checkSpreadEquivalence function

Checking the Equivalence of Two Spreads

This function checks the equivalence of two (t-1)-spreads of PG(n-1,2) by comparing their sorted bitstring representations.

checkSpreadEquivalence(spread1, spread2)

Arguments

• spread1: A (t-1)-spread of PG(n-1,2) stored as a three dimensional binary array (see Details and Examples).
• spread2: A (t-1)-spread of PG(n-1,2) stored as a three dimensional binary array (see Details and Examples).

Details

This code checks if two (t-1)-spreads of PG(n-1,2) are equivalent using the bitstring representation of Spencer et al. (2019). Both input spreads should be formatted as 3-dimensional arrays, for example, spread1[i,j,k] indicates whether or not the ith basic factor is present in the jth effect of the kth flat of spread1.

Returns

A Boolean indicating whether or not the two spreads are equivalent.

References

Spencer, N.A., Ranjan, P., and Mendivil, F., (2019), "Isomorphism Check for $2^n$ Factorial Designs with Randomization Restrictions", Journal of Statistical Theory and Practice, 13(60),1-24 [https://doi.org/10.1007/s42519-019-0064-5]

See Also

checkSpreadIsomorphism for checking the isomorphism of spreads.

checkStarEquivalence for checking the equivalence of two stars.

Author(s)

Neil Spencer, Pritam Ranjan, Franklin Mendivil

Examples

## Example 1: two non-equivalent 1-spreads of PG(3,2)
data(spreadn4t2a)
data(spreadn4t2b)

# test their equivalence
(test1 <- checkSpreadEquivalence(spreadn4t2a, spreadn4t2b))

# direct instantiation of a spread
spreadn4t2c <- array(NA, c(4,3,5))

spreadn4t2c[,1,1] <- c(0, 0, 0, 1)
spreadn4t2c[,2,1] <- c(0, 1, 1, 0)
spreadn4t2c[,3,1] <- c(0, 1, 1, 1)
spreadn4t2c[,1,2] <- c(0, 0, 1, 0)
spreadn4t2c[,2,2] <- c(1, 1, 0, 0)
spreadn4t2c[,3,2] <- c(1, 1, 1, 0)
spreadn4t2c[,1,3] <- c(0, 1, 0, 0)
spreadn4t2c[,2,3] <- c(1, 0, 1, 1)
spreadn4t2c[,3,3] <- c(1, 1, 1, 1)
spreadn4t2c[,1,4] <- c(1, 0, 0, 0)
spreadn4t2c[,2,4] <- c(0, 1, 0, 1)
spreadn4t2c[,3,4] <- c(1, 1, 0, 1)
spreadn4t2c[,1,5] <- c(0, 0, 1, 1)
spreadn4t2c[,2,5] <- c(1, 0, 1, 0)
spreadn4t2c[,3,5] <- c(1, 0, 0, 1)

(test2 <- checkSpreadEquivalence(spreadn4t2a, spreadn4t2c))

## Example 2: two equivalent 2-spreads of PG(5,2)
data(spreadn6t3a)

# permute the flats and flat order of spreadn6t3a to create a 
# second equivalent spread equiv_spreadn6t3a.

equiv_spreadn6t3a <- spreadn6t3a
dims <- dim(equiv_spreadn6t3a)
for(i in 1:(dims[3])){
   equiv_spreadn6t3a[,,i] <- equiv_spreadn6t3a[,sample(1:dims[2], dims[2]),i]
}
equiv_spreadn6t3a <- equiv_spreadn6t3a[,,sample(1:dims[3], dims[3])]

(test3 <- checkSpreadEquivalence(spreadn6t3a, equiv_spreadn6t3a))