The last packing number of quadruples, and cyclic SQS

Abstract

The packing number of quadruples without common triples of ann-set, or the maximum number of codewords of a code of lengthn, constant weight 4, and minimum Hamming distance 4, is an old problem. The only unsolved case isn≡5 (mod 6). For 246 values of the formn≡5 (mod 6), we present constant weight codes with these parameters, of size [(n−1)(n 2 −3n−4)]/24, which is greater by (4n-20/24) from the previous lower bound and leaves a gap of [(n−5)/12] to the known upper bound. For infinitely many valuesn≡5 (mod 6) we give enough evidence to believe that such codes exist. The constructed codes are optimal extended cyclic codes with these parameters. The construction of the code is done by a new approach of analyzing the Kohler orbit graph. We also use this analysis to construct new S-cyclic Steiner Quadruple Systems. Another important application of the analysis is in the design of optical orthogonal codes.