A Family of Binary Sequences with Optimal Correlation Property and Large Linear Span

A family of binary sequences is presented and proved to have optimal correlation property and large linear span. It includes the small set of Kasami sequences, No sequence set and TN sequence set as special cases. An explicit lower bound expression on the linear span of sequences in the family is given. With suitable choices of parameters, it is proved that the family has exponentially larger linear spans than both No sequences and TN sequences. A class of ideal autocorrelation sequences is also constructed and proved to have large linear span.Comment: 21 page

A Family of pp-ary Binomial Bent Functions

For a prime $p$ with $p\equiv 3\,({\rm mod}\, 4)$ and an odd number $m$, the Bentness of the $p$-ary binomial function $f_{a,b}(x)={\rm Tr}_{1}^n(ax^{p^m-1})+{\rm Tr}_{1}^2(bx^{\frac{p^n-1}{4}})$ is characterized, where $n=2m$, a\in \bF_{p^n}^*, and b\in \bF_{p^2}^*. The necessary and sufficient conditions of $f_{a,b}(x)$ being Bent are established respectively by an exponential sum and two sequences related to $a$ and $b$. For the special case of $p=3$, we further characterize the Bentness of the ternary function $f_{a,b}(x)$ by the Hamming weight of a sequence