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# Pairwise Balanced Designs as Linear Spaces

DOI link for Pairwise Balanced Designs as Linear Spaces

Pairwise Balanced Designs as Linear Spaces book

# Pairwise Balanced Designs as Linear Spaces

DOI link for Pairwise Balanced Designs as Linear Spaces

Pairwise Balanced Designs as Linear Spaces book

## ABSTRACT

In particular, a cubic graph on n vertices (n even) corresponds to a regular linear space of type (3n/2

∣∣ 3n, 2c), where c = 3n(3n− 10)/8, and conversely. 6.23 Remark Let Kn denote the complete graph on n vertices (which is regular of degree

n−1). Then Constructions 6.19 and 6.22 give regular linear spaces of type ( n ∣∣ 2(n2))

and (( n 2

) ∣∣ (n− 1)n, 2c) , where c = ((n2) 2

)− n(n−12 ) = n(n−1)8 ((n− 1)(n− 4) + 2). 6.24 Construction Let L be a latin square of side n > 3 (abbreviated LS(n)) with symbol

set {0, . . . , n− 1}. Define a regular linear space of type (3n ∣∣ n3, 3n2) as follows. The three lines of length n are the sets {1, 2, . . . , n}, {n+1, . . . , 2n}, and {2n+1, . . . , 3n}. For any pair i, j with 1 ≤ i, j ≤ n, define a line {i, n+ j, 2n+ 1+L(i, j)}. Conversely, a regular linear space of type (3n

∣∣ n3, 3n2) determines a latin square of side n. The isomorphism types of linear spaces of this line type correspond in a one-to-one manner to the main classes of latin squares of side n.