Necessary conditions for a balanced incomplete block design

Counting incidences two ways forces \(r = \lambda(v-1)/(k-1)\) and \(b = vr/k\), both whole; Fisher's inequality adds \(b \ge v\).

Usage

morie_bibd_parameters(v, k, lam)

Arguments

v, k, lam

Design parameters.

Value

A list with r, b, divisibility_ok, fisher_ok, feasible, exists, warnings.

Details

THESE ARE NECESSARY, NOT SUFFICIENT. (22, 7, 2) passes every one of them and cannot exist, by Bruck-Ryser-Chowla. feasible and exists are therefore separate, and exists is left NULL wherever the arithmetic cannot settle it rather than reporting feasibility as existence.

References

Fisher RA (1940). Stinson DR (2004).

Examples

morie_bibd_parameters(v = 5L, k = 5L, lam = 5L)
#> $v
#> [1] 5
#> 
#> $k
#> [1] 5
#> 
#> $lambda
#> [1] 5
#> 
#> $r
#> [1] 5
#> 
#> $b
#> [1] 5
#> 
#> $divisibility_ok
#> [1] TRUE
#> 
#> $r_integral
#> [1] TRUE
#> 
#> $b_integral
#> [1] TRUE
#> 
#> $fisher_ok
#> [1] TRUE
#> 
#> $feasible
#> [1] TRUE
#> 
#> $exists
#> [1] TRUE
#> 
#> $note
#> [1] "trivial: the single block containing every point"
#> 
#> $n
#> [1] 5
#> 
#> $warnings
#> character(0)
#> 
#> $method
#> [1] "BIBD necessary conditions with Fisher's inequality"
#>