block design

Given positive integers v,β,r,k,λv,\beta,r,k,\lambda, a block 2-design of type (v,β,r,k,λ)(v,\beta,r,k,\lambda) is a combinatorial structure consisting of

  • a finite set XX whose elements are declared points
  • set BB of kk-element subsets of XX, called blocks

such that

  • (i) for each xXx\in X the number of vBv\in B such that xbx\in b is independent of xx and equals rr
  • (ii) for each pair of distinct elements x,yXx,y\in X, the number of bBb\in B such that {x,y}b\{x,y\}\subset b is independent of (x,y)(x,y) and equals λ\lambda

Block ss-designs of type (v,β,r,k,λ)(v,\beta,r,k,\lambda) for s>2s\gt 2 have (ii) changed to

(ii’) for each subset {x 1,,x s}X\{x_1,\ldots,x_s\}\subset X of cardinality precisely ss, the number of bBb\in B such that {x 1,,x s}b\{x_1,\ldots,x_s\}\subset b is independent of the choice of such a subset and equals λ\lambda.

Examples: Steiner systems, finite projective planes.

Applications: cryptography, codes, finite geometries, finite group theory.

Created on August 30, 2011 at 16:05:31. See the history of this page for a list of all contributions to it.