Abstract: Isomorphism examination determines whether two design matrices are equivalent subject to some row, column, and level permutations. The purpose of this paper is to study the isomorphism problem from the viewpoint of the count vector. We find that two designs are isomorphic if and only if there exists a special type of linear transformation between their count vectors. The transformation can be characterized in terms of set operations for the subscripts of elements in the count vector. Besides, we propose an initial screening method based on the count vector, called the split-count matrix. We prove that the split-count matrix is more efficient than most existing initial screening methods. Some modified versions of the split-count matrix, including a projection version and some simplified versions, are discussed. Some examples and comparisons are given to demonstrate the power of the split-count matrix.
Key words and phrases: Generalized word length pattern, Hamming distance, indicator function, J-characteristics, power moment, projection, set operations, split-count matrix, squared centered L₂.