Vectorial bent functions and linear codes from quadratic forms

Abstract

In this paper, we study the vectorial bentness of an arbitrary quadratic form and construct two classes of linear codes of few weights from the quadratic forms. Let q be a prime power, m be a positive integer and $Q:\mathbb {F}_{q^m}\rightarrow \mathbb {F}_q$ be a quadratic form. We first show that Q is a vectorial bent function if and only if Q is non-degenerate and $(q+1)m$ is even (i.e. either q is odd or m is even). Furthermore, if $2\mid (q+1)m$ and $Q(x)= \sum \limits _{i=0}^{m-1} \textrm{Tr}_{q^m/q}(a_i x^{q^i+1})\ (a_i\ne 0)$, we show that Q is vectorial bent if and only if the associated additive polynomial $L_Q(x)=\sum _i (a_i + a_{m-i}^{q^{i}}) x^{q^i}$ is a permutation polynomial over $\mathbb {F}_{q^m}$. If there is only one $a_i\ne 0$, we recover the constructions of Sidelnikov, Dembowski-Ostrom and Kasami of quadratic vectorial bent functions. We then construct two classes of linear codes $\mathcal {C}'_Q$ and $\mathcal {C}_Q$ over $\mathbb {F}_q$ from Q and completely determine the weight distributions of our codes, showing that they are two-, three- or four-weight codes and contain optimal codes satisfying the Griesmer and Singleton bounds.