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 Köhler 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.
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References
H. Hanani, On quadruple systems.Canad. J. Math, Vol. 12, (1960) pp. 145–157.
J. Spencer, Maximal consistent families of triples,J. of Combinatorial Theory, Vol. 5, (1968) pp. 1–8.
S.M. Johnson, Upper bounds for constant weight error-correcting codes.Discrete Mathermatics, Vol. 3, (1972) pp. 109–124.
J.G. Kalbfleish and R.G. Stanton, Maximal and minimal coverings of (k−1)-tuples by κ-tuples,Pacific J. Math., Vol. 26, (1968) pp. 131–140.
W.H. Mills, On the covering of triples by quadruples,Proc. of the 5th Southeastern Conference on Combinatorics Graph Theory and Computation, Boca Raton, (1974) pp. 563–581.
A.E. Brouwer, On the packing of quadruples without common triples,Ars Combinatoria, Vol. 5, (1978) pp. 3–6.
W.H. Mills and R.C. Mullin, Covering and packings. In Jeffry H. Dinitz and Douglas R. Stinton, editors,Contemporary Design Theory, John Wiley, New York (1992) pp. 371–399.
M.R. Best, A(11, 4, 4)=35 or some new optimal constant weight codes. Technical Report ZN 71/77, Math. Centr. Amsterdam, (1977).
J.H. Conway and N.J.A. Sloane,Sphere Packings, Lattices and Groups. Springer-Verlag, Berlin-Heidelberg, New York, (1988).
A.E. Brouwer, J.B. Shearer, N.J.A. Sloane, and W.D. Smith, A new table of constant weight codes,IEEE Trans. on Inform. Theory, IT-36, (1990) pp. 1334–1380.
C.C. Lindner and A. Rosa, Steiner quadruple systems—a survery,Discrete Math., Vol. 22, (1978) pp. 148–171.
C.L. van Pul and T. Etzion, New lower bounds for constant weight codes,IEEE Trans. on Inform. Theory, IT-35, (1989) pp. 1324–1329.
K.T. Phelps. Rotational Steiner quadruple systems,Ars Combinatoria, Vol. 4, (1977) pp. 177–185.
Alan Hartman and Kevin T. Phelps, Steiner quadruple systems. In Jeffry H. Dinitz and Douglas R. Stinton, editors,Contemporary Design Theory, pp. 205–240. John Wiley, New York (1992).
E. Kohler, Zyklische quadrupelsysteme.Abh. Sem. Univ. Hamburg, No. 48, (1978) pp. 1–24.
H. Siemon, Some remarks on the construction of cyclic Steiner quadruple system,Arch. der Math. Vol. 49, (1987) pp. 166–178.
H. Siemon, Infinite families of strictly cyclic Steiner quadruple systems,Discrete Mathematics, Vol. 77, (1989) pp. 307–316.
H. Siemon, Cyclic Steiner quadruple systems and Köhler's orbit graphs,Designs, Codes and Cryptography, Vol. 1, (1991) pp. 121–132.
H. Siemon, On the existence of cyclic Steiner quadruple systemsSQS(2p), Discrete Mathematics, Vol. 97, (1991) pp. 377–385.
F.R.K. Chung, J.A. Salehi, and V.K. Wei, Optical orthogonal codes: design, analysis, and applications,IEEE Transactions on Information Theory, 35(3), (1989) pp. 595–604.
K.T. Phelps, On cyclic Steiner systemS(3,4,20),Annals of Discrete Mathematics, Vol. 7, (1980) pp. 277–300.
I. Niven and H.S. Zuckerman,An Introduction to the Theory of Numbers, John Wiley, New York, fourth edition, (1980).
C. Berge,Graphs and Hyper-Graphs, North-Holland, Amsterdam, (1970).
I. Diener, On cyclic Steiner systemsS(3,4,22).Annals of Discrete Mathematics, Vol. 7, (1980) pp. 301–313.
E.F. Brickell and V.K. Wei, Optical orthogonal codes and difference families,Proc. of the Southeastern Conference on Combinatorics Graph Theory and Algorithms, (1987).
H. Chung and P.V. Kumar, Optical orthogonal codes—new bounds and an optimal construction,IEEE Transactions on Information Theory, 36(4), (1990) pp. 866–873.
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Communicated by D. Jungnickel
This research was supported in part by the Technion V.P.R. Fund.
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Bitan, S., Etzion, T. The last packing number of quadruples, and cyclic SQS. Des Codes Crypt 3, 283–313 (1993). https://doi.org/10.1007/BF01418528
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DOI: https://doi.org/10.1007/BF01418528