checkSpreadIsomorphism function

Checking the Isomorphism of Two Spreads

This function checks the isomorphism of two (t-1)-spreads of PG(n-1,2). If they are isomorphic, it returns the list of isomorphism establishing collineations (IECs). The option is provided to enumerate all IECs or to terminate after the first one is found.

checkSpreadIsomorphism(spread1, spread2, returnfirstIEC = FALSE, printstatement = TRUE)

Arguments

• spread1: A (t-1)-spread of PG(n-1,2) stored as a three dimensional binary array (see Details and Examples of checkSpreadEquivalence).
• spread2: A (t-1)-spread of PG(n-1,2) stored as a three dimensional binary array (see Details and Examples of checkSpreadEquivalence).
• returnfirstIEC: An indicator to indicate whether all isomorphism establishing collineations should be returned (default), or terminate only after the first one is found.
• printstatement: If set to true (default), running the function also prints a sentence declaring the isomorphism of the spreads.

Details

This code considers all possible collineations of PG(n-1,2) to search for isomorphism establishing collineations (IECs) from spread1 to spread2. The search is conducted over the reduced space described in Algorithm 1 of Spencer et al. (2019). Equivalence is assessed using the bitstring comparison method described in Spencer et al. (2019).

Both input spreads should be formatted as 3-dimensional arrays with spread1[i,j,k] indicating whether or not the ith basic factor is present in the jth effect of the kth flat of spread1.

Returns

A list containing two objects. The first object is a Boolean indicating whether or not spread1 is isomorphic to spread2. If isomorphic, the second object is a list of isomorphism establishing collineation matrices. If not isomorphic, the second object is NA.

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

checkStarIsomorphism for checking the isomorphism of balanced covering stars.

checkSpreadEquivalence for checking the equivalence of balanced spreads.

Author(s)

Neil Spencer, Pritam Ranjan, Franklin Mendivil

Examples

## Example 1: two 1-spreads of PG(3,2)
data(spreadn4t2a)
data(spreadn4t2b)
# test their isomorphism
test1 <- checkSpreadIsomorphism(spreadn4t2a, spreadn4t2b)
test1$result # the test indicates that they are isomorphic
(IEC1 <- (test1$IECs)[[1]])
# we store the first isomorphism establishing collineation as IEC1

## Example 2: two 2-spreads of PG(5,2) using returnfirstIEC to cut down on runtime
data(spreadn6t3a)
data(spreadn6t3b)
test2 <- checkSpreadIsomorphism(spreadn6t3a, spreadn6t3b, returnfirstIEC = TRUE)
test2$result # the test indicates that they are isomorphic

## Example 3: non-isomorphic 1-spreads of PG(5,2)
data(spreadn6t2a)
data(spreadn6t2c)

# A bit slow for official example
# test3 <- checkSpreadIsomorphism(spreadn6t2a, spreadn6t2c, returnfirstIE#C = TRUE)
# test3$result

## Example 4: isomorphic 1-spreads of PG(5,2)
data(spreadn6t2a)
data(spreadn6t2b)
test4 <- checkSpreadIsomorphism(spreadn6t2a, spreadn6t2b, returnfirstIEC = TRUE)
test4$result # the test indicates that they are isomorphic