applyCollineation function

Relabel a Spread or Star with a Collineation Matrix

This function relabels a balanced (t-1)-spread or a covering star $St(n, \mu, t, t_0)$ of PG(n-1,2) according to the specified collineation matrix.

applyCollineation(C, spr)

Arguments

• C: A binary n by n matrix representing a collineation of PG(n-1,2).
• spr: A balanced spread or star of PG(n-1,2) stored as a three dimensional binary array (see Details and Examples of checkSpreadEquivalence).

Details

This code applies the relabelling corresponding to a collineation matrix C to any given balanced spread or star of PG(n-1, 2). The spread should be formatted as a 3-dimensional array with spr[i,j,k] indicating whether or not the ith basic factor is present in the jth effect of the kth flat of spr. The collineation is applied via a matrix multiplication modulo 2 (i.e., the calculations are done over GF(2)). See Spencer et al. (2019) for details.

Returns

A spread or star of the same dimensions as spr.

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 balanced spreads.

checkStarIsomorphism for checking the isomorphism of balanced covering stars.

Author(s)

Neil Spencer, Pritam Ranjan, Franklin Mendivil

Examples

## Example 1: relabelling a 1-spread of PG(3,2)
data(spreadn4t2a)
Collin <- cbind(c(1,0,0,1), c(0,0,1,1), c(1,1,1,1), c(0,1,1,1))
# Collin is the collineation matrix corresponding to
# A -> AD, B -> CD, C -> ABCD, D -> BCD
applyCollineation(Collin, spreadn4t2a)


## Example 2: Relabelling a star of PG(4,2) consisting of 4-flats.
data(starn5t3a)
Collin2 <- cbind(c(0,0,0,0,1), c(1,0,0,0,0), c(0,1,0,0,0), c(0,0,0,1,0), c(0,0,1,0,0))
# Collin2 is the collineation matrix corresponding to
# A -> E, B -> A, C -> B, D -> D, E -> C
applyCollineation(Collin2, starn5t3a)