Orthomorphism Polynomials of degree $7$ over finite fields

TL;DR

The paper addresses the explicit classification of orthomorphism polynomials of degree $7$ over finite fields by leveraging Xiang Fan's degree $7$ permutation polynomial results, together with linear-relatedness and computational enumeration. It delivers complete lists of non-exceptional and exceptional degree-$7$ OPs for select fields, with precise counts (e.g., $7260$ OPs over $\mathbb{F}_{11}$ and $4624$ over $\mathbb{F}_{17}$) and nonexistence results for others (e.g., $\mathbb{F}_{23}$, $\mathbb{F}_{31}$). A central finding is that non-exceptional OPs of degree $7$ exist if and only if $q \in \{11,13,17,19,25\}$, while all OPs over $\mathbb{F}_{49}$ are exceptional and numerous in number; the work also confirms consistency with Shallue–Wanless data and extends the classification to additional fields. The results advance explicit understanding of OPs, contributing to the broader design-theoretic and permutation-polynomial literature and outlining directions to complete the remaining cases, notably for fields $\mathbb{F}_{7^r}$ with $r\ge 3$.

Abstract

In 2019, Xiang Fan \cite{xfan} classified all permutation polynomials of degree $7$ over finite fields of odd characteristics. In this paper, we use this classification to determine the complete list of degree $7$ orthomorphism polynomials over finite fields of order $q\in\{11,~13,~17,~19,~25,~49\}.$ In addition, the non-existence of these polynomials is established for certain fields.

Theorems & Definitions (18)

• Theorem 1.1
• Lemma 1
• Lemma 2
• Remark 1
• Remark 2
• Lemma 3
• Lemma 4
• proof
• Theorem 3.1
• Corollary 3.2