turkmath.org

Bent functions from a vector space $V_n$ over $F_2$ of even dimension $n= 2m$ into the cyclic group $Z_{2^k}$, or equivalently, relative difference sets in $V_n × Z_{2^k} $ with forbidden subgroup $Z_{2^k}$, can be obtained from spreads of $V_n$ for any $ k≤n/2$. In this talk we show the existence of bent functions from $V_n$ into $Z_{2^k}$,$k≥3$, which do not come from the spread construction. We present a construction of bent functions from $V_n into Z_{2^k}$ $≤n/6$, (and more general,into any abelian group of order $2^k$) obtained from partitions of $F_{2^m}×F_{2^m}$,which can be seen as a generalization of the Desarguesian spread. As for the spreads, the union of a certain fixed number of sets of these partitionsis always the support of a Boolean bent function