A new upper bound for Sidon sets in ${\mathbb{F}}_2^{4k+3}$

Abstract

A subset $S\subseteq {\mathbb{F}}_2^n$ is called a Sidon set if no four distinct points of $S$ have zero sum. It is shown that if $n\ge 7$ and $n\equiv 3\mathrm{mod}4$, then $|S|\le 2^{\frac{n+1}{2}}-3$. As a consequence, for any even $t\ge 4$, there does not exist a binary linear $[2^t-3,2^t-2t-2,5]$-code, strengthening a nonexistence result of Brouwer and Tolhuizen from 1993. As a second consequence, we also show that for an even integer $n\ge 4$, every almost perfect nonlinear (APN) function $F:{\mathbb{F}}_2^n\to {\mathbb{F}}_2^n$ has nonlinearity at least $3$.

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