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By finding explicit PD sets, we show that permutation decoding can be used for the binary code obtained from the row span over the field F 2 of an adjacency matrix of the lattice graph L 2 ( n ) for any n ⩾ 5 .
We give some constructions of new infinite families of group divisible designs, GDD ( n , 2 , 4 ; λ 1 , λ 2 ) , including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3 ⩽ n ⩽ 8 . For n = 10 there is one missing critical design. If...
In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly ( v , 4 , 1 ) RBIBDs. The problem we investigate is the existence of ( v , 4 , 1 ) RBIBDs containing a ( w , 4 , 1 ) RBIBD as a subdesign. We also require that each parallel class...
Let A denote a set of order m and let X be a subset of A k + 1 . Then X will be called a blocker (of A k + 1 ) if for any element say ( a 1 , a 2 , … , a k , a k + 1 ) of A k + 1 , there is some element ( x 1 , x 2 , … , x k , x k + 1 ) of X such that x i equals a i for at least two i . The...
We continue our study of partitions of the full set of v 3 triples chosen from a v -set into copies of the Fano plane PG ( 2 , 2 ) (Fano partitions) or copies of the affine plane AG ( 2 , 3 ) (affine partitions) or into copies of both of these planes (mixed partitions). The smallest cases for...
We introduce power Hadamard matrices , in order to study the structure of (group) generalized Hadamard matrices, Butson (generalized) Hadamard matrices and other related orthogonal matrices, with which they share certain common characteristics. The new objects turn out to be as interesting, and...
Let G be a graph. A G -trade of volume m is a pair ( T , T ′ ) , where each of T and T ′ consists of m graphs, pairwise edge-disjoint, isomorphic to G , such that T ∩ T ′ = ∅ and the union of the edge sets of the graphs in T is identical to the union of the edge sets of the graphs in T...
The complete multipartite graph K n ( m ) with n parts of size m is shown to have a decomposition into n -cycles in such a way that each cycle meets each part of K n ( m ) ; that is, each cycle is said to be gregarious . Furthermore, gregarious decompositions are given which are also resolvable.
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