Abstract
Two-weight codes and projectivesets having two intersection sizes with hyperplanes are equivalentobjects and they define strongly regular graphs. We construct projective sets in PG(2m − 1,q) that have the sameintersection numbers with hyperplanes as the hyperbolic quadricQ+(2m − 1,q). We investigate these sets; we provethat if q = 2 the corresponding strongly regular graphsare switching equivalent and that they contain subconstituentsthat are point graphs of partial geometries. If m = 4the partial geometries have parameters s = 7, t = 8,α = 4 and some of them are embeddable in Steinersystems S(2,8,120).
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References
A. E. Brouwer, W. H. Haemers and V. D. Tonchev, Embedding partial geometries in Steiner designs, In Geometry, Combinatorial Designs and Related Structures (J. W. P. Hirschfeld, S. S. Magliveras, and M. J. de Resmini, eds.), Cambridge (1997) pp. 33–41.
A. E. Brouwer, A. V. Ivanov and M. H. Klin, Some new strongly regular graphs, Combinatorica, Vol. 9 (1989) pp. 339–344.
A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., Vol. 18 (1986) pp. 97–122.
F. De Clerck, New partial geometries derived from old ones, Bull. Belg. Math. Soc.-Simon Stevin, Vol. 5 (1998) pp. 255–263.
F. De Clerck and M. Delanote, Partial geometries and the triality quadric, J. Geom., Vol. 68 (2000) pp. 34–47.
F. De Clerck, R. H. Dye and J. A. Thas, An infinite class of partial geometries associated with the hyperbolic quadric in PG(4n-1; 2), European J. Combin., Vol. 1 (1980) pp. 323–326.
F. De Clerck, H. Gevaert and J. A. Thas, Partial geometries and copolar spaces, In Combinatorics’ 88 (A. Barlotti, G. Lunardon, F. Mazzocca, N. Melone, D. Olanda, A. Pasini and G. Tallini, eds.) Rende (I) (1991) pp. 267–280.
F. De Clerck, N. Hamilton, C. O'Keefe and T. Penttila, Quasi-quadrics and related structures, Australas. J. Combin., Vol. 22 (2000) pp. 151–166.
P. Delsarte, Weights of linear codes and strongly regular normed spaces, Discrete Math., Vol. 3 (1972) pp. 47–64.
R. H. Dye, Partitions and their stabilizers for line complexes and quadrics, Ann. Mat. Pura Appl. (4), Vol. 114 (1977) pp. 173–194.
W. M. Kantor, Symplectic groups, symmetric designs and line ovals, J. Algebra, Vol. 33 (1975) pp. 43–58.
W. M. Kantor, Strongly regular graphs defined by spreads, Israel J. Math., Vol. 41 (1982) pp. 298–312.
M. H. Klin, On new partial geometries pg(8; 9; 4), Contributed talk, 16th British Combinatorial Conference (1998).
R. Mathon and A. P. Street, Overlarge sets and partial geometries, J. Geom., Vol. 60, No. 1–2 (1997) pp. 85–104.
N. J. Patterson, A four-dimensional Kerdock set over GF(3), J. Combin. Theory Ser. A, Vol. 20 (1976) pp. 365–366.
J. J. Seidel, On two-graphs and Shult' characterization of symplectic and orthogonal geometries over GF(2), T.H.-Report 73-WSK-02, Tech. Univ. Eindhoven (1973).
J. A. Thas, Some results on quadrics and a new class of partial geometries, Simon Stevin, Vol. 55 (1981) pp. 129–139.
V. D. Tonchev, Quasi-symmetric designs, codes, quadrics, and hyperplane sections, Geom. Dedicata, Vol. 48 (1993) pp. 295–308.
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Clerck, F.D., Delanote, M. Two-Weight Codes, Partial Geometries and Steiner Systems. Designs, Codes and Cryptography 21, 87–98 (2000). https://doi.org/10.1023/A:1008383510488
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DOI: https://doi.org/10.1023/A:1008383510488