Consider a function .

Such functions are often studied in terms of their *Walsh(-Hadamard) transforms* – a complex-valued function on that can be defined as

where . We call *bent* if

for all (here ). Such ternary (or -ary) bent functions have been studied by a number of authors. Here we shall only consider those ternary bent functions of two variables which are *even* (in the sense ) and .

Thanks to a Sage computation, there are precisely such functions.

The (edge-weighted) Cayley graph of the second function is shown here (thanks to Sage):

cayley graph GF3 bent

When I showed this (actually, not just this graph but the print-outs of most of these Cayley graphs) to my colleague T.S. Michael, he observed that there is only one (up to iso) strongly regular (unweighted) graph of degree with vertices, and this is it!

Here are some relationships they satisfy:

Their supports are given as follows:

Note that all these functions are weight or weight .

In fact, my colleague David Phillips and I found that if you consider their set of supports (together with the emptyset ),

then forms a group under the symmetric difference operator ! In fact, .

I told you this was strange!

The Sage code for this is rather involved but it can be found here: hadamard_transform.sage.

[1] Sage, mathematics software, sagemath.com.

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*Related*

Regarding , a nicer/simpler picture would be to put the vertices onto a 3×3 grid, and then the maximum cliques containing the edges of the graph correspond to vertical and horisontal lines. Moreover, edge weights 1 (resp. 2) correspond to vertical (resp. horisontal) lines.

Indeed, TS mentioned this. I don’t know of a way that Sage can plot this. Do you?

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