New constructions of $2$-to-$1$ mappings over $\gf_{2^n}$ and their applications to binary linear codes

The $2$-to-$1$ mapping over finite fields has a wide range of applications, including combinatorial mathematics and coding theory. Thus, constructions of $2$-to-$1$ mappings have attracted considerable attention recently. Based on summarizing the existing construction results of all $2$-to-$1$ mappings over finite fields with even characteristic, this article first applies the generalized switching method to the study of $2$-to-$1$ mappings, that is, to construct $2$-to-$1$ mappings over the finite field $\mathbb{F}_{q^l}$ with $F(x)=G(x)+{\rm Tr}_{q^l/q}(R(x))$, where $G$ is a monomial and $R$ is a monomial or binomial. Using the properties of Dickson polynomial theory and the complete characterization of low-degree equations, we construct a total of $16$ new classes of $2$-to-$1$ mappings, which are not QM-equivalent to any existing $2$-to-$1$ polynomials. Among these, $9$ classes are of the form $cx + {\rm Tr}_{q^l/q}(x^d)$, and $7$ classes have the form $cx + {\rm Tr}_{q^l/q}(x^{d_1} + x^{d_2})$. These new infinite classes explain most of numerical results by MAGMA under the conditions that $q=2^k$, $k>1$, $kl<14$ and $c \in \gf_{q^l}^*$. Finally, we construct some binary linear codes using the newly proposed $2$-to-$1$ mappings of the form $cx + {\rm Tr}_{q^l/q}(x^d)$. The weight distributions of these codes are also determined. Interestingly, our codes are self-orthogonal, minimal, and have few weights.