Equivalence Relations for Computing Permutation Polynomials

TL;DR

This paper introduces a novel method using equivalence relations and normalization to efficiently compute and classify permutation polynomials over finite fields, enabling extensive enumeration and new bounds on permutation sets.

Contribution

The paper develops a new normalization-based technique for computing permutation polynomials, reducing search space and classifying PPs into equivalence classes for arbitrary finite fields and degrees.

Findings

Computed almost all PPs of degree ≤10 over GF(q) for q ≤ 97

Enumerated representative PPs for various degrees and fields

Derived new lower bounds for maximum permutation sets with given Hamming distance

Abstract

We present a new technique for computing permutation polynomials based on equivalence relations. The equivalence relations are defined by expanded normalization operations and new functions that map permutation polynomials (PPs) to other PPs. Our expanded normalization applies to almost all PPs, including when the characteristic of the finite field divides the degree of the polynomial. The equivalence relations make it possible to reduce the size of the space, when doing an exhaustive search. As a result, we have been able to compute almost all permutation polynomials of degree $d$ at most 10 over $GF(q)$, where $q$ is at most 97. We have also been able to compute nPPs of degrees 11 and 12 in a few cases. The techniques apply to arbitrary $q$ and $d$. In addition, the equivalence relations allow the set all PPs for a given degree and a given field $GF(q)$ to be succinctly described by their representative nPPs. We give several tables at the end of the paper listing the representative nPPs (\ie the equivalence classes) for several values of $q$ and $d$. We also give several new lower bounds for $M(n,D)$, the maximum number of permutations on $n$ symbols with pairwise Hamming distance $D$, mostly derived from our results on PPs.