Extending the $t$-design concept
HTML articles powered by AMS MathViewer
- by A. R. Calderbank and P. Delsarte PDF
- Trans. Amer. Math. Soc. 338 (1993), 941-952 Request permission
Abstract:
Let $\mathfrak {B}$ be a family of $k$-subsets of a $v$-set $V$, with $1 \leq k \leq v/2$. Given only the inner distribution of $\mathfrak {B}$, i.e., the number of pairs of blocks that meet in $j$ points (with $j = 0,1, \ldots ,k$), we are able to completely describe the regularity with which $\mathfrak {B}$ meets an arbitrary $t$-subset of $V$, for each order $t$ (with $1 \leq t \leq v/2$). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters $v$, $k$, $t$. The main regularity parameter is the dimension of a well-defined subspace of ${\mathbb {R}^{t + 1}}$, called the $t$-form space of $\mathfrak {B}$. (This subspace coincides with ${\mathbb {R}^{t + 1}}$ if and only if $\mathfrak {B}$ is a $t$-design.) We show that the $t$-form space has the structure of an ideal, and we explain how to compute its canonical generator.References
- A. R. Calderbank and P. Delsarte, On error-correcting codes and invariant linear forms, SIAM J. Discrete Math. 6 (1993), no.Β 1, 1β23. MR 1201986, DOI 10.1137/0406001
- A. R. Calderbank, P. Delsarte, and N. J. A. Sloane, A strengthening of the Assmus-Mattson theorem, IEEE Trans. Inform. Theory 37 (1991), no.Β 5, 1261β1268. MR 1136663, DOI 10.1109/18.133244
- P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. 10 (1973), vi+97. MR 384310
- Ph. Delsarte, Pairs of vectors in the space of an association scheme, Philips Res. Rep. 32 (1977), no.Β 5-6, 373β411. MR 498190
- Ph. Delsarte, Hahn polynomials, discrete harmonics, and $t$-designs, SIAM J. Appl. Math. 34 (1978), no.Β 1, 157β166. MR 460158, DOI 10.1137/0134012
- Charles F. Dunkl, Spherical functions on compact groups and applications to special functions, Symposia Mathematica, Vol. XXII (Convegno sullβAnalisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976) Academic Press, London, 1977, pp.Β 145β161. MR 0622207
- Charles F. Dunkl, An addition theorem for Hahn polynomials: the spherical functions, SIAM J. Math. Anal. 9 (1978), no.Β 4, 627β637. MR 486704, DOI 10.1137/0509043
- R. L. Graham, S.-Y. R. Li, and W. C. W. Li, On the structure of $t$-designs, SIAM J. Algebraic Discrete Methods 1 (1980), no.Β 1, 8β14. MR 563008, DOI 10.1137/0601002
- J. E. Graver and W. B. Jurkat, The module structure of integral designs, J. Combinatorial Theory Ser. A 15 (1973), 75β90. MR 329930, DOI 10.1016/0097-3165(73)90037-x
- Samuel Karlin and James L. McGregor, The Hahn polynomials, formulas and an application, Scripta Math. 26 (1961), 33β46. MR 138806
- D. K. Ray-Chaudhuri and N. M. Singhi, On existence of $t$-designs with large $v$ and $\lambda$, SIAM J. Discrete Math. 1 (1988), no.Β 1, 98β104. MR 936611, DOI 10.1137/0401011
- Richard M. Wilson, Inequalities for $t$-designs, J. Combin. Theory Ser. A 34 (1983), no.Β 3, 313β324. MR 700037, DOI 10.1016/0097-3165(83)90065-1 β, On the theory of $t$-designs, Enumeration and Designs (D. M. Jackson and S. A. Vanstone, eds.), Academic Press, New York, 1984, pp. 19-49.
Additional Information
- © Copyright 1993 American Mathematical Society
- 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