Extending the -design concept

Authors:
A. R. Calderbank and P. Delsarte

Journal:
Trans. Amer. Math. Soc. **338** (1993), 941-952

MSC:
Primary 05E30; Secondary 05B05, 33C45

DOI:
https://doi.org/10.1090/S0002-9947-1993-1134756-0

MathSciNet review:
1134756

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Abstract | References | Similar Articles | 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 well-defined 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.

**[1]**A. R. Calderbank and P. Delsarte,*On error-correcting codes and invariant linear forms*, SIAM J. Discrete Math.**6**(1993), 1-23. MR**1201986 (93m:94025)****[2]**A. R. Calderbank, P. Delsarte, and N. J. A. Sloane,*A strengthening of the Assmus-Mattson theorem*, IEEE Trans. Information Theory**IT-37**(1991), 1261-1268. 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), 373-411. MR**0498190 (58:16345)****[5]**-,*Hahn polynomials, discrete harmonics, and*-*designs*, SIAM J. Appl. Math.**34**(1978), 157-166. MR**0460158 (57:154)****[6]**C. F. Dunkl,*Spherical functions on compact groups and applications to special functions*, Sympos. Math.**22**(1977), 145-161. MR**0622207 (58:29865)****[7]**-,*An addition theorem for Hahn polynomials*:*the spherical functions*, SIAM J. Math. Anal.**9**(1978), 627-637. 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), 8-14. 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), 75-90. MR**0329930 (48:8270)****[10]**S. Karlin and J. L. McGregor,*The Hahn polynomials, formulas and an application*, Scripta Math.**26**(1961), 33-46. MR**0138806 (25:2249)****[11]**D. K. Ray-Chaudhuri and N. M. Singhi,*On existence of*-*designs with large**and*, SIAM J. Discrete Math.**1**(1988), 98-104. MR**936611 (89e:05034)****[12]**R. M. Wilson,*Inequalities for*-*designs*, J. Combin. Theory Ser. A**34**(1983), 313-324. 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. 19-49.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1134756-0

Keywords:
-design,
-form space,
-distribution matrix,
dual Hahn polynomials

Article copyright:
© Copyright 1993
American Mathematical Society