  We show that the there exists an infinite family of APN functions of the form
$F(x)=x^{2^{s}+1} + x^{2^{k+s}+2^k} + cx^{2^{k+s}+1} + c^{2^k}x^{2^k + 2^s} +
\delta x^{2^{k}+1}$, over $\gf_{2^{2k}}$, where $k$ is an even integer and
$\gcd(2k,s)=1, 3\nmid k$. This is actually a proposed APN family of Lilya
Budaghyan and Claude Carlet who show in \cite{carlet-1} that the function is
APN when there exists $c$ such that the polynomial
$y^{2^s+1}+cy^{2^s}+c^{2^k}y+1=0$ has no solutions in the field $\gf_{2^{2k}}$.
In \cite{carlet-1} they demonstrate by computer that such elements $c$ can be
found over many fields, particularly when the degree of the field is not
divisible by 3. We show that such $c$ exists when $k$ is even and $3\nmid k$
(and demonstrate why the $k$ odd case only re-describes an existing family of
APN functions). The form of these coefficients is given so that we may write
the infinite family of APN functions.
