remaining cases is completed, and the weight enumerator of the code is determined. No plane of order 10 was found. The weight enumerator shows that a weight 19 code word must exist if a plane exists. The authors thus recommend a search starting from a 19-point configuration.
scientific article; zbMATH DE number 3949156
scientific article