We prove a conjecture that classifies exceptional numbers. This conjecturearises in two different ways, from cryptography and from coding theory. An oddinteger $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (AlmostPerfect Nonlinear) over $\mathbb{F}_{2^n}$ for infinitely many values of $n$.Equivalently, $t$ is exceptional if the binary cyclic code of length $2^n-1$with two zeros $\omega, \omega^t$ has minimum distance 5 for infinitely manyvalues of $n$. The conjecture we prove states that every exceptional number hasthe form $2^i+1$ or $4^i-2^i+1$.
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机译：我们证明了一个对异常数进行分类的猜想。从密码学和编码理论以两种不同的方式进行推测。如果$ f（x）= x ^ t $是APN（AllowPerfect Nonlinear）超过$ \ mathbb {F} _ {{2 ^ n} $等效地，如果长度为$ 2 ^ n-1 $的二进制循环码具有两个零$ \ omega，\ omega ^ t $的无限多个$ n $值的最小距离为5，则$ t $是例外。我们证明的猜想表明，每个例外数的形式为$ 2 ^ i + 1 $或$ 4 ^ i-2 ^ i + 1 $。
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