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.

A particularly simple puzzles on round pegs

I just thought of this simple (at least I think it is simple) puzzle.

Consider a long loop of string and a number (at least 2) of round pegs of radius 1 inch each, parallel to each other. Drape the string around the pegs and pull the pegs so that the string is tight, as in the picture (which has only 3 pegs). Notice some sections of the string are straight and some are curved (shown in red in the picture).

Why is the total length of the curved sections equal to 2\pi?

2012-12-23-belt-loop_reddish

This is related to a pulley puzzle of Harry Langman, published in 1949.