Weighing matrix

In mathematics, a weighing matrix W of order n and weight w is an n × n (0,1,-1)-matrix such that $WW^{T}=wI_n$, where $W^T$ is the transpose of $W$ and $I_n$ is the identity matrix of order $n$.

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:

• The rows of W are pairwise orthogonal (that is, every pair of rows you pick from W will be orthogonal). Similarly, the columns are pairwise orthogonal.
• Each row and each column of W has exactly w non-zero elements.
• $W^{T}W=wI$, since the definition means that $W^{-1} = w^{-1}W^{T}$, where $W^{-1}$ is the inverse of $W$.
• $\operatorname{det}(W)=\pm w^{n/2}$ where $\operatorname{det}(W)$ is the determinant of $W$.

Examples

Note that when weighing matrices are displayed, the symbol $-$ is used to represent -1. Here are two examples:

This is a W(2,2):

$\begin{pmatrix}1 & 1 \\ 1 & -\end{pmatrix}$

This is a W(7,4):

$\begin{pmatrix} 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & - & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & - & 0 & - & 0 & 1 \\ 1 & 0 & 0 & - & 0 & - & - \\ 0 & 1 & - & 0 & 0 & 1 & - \\ 0 & 1 & 0 & - & 1 & 0 & 1 \\ 0 & 0 & 1 & - & - & 1 & 0 \end{pmatrix}$

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.[1] However, very little has been done beyond this with exception to classifying circulant weighing matrices.[2][3]

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?

References

1. ^ M. Harada, A. Munemasa, On the classification of weighing matrices and self-orthogonal codes, 2011, http://arxiv.org/abs/1011.5382.
2. ^ Ang, Miin Huey, et al. "Study of proper circulant weighing matrices with weight 9." Discrete Mathematics 308.13 (2008): 2802-2809.
3. ^ Arasu, K. T., et al. "Determination of all possible orders of weight 16 circulant weighing matrices." Finite Fields and Their Applications 12.4 (2006): 498-538.