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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 $\\gf_{2^{2k}}$. 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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 $\\gf_{2^{2k}}$. 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