Using permutation rational functions to obtain permutation arrays with large hamming distance

• Published in: Designs, codes, and cryptography Vol. 90; no. 7; pp. 1659 - 1677
• Main Authors: Bereg, Sergey, Malouf, Brian, Morales, Linda, Stanley, Thomas, Sudborough, I. Hal
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2022; Springer Nature B.V
• Subjects: Coding and Information Theory; Computer Science; Cryptology; Discrete Mathematics in Computer Science; Error correcting codes; Error correction; Fields (mathematics); Lower bounds; Permutations; Polynomials; Rational functions; Hamming distance; Permutation rational functions; Permutation codes; Permutation polynomials; 94B60; Permutation arrays; 05A05; 94B65
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-022-01039-x

We consider permutation rational functions ( PRFs ), V ( x )/ U ( x ), where both V ( x ) and U ( x ) are polynomials over a finite field F q . Permutation rational functions have been the subject of several recent papers. Let M ( n ,  D ) denote the maximum number of permutations on n symbols with pairwise Hamming distance D . Computing lower bounds for M ( n ,  D ) is the subject of current research with applications in error correcting codes. Using PRFs of specified degrees d we obtain improved lower bounds for M ( q , q - k ) for prime powers q and k ∈ { 5 , 6 , 7 , 8 , 9 } , and for M ( q + 1 , q - k ) for prime powers q and k ∈ { 4 , 5 , 6 , 7 , 8 , 9 } .