A family of crooked functions

It has been proved in Bierbrauer and Kyureghyan (Des. Codes Cryptogr. 46:269–301, 2008) that a binomial function aXi + bX j can be crooked only if both exponents i, j have 2-weight  ≤2. In the present paper we give a brief construction for all known examples of crooked binomial functions. These consist of an infinite family and one sporadic example. The construction of the sporadic example uses the properties of an algebraic curve of genus 3. Computer experiments support the conjecture that each crooked binomial is equivalent either to a member of the family or to the sporadic example.