Geodesic
Vectorial Boolean functions on distance one from APN functions
Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 36-37.

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The metric properties of the class of vectorial Boolean functions are studied. A vectorial Boolean function $F$ in $n$ variables is called a differential $\delta$-uniform function if the equation $F(x)\oplus F(x\oplus a)=b$ has at most $\delta$ solutions for any vectors $a,b$, where $a\neq0$. In particular, if it is true for $\delta=2$, then the function $f$ is called APN. The distance between vectorial Boolean functions $F$ and $G$ is the cardinality of the set $\{x\in\mathbb Z_2^n\colon F(x)\neq G(x)\}$. It is proved that there are only differential $4$-uniform functions which are on the distance 1 from an APN function.
Keywords: vectorial Boolean function, differentially $\delta$-uniform function, APN function. 
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     author = {G. I. Shushuev},
     title = {Vectorial {Boolean} functions on distance one from {APN} functions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {36--37},
     publisher = {mathdoc},
     number = {7},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2014_7_a14/}
}
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G. I. Shushuev. Vectorial Boolean functions on distance one from APN functions. Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 36-37. http://geodesic.mathdoc.fr/item/PDMA_2014_7_a14/

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