Weighing matrix

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

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.
• ${\displaystyle W^{T}W=wI}$, since the definition means that ${\displaystyle W^{-1}=w^{-1}W^{T}}$, where ${\displaystyle W^{-1}}$ is the inverse of ${\displaystyle W}$.
• ${\displaystyle \operatorname {det} (W)=\pm w^{n/2}}$ where ${\displaystyle \operatorname {det} (W)}$ is the determinant of ${\displaystyle W}$.

Examples

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

This is a W(2,2):

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

This is a W(7,4):

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

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?

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.

References

1. Harada, Masaaki; Munemasa, Akihiro (2012). "On the classification of weighing matrices and self-orthogonal codes". J. Combin. Designs. 20: 40–57. arXiv:1011.5382. doi:10.1002/jcd.20295. S2CID 1004492.
2. Ang, Miin Huey; Arasu, K.T.; Lun Ma, Siu; Strassler, Yoseph (2008). "Study of proper circulant weighing matrices with weight 9". Discrete Mathematics. 308 (13): 2802–2809. doi:10.1016/j.disc.2004.12.029.
3. Arasu, K.T.; Hin Leung, Ka; Lun Ma, Siu; Nabavi, Ali; Ray-Chaudhuri, D.K. (2006). "Determination of all possible orders of weight 16 circulant weighing matrices". Finite Fields and Their Applications. 12 (4): 498–538. doi:10.1016/j.ffa.2005.06.009.