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Orthomorphism Polynomials of degree $7$ over finite fields

Building upon Xiang Fan's 2019 classification of degree 7 permutation polynomials, this paper determines the complete list of degree 7 orthomorphism polynomials over finite fields of specific orders while also establishing their non-existence in certain other cases.

Original authors: Bhitali Kousik, Dhiren Kumar Basnet

Published 2026-01-30

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CC BY 4.0

Original authors: Bhitali Kousik, Dhiren Kumar Basnet

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect working with a very specific set of building blocks. These blocks live in a world called a "finite field," which is like a tiny, self-contained universe with a fixed number of elements (let's say 11, 13, or 17). In this universe, you have a special rule: you can only use numbers that exist within that specific universe.

The paper you are asking about is a detailed map created by two architects, Kousik Bhitali and Dhiren Kumar Basnet. They are trying to find a very specific type of structure called an Orthomorphism Polynomial.

Here is the simple breakdown of what they did and what they found:

The Two Rules of the Game

To build an "Orthomorphism" structure, you must follow two strict rules simultaneously:

Rule A: If you take your structure and plug in every single number from your universe, the output must be a perfect shuffle. Every number comes out exactly once, with no duplicates and no missing pieces. (In math terms, this is a Permutation Polynomial).
Rule B: If you take your structure, subtract the original number you put in, and look at the result, that new result must also be a perfect shuffle of the universe.

If a structure passes both tests, it's a winner. If it fails even one, it's a loser.

The Challenge: The "Degree 7" Tower

The authors were specifically interested in building towers of a specific height: Degree 7. Think of this as a polynomial equation where the highest power of $x$ is 7 (like $x^7 + \dots$ ).

They wanted to know: "In which of these tiny universes (finite fields) can we actually build a Degree 7 tower that passes both Rule A and Rule B?"

The Detective Work

The authors didn't just guess. They used a mix of mathematical logic and computer power (specifically a program called SageMath) to act as detectives.

The Filter: They knew from previous research that most Degree 7 towers are "impossible" to build in most universes. They had a list of "candidate" universes to check.
The Transformation: They realized that many towers look different but are actually the same structure just rotated or flipped (mathematically called "linear transformations"). Instead of checking every single variation, they checked the "base" versions and then figured out how many variations existed.
The Test: For each candidate, they ran the two rules. They asked the computer: "Does this polynomial shuffle the numbers? And does the polynomial minus $x$ also shuffle the numbers?"

The Big Discovery

After checking many universes, they found a very specific pattern.

The "Yes" List:
Orthomorphism polynomials of degree 7 only exist in universes with these specific sizes:

11, 13, 17, 19, and 25.

For these sizes, they didn't just say "yes"; they wrote down the exact blueprints for every single possible winning structure.

For the universe of size 13, they found exactly 6,422 unique winning structures.
For the universe of size 11, they found 7,260.
For size 25, they found a massive 60,000.

The "No" List:
They also proved that in many other universes, it is impossible to build such a structure.

If the universe size is 23 or 31, no such structure exists.
If the universe size is 49 (which is $7 \times 7$ ), structures do exist, but they are "special" or "exceptional" (meaning they behave differently than the standard ones found in the smaller fields).
If the universe size is 27 or larger multiples of 7, they found no standard structures.

The "Exceptional" vs. "Non-Exceptional"

The paper makes a distinction between two types of winners:

Non-Exceptional: These are the "standard" winners found in the sizes 11, 13, 17, 19, and 25. These are the main focus of the paper.
Exceptional: These are rare, special cases that appear in larger fields (like 49) or specific conditions. The authors mapped these out too, noting that in fields like $F_{49}$ , all degree 7 orthomorphisms are of this "exceptional" type.

The Conclusion

In simple terms, this paper is a complete catalog. It tells us exactly where we can find these special mathematical shuffling structures of degree 7 and what they look like.

If you are in a universe of size 11, 13, 17, 19, or 25: You can build them, and here is the list of every single one.
If you are in a universe of size 23, 31, or 27: You cannot build them (at least not the standard kind).
If you are in a universe of size 49: You can build them, but they are all "exceptional" types, and there are nearly 4 million of them.

The authors have essentially closed the book on this specific puzzle for all the most common finite fields, providing a definitive answer to the question: "Where do these specific mathematical shuffles exist?"

Technical Summary: Orthomorphism Polynomials of Degree 7 Over Finite Fields

Problem Statement
The paper addresses the classification of orthomorphism polynomials (OPs) of degree 7 over finite fields $\mathbb{F}_q$ . A polynomial $f \in \mathbb{F}_q[x]$ is defined as an orthomorphism polynomial if both $f(x)$ and $f(x) - x$ are permutation polynomials (PPs) over $\mathbb{F}_q$ . While previous works by Neiderreiter, Robinson, Shallue, and Wanless had classified complete permutation polynomials (CPPs) and OPs for degrees less than 7 and specific degree 6 cases, the explicit forms and complete enumeration of degree 7 OPs remained an open problem for various field orders. The authors aim to determine the complete list of non-exceptional OPs of degree 7 and provide a partial characterization of exceptional ones.

Methodology
The authors employ a combination of theoretical classification results and computational enumeration:

Reduction via Linear Transformations: The study relies on the work of Xiang Fan [2], which establishes that any degree 7 PP over $\mathbb{F}_q$ is linearly related to a normalized form $x^7 + \sum_{i=1}^5 f_i x^i$ . Two polynomials are "linearly related" if one can be transformed into the other via $g(x) = af(bx+c)+d$.
Distinction Between PPs and OPs: A critical methodological challenge noted is that while linear transformations preserve the property of being a PP, they do not necessarily preserve the property of being an OP. Therefore, the authors cannot simply check if a normalized PP is an OP; they must determine the specific scaling factors $\alpha$ and $\beta$ such that $\alpha f(\beta x)$ (and its shifts) satisfies the OP condition.
Systematic Enumeration:
- The authors utilize Lemma 2 [2] to identify all non-exceptional PPs of degree 7 for specific field orders ( $q \in \{11, 13, 17, 19, 23, 25, 31\}$ ).
- For each candidate polynomial family, they construct systems of equations derived from the condition that $h(x) = \alpha f(\beta x) - x$ must also be a PP (and thus linearly related to one of the known normalized forms).
- These systems are solved using the computer algebra system SageMath to find valid pairs $(\alpha, \beta) \in (\mathbb{F}_q^*)^2$ .
- Lemma 3 [5] is used to generate the full set of OPs from valid base polynomials via translations $f(x+\gamma) + \delta$ .
- Lemma 4 ensures that distinct pairs $(\alpha, \beta)$ yield distinct polynomials, allowing for an exact count.

Key Contributions and Results

Existence and Classification for Non-Exceptional OPs:
The paper proves that a non-exceptional orthomorphism polynomial of degree 7 exists over $\mathbb{F}_q$ (for $q > 7$ ) if and only if $q \in \{11, 13, 17, 19, 25\}$ .
- Explicit Lists: The authors provide explicit forms of these polynomials for each valid $q$ , parameterized by the pairs $(\alpha, \beta)$ .
- Exact Counts:
  - $\mathbb{F}_{11}$ : 7,260 OPs (20 exceptional).
  - $\mathbb{F}_{13}$ : 6,422 OPs (4 exceptional).
  - $\mathbb{F}_{17}$ : 4,624 OPs (all non-exceptional).
  - $\mathbb{F}_{19}$ : 4,332 OPs (4 exceptional).
  - $\mathbb{F}_{25}$ : 60,000 OPs (20 exceptional).
- These counts for $q=11, 13, 17$ align perfectly with the data provided by Shallue and Wanless [7].
Non-Existence Results:
The study establishes the non-existence of degree 7 OPs over $\mathbb{F}_{23}$ and $\mathbb{F}_{31}$ . Furthermore, it confirms that no OPs exist for fields where $q \equiv 6 \pmod 7$ (except $q=13$ ), extending previous results on characteristic 2 fields.
Exceptional Polynomials:
The paper characterizes exceptional OPs (those that are PPs over infinitely many extensions).
- For $\mathbb{F}_{49}$ (and generally $\mathbb{F}_{7^t}, t \ge 2$ ), all degree 7 OPs are exceptional.
- The authors provide an explicit characterization for $\mathbb{F}_{49}$ , noting a total of 3,937,640 such polynomials.
- It is shown that for $\mathbb{F}_{27}$ , no OPs of degree 7 exist.

Significance
The paper claims to complete the classification of non-exceptional orthomorphism polynomials of degree 7 for all finite fields where such polynomials exist. By verifying the counts against existing data for smaller fields and extending the analysis to $q=19, 25, 49$ , the authors fill gaps in the literature regarding degree 7 OPs. The work provides a definitive list of explicit forms for these polynomials, which are relevant to design theory, particularly in the construction of Latin squares. The authors note that future work would involve examining fields $\mathbb{F}_{7^r}$ ( $r \ge 3$ ) and fields with $q > 31$ where $q \not\equiv 0, 1, 6 \pmod 7$ , though they anticipate only exceptional polynomials will be found in those cases.