Weighing matrix

In mathematics, a weighing matrix "W" of order "n" and weight "w" is an "n" × "n" (0,1,−1)-matrix such that formula_1, where formula_2 is the transpose of formula_3 and formula_4 is the identity matrix of order formula_5.
For convenience, a weighing matrix of order "n" and weight "w" is often denoted by "W"("n","w"). A "W"("n","n") is a Hadamard matrix and a "W"("n","n"−1) is equivalent to a conference matrix.
Properties.
Some properties are immediate from the definition. If "W" is a "W"("n","w"), then:
Examples.
Note that when weighing matrices are displayed, the symbol formula_10 is used to represent −1. Here are two examples:
This is a "W"(2,2):
This is a "W"("7","4"):
Equivalence.
Two weighing matrices are considered to be equivalent if one can be obtained from the other by a series of permutations and negations of the rows and columns of the matrix. The classification of weighing matrices is complete for cases where "w" ≤ 5 as well as all cases where "n" ≤ 15 are also completed. However, very little has been done beyond this with exception to classifying circulant weighing matrices.
Open Questions.
There are many open questions about weighing matrices. The main question about weighing matrices is their existence: for which values of "n" and "w" does there exist a "W"("n","w")? A great deal about this is unknown. An equally important but often overlooked question about weighing matrices is their enumeration: for a given "n" and "w", how many "W"("n","w")'s are there?
This question has two different meanings. Enumerating up to equivalence and enumerating different matrices with same n,k parameters. Some papers were published on the first question but none were published on the second important question.