New constructions of 2-to-1 mappings over \gf_2ⁿ and their applications to binary linear codes

2025/07/11 by Yaqin Li, Li, Yaqin, Kangquan Li +3
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2507.08315

openalex publication_date 2025/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbF_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^d₁ + x^d₂). These new infinite classes explain most of numerical results by MAGMA under the conditions that q=2^k, k>1, kl<14 and c ∈ \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.

Citations

• Balanced Boolean functions with few-valued Walsh spectra parameterized by P(x²+x)
• On many-to-one mappings over finite fields
• Low c-differentially uniform functions via an extension of Dillon's switching method
• Two-to-one mappings and involutions without fixed points over \bF_2ⁿ
• New PcN and APcN functions over finite fields
• Binary linear codes with few weights from two-to-one functions
• On two-to-one mappings over finite fields
• Constructing permutations of finite fields via linear translators

Related

• Further study of 2-to-1 mappings over \mathbbF_2ⁿ
• On two-to-one mappings over finite fields
• On many-to-one mappings over finite fields
• Large class of many-to-one mappings over quadratic extension of finite fields
• Linear Codes over \mathbbF_q[x]/(x²) and GR(p²,m) Reaching the Griesmer Bound