The strange story of ternary bent functions in 2 variables

Consider a function f:GF(3)^2\to GF(3).

Such functions are often studied in terms of their Walsh(-Hadamard) transforms – a complex-valued function on V=GF(3)^2 that can be defined as

W_f(u) =  \sum_{x \in V}  \zeta^{f(x)- \langle u,x\rangle},
where \zeta = e^{2\pi i/3}. We call f bent if
|W_f(u)|=3^{n/2},
for all u\in V (here n=2). Such ternary (or 3-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 f(-x)=f(x)) and f(\vec{0})=0.

Thanks to a Sage computation, there are precisely 18 such functions.

\begin{array}{cccccccccc}  {\rm bents}\backslash GF(3)^2 & (0, 0) & (1, 0) & (2, 0) & (0, 1) & (1, 1) & (2, 1) & (0,  2) & (1, 2) & (2, 2) \\ \hline  b1 & 0 & 1 & 1 & 1 & 2 & 2 & 1 & 2 & 2 \\  b2 & 0 & 2 & 2 & 1 & 0 & 0 & 1 & 0 & 0 \\  b3 & 0 & 1 & 1 & 2 & 0 & 0 & 2 & 0 & 0 \\  b4 & 0 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 1 \\  b5 & 0 & 0 & 0 & 2 & 1 & 0 & 2 & 0 & 1 \\  b6 & 0 & 1 & 1 & 0 & 2 & 0 & 0 & 0 & 2 \\  b7 & 0 & 0 & 0 & 1 & 2 & 0 & 1 & 0 & 2 \\  b8 & 0 & 2 & 2 & 0 & 0 & 1 & 0 & 1 & 0 \\  b9 & 0 & 0 & 0 & 2 & 0 & 1 & 2 & 1 & 0 \\  b10 & 0 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 1 \\  b11 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 2 \\  b12 & 0 & 2 & 2 & 1 & 2 & 1 & 1 & 1 & 2 \\  b13 & 0 & 1 & 1 & 2 & 2 & 1 & 2 & 1 & 2 \\  b14 & 0 & 1 & 1 & 0 & 0 & 2 & 0 & 2 & 0 \\  b15 & 0 & 0 & 0 & 1 & 0 & 2 & 1 & 2 & 0 \\  b16 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 2 & 1 \\  b17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  b18 & 0 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 1 \\  \end{array}

The (edge-weighted) Cayley graph of the second function b_2 is shown here (thanks to Sage):

cayley graph GF3 bent

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 4 with 9 vertices, and this is it!

Here are some relationships they satisfy:

b_1=-b_{10}, \ \  b_2=-b_3, \ \  b_4 = -b_6, \ \  b_5 = -b_7, \ \  b_8=-b_{14},

b_9=-b_{15}, \ \  b_{11}=-b_{16}, \ \  b_{12}=-b_{18}, \ \  b_{13}=-b_{17},

b_1 = b_7+b_{14} = b_6+b_{15}, \ \  b_{10} = b_{4}+b_{9} = b_{5}+b_{8}, \ \  b_{12} = b_{2}+b_{11} = b_{7}+b_{8},

b_{13} = b_{3}+b_{11} = b_{6}+b_{9}, \ \  b_{17} = b_{2}+b_{16} = b_{4}+b_{15,}\ \  b_{18} = b_{3}+b_{16} = b_{5}+b_{14}.

Their supports are given as follows:

\{1,2,3,6\} = supp(b_2)=supp(b_3), \ \  \{1,2,4,8\} = supp(b_4)=supp(b_6),

\{1,2,5,7\} = supp(b_8)=supp(b_{14}), \ \  \{3,5,6,7\} = supp(b_9)=supp(b_{15}),

\{3,4,6,8\} = supp(b_{5})=supp(b_{7}),\ \  \{4,5,7,8\} = supp(b_{11})=supp(b_{16}),

\{1,2,3,4,5,6,7,8\} = supp(b_{1})=supp(b_{10})  = supp(b_{12})=supp(b_{13})  = supp(b_{17})=supp(b_{18}).

Note that all these functions are weight 4 or weight 8.

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

S = \{ \emptyset \} \cup \{ {\rm supp}(f)\ |\ f:GF(3)^2\to GF(3), \ f(0)=0,\ f\ {\rm bent} \},
then S forms a group under the symmetric difference operator \Delta! In fact, S\cong GF(2)^3.

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.

3 thoughts on “The strange story of ternary bent functions in 2 variables

  1. Regarding b_2, 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.

  2. Pingback: More odd examples of p-ary bent functions | Yet Another Mathblog

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