# Reference request: about “SNF” (Smith Normal Form)

I've read about some studies on the Paley I Construction. Among them I found the following notations ( See this page: https://documents.uow.edu.au/~jennie/matrices/32P02.html ).

$$SNF:1,2^a,4^{b},8^{b},16^a,32$$

I am not familiar with these notations. Could you please recommend me some books or articles about it? Thanks in advance!

• Smith Normal Form, my first guess, as in the answer. Apr 21 at 14:15
• If you want the very basics about some topic in math, Wikipedia is always a good place to check: en.wikipedia.org/wiki/Smith_normal_form Apr 21 at 15:36
• To complete the guess of @Will Jagy, in your example the SNF is given by the diagonal matrix $\operatorname{diag}(1,2^a,4^b,8^b,16^a,32,0,\ldots)$ for adequate non-negative integers $a,b$, where for each diagonal entry $d_i$ divides $d_{i+1}$. You can find additional details on the same page where you found the example: documents.uow.edu.au/~jennie/WEBPDF/1996_08.pdf Apr 21 at 17:18
• Brouwer and Haemers' "Spectra of Graphs" has a chapter about p-ranks, it is Chapter 13. In the last section of that chapter they discuss SNFs. Not the most basic reference but I think more in line with the type of SNF you are interested in, as a graph invariant. win.tue.nl/~aeb/2WF05/spectra.pdf By the way, the exponents a, b in your example are meant to indicate the multiplicity of the entry. So "2" occurs a times.
– Josh
Apr 22 at 19:24