ILLUSTRATION

Let $L$ be an extension of degree $m$ of${\bf F}_2$
let $a\in{L}^\times$
let$d$ be an integer, and consider the boolean function

$$f_d(x)={\rm tr}_{{L}/{{\bf F}_2}} (x^ d)$$
 

    If $m=2t+1$ is odd the $$\min_{d} A(f_d) = 2^{t+1}$$

 

    If $m=2t$ is even

$$\min_{d} A(f_d) \leq 2^{t+1}$$
 

The Fourier tansform of $f_d$ is given by

$$\begin{split}\widehat{f_\chi}(a) & = 1 + \sum_{x\in{L}^\time......_L(\chi,\mu_L)G_L(\bar\chi^d,\mu_L)\bar\chi^d(a)\end{split}$$

where $\mu_L(x)$ is the canonical additive character of $L$. Note that by Fourier inversion :

$$\mu_L(x)=\frac 1{q-1} \sum_{\chi\in\widehat{L^{\times}}}G_L(\chi,\mu_L)\bar\chi(x)$$
 
  $$- \widehat{f}(a) \equiv \sum_{\chi\not = 1} G_L(\chi,\mu_L)G_L(\bar\chi^d,\mu_L)\bar\chi^d(a)\bmod q$$
 

    We isolate the parameters

$$w_d = \min_{j} S(j)+S(-dj), \qquad J_d=\{j\mid S(j)+S(-dj)=w\}$$
 
  $$\widehat{f_\chi}(a) = 2^w\big[\sum_{j\in J_d} \bar\omega^{jn}(a)\big] \pmod{2^{w_d}\wp}$$

For all $a$,

$${\rm ord}_2\big(\widehat{f_\chi}(a)\big) \geq w_d$$

Moreover, $P(X)=\sum_{j\in J} X^{dj}$ is an idempotent whose weight is equal to the number of $a$ such that equality occurs...

[PL, unpub.] Assume $m$ odd and $w_d=t+1$. If the number of roots of $P_d(X)$ is less than $2^{m-1}$ then $A(f_d) = 2^{t+1}$.

 

    open problem : weight distribution of

  $$\begin{split}\{0,1\}^m & \to \{0,1\}^{2m}\\x & \mapsto [x, -dx\mod n]\end{split}$$