Extending the design concept
Authors:
A. R. Calderbank and P. Delsarte
Journal:
Trans. Amer. Math. Soc. 338 (1993), 941952
MSC:
Primary 05E30; Secondary 05B05, 33C45
MathSciNet review:
1134756
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: Let be a family of subsets of a set , with . Given only the inner distribution of , i.e., the number of pairs of blocks that meet in points (with ), we are able to completely describe the regularity with which meets an arbitrary subset of , for each order (with ). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters , , . The main regularity parameter is the dimension of a welldefined subspace of , called the form space of . (This subspace coincides with if and only if is a design.) We show that the form space has the structure of an ideal, and we explain how to compute its canonical generator.
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 [1]
 A. R. Calderbank and P. Delsarte, On errorcorrecting codes and invariant linear forms, SIAM J. Discrete Math. 6 (1993), 123. MR 1201986 (93m:94025)
 [2]
 A. R. Calderbank, P. Delsarte, and N. J. A. Sloane, A strengthening of the AssmusMattson theorem, IEEE Trans. Information Theory IT37 (1991), 12611268. MR 1136663 (92k:94021)
 [3]
 P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. 10 (1973). MR 0384310 (52:5187)
 [4]
 , Pairs of vectors in the space of an association scheme, Philips Res. Rep. 32 (1977), 373411. MR 0498190 (58:16345)
 [5]
 , Hahn polynomials, discrete harmonics, and designs, SIAM J. Appl. Math. 34 (1978), 157166. MR 0460158 (57:154)
 [6]
 C. F. Dunkl, Spherical functions on compact groups and applications to special functions, Sympos. Math. 22 (1977), 145161. MR 0622207 (58:29865)
 [7]
 , An addition theorem for Hahn polynomials: the spherical functions, SIAM J. Math. Anal. 9 (1978), 627637. MR 0486704 (58:6405)
 [8]
 R. L. Graham, S.Y. R. Li, and W.C. W. Li, On the structure of designs, SIAM J. Algebraic Discrete Methods 1 (1980), 814. MR 563008 (83b:05042)
 [9]
 J. E. Graver and W. B. Jurkat, The module structure of integral designs, J. Combin. Theory Ser. A 15 (1973), 7590. MR 0329930 (48:8270)
 [10]
 S. Karlin and J. L. McGregor, The Hahn polynomials, formulas and an application, Scripta Math. 26 (1961), 3346. MR 0138806 (25:2249)
 [11]
 D. K. RayChaudhuri and N. M. Singhi, On existence of designs with large and , SIAM J. Discrete Math. 1 (1988), 98104. MR 936611 (89e:05034)
 [12]
 R. M. Wilson, Inequalities for designs, J. Combin. Theory Ser. A 34 (1983), 313324. MR 700037 (84j:05023)
 [13]
 , On the theory of designs, Enumeration and Designs (D. M. Jackson and S. A. Vanstone, eds.), Academic Press, New York, 1984, pp. 1949.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199311347560
PII:
S 00029947(1993)11347560
Keywords:
design,
form space,
distribution matrix,
dual Hahn polynomials
Article copyright:
© Copyright 1993
American Mathematical Society
