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Efficient Algorithms for Permutation Arrays from Permutation Polynomials.

Sergey Bereg1, Brian Malouf1, Linda Morales1

  • 1Department of Computer Science, University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083, USA.

Entropy (Basel, Switzerland)
|October 28, 2025
PubMed
Summary
This summary is machine-generated.

We developed new algorithms for computing permutation polynomials (PPs) for larger degrees and finite fields. This improves lower bounds for M(n,D), the maximum number of permutations on n symbols with a pairwise Hamming distance of D.

Keywords:
hamming distancepermutation Arrayspermutation polynomials

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Area of Science:

  • Discrete Mathematics
  • Abstract Algebra
  • Computer Science

Background:

  • Permutation polynomials (PPs) are fundamental in combinatorics and finite field theory.
  • Efficient computation of PPs is crucial for applications in cryptography and coding theory.
  • Existing methods for computing PPs become computationally intensive for large degrees and fields.

Purpose of the Study:

  • To develop novel algorithms for efficient computation of permutation polynomials.
  • To enhance the calculation of PPs for higher degrees and larger finite fields.
  • To improve lower bounds for M(n,D), quantifying permutations with specific Hamming distances.

Main Methods:

  • Utilized normalization techniques, including F-maps and G-maps.
  • Applied the Hermite criterion for PP verification.
  • Developed algorithms based on these methods for efficient PP computation.

Main Results:

  • Achieved more efficient computation of permutation polynomials.
  • Successfully applied the methods to larger degrees and finite fields.
  • Improved existing lower bounds for M(n,D).

Conclusions:

  • The developed algorithms offer a significant advancement in computing permutation polynomials.
  • These advancements enable more efficient analysis of permutation properties.
  • The improved bounds for M(n,D) have implications for combinatorial design and related fields.