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

2025-07-11

AI Summary

This work addresses the construction of novel $2$-to-$1$ mappings over the finite field $mathbb{F}_{2^n}$ and their application to binary linear code design. We introduce a generalized switching method that integrates the trace function ${ m Tr}_{q^l/q}$ with monomial and binomial structures, systematically yielding 16 new families of $2$-to-$1$ mappings: nine of the form $cx + { m Tr}_{q^l/q}(x^d)$ and seven of the form $cx + { m Tr}_{q^l/q}(x^{d_1} + x^{d_2})$, all inequivalent to known quadratic-monomial (QM) classes and covering most known instances verified in MAGMA experiments. Leveraging these mappings, we construct several families of binary linear codes that are self-orthogonal, minimal, and low-weight, and fully determine their weight distributions. These results strengthen the theoretical connection between structured mappings and coding theory, while enhancing the constructibility and controllability of binary linear codes.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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 $F(x)=G(x)+{ m Tr}_{q^l/q}(R(x))$, 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 $cx + { m Tr}_{q^l/q}(x^d)$, and $7$ classes have the form $cx + { m Tr}_{q^l/q}(x^{d_1} + x^{d_2})$. 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 $cx + { m Tr}_{q^l/q}(x^d)$. The weight distributions of these codes are also determined. Interestingly, our codes are self-orthogonal, minimal, and have few weights.

Problem

• Construct new 2-to-1 mappings over finite fields
• Apply generalized switching method to 2-to-1 mappings
• Build binary linear codes using new 2-to-1 mappings

Innovation

• Generalized switching method for 2-to-1 mappings
• Dickson polynomial theory for low-degree equations
• New binary linear codes with self-orthogonal properties