On many-to-one property of generalized cyclotomic mappings

2025/03/09 by Zheng, Yanbin, Zhang, Yang, Zha, Zhengbang +2
#FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2503.06654

Abstract

The generalized cyclotomic mappings over finite fields \mathbbF_q are those mappings which induce monomial functions on all cosets of an index ℓ subgroup C₀ of the multiplicative group \mathbbF_q^*. Previous research has focused on the one-to-one property, the functional graphs, and their applications in constructing linear codes and bent functions. In this paper, we devote to study the many-to-one property of these mappings. We completely characterize many-to-one generalized cyclotomic mappings for 1 ≤ ℓ ≤ 3. Moreover, we completely classify 2-to-1 generalized cyclotomic mappings for any divisor ℓ of q-1. In addition, we construct several classes of many-to-one binomials and trinomials of the form x^r h(x^q-1) on \mathbbF_q², where h(x)^q-1 induces monomial functions on the cosets of a subgroup of U_q+1.

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