Constructing permutation arrays using partition and extension

Bereg, Sergey; Mojica, Luis Gerardo; Morales, Linda; Sudborough, Hal

doi:10.1007/s10623-019-00684-z

Cited by 7 publications

(8 citation statements)

References 19 publications

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“…• The degree-10 PP x 10 + x 9 + x 7 + 26x 5 + 30x 4 + 21x 2 + 31x over GF (2 5 ) is b-normalized, as a 10 = 1, a 6 = 0 and a 0 = 0 (with i = 3 and r = 2 i − 1 = 7).…”

Section: Normalized Permutation Polynomials and Equivalence Relationsmentioning

confidence: 99%

“…R. Sobhani, et al [26] computed some values of N d (q) and used these to give lower bounds for some values of M (n, D). Bereg, et al [5] give a table with several new lower bounds for M (n, D) for n ≤ 550 and a table for M (n, n − 1) for prime powers n ≤ 600.…”

Section: Introductionmentioning

confidence: 99%

“…New lower bounds for M (n, D). Note: * denotes values that were obtained by coset search[5], not by permutation polynomials. We include these for the sake of completeness.…”

mentioning

confidence: 99%

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Equivalence Relations for Computing Permutation Polynomials

Bereg,

Malouf,

Morales

et al. 2019

Preprint

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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 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 (i.e., 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.

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“…• The degree-10 PP x 10 + x 9 + x 7 + 26x 5 + 30x 4 + 21x 2 + 31x over GF (2 5 ) is b-normalized, as a 10 = 1, a 6 = 0 and a 0 = 0 (with i = 3 and r = 2 i − 1 = 7).…”

Section: Normalized Permutation Polynomials and Equivalence Relationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivalence Relations for Computing Permutation Polynomials

Bereg,

Malouf,

Morales

et al. 2019

Preprint

Self Cite

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show abstract

“…There are lower bounds for

(in particular, Gilbert–Varshamov bounds and their improvements) as well as algebraic techniques for constructing

-codes [ 39 , 41 , 42 , 43 , 44 , 45 , 46 , 47 ]. For example,

, if

is a prime power then

and

[ 41 ],

and

[ 42 ].…”

Section: Introductionmentioning

confidence: 99%

Permutation-Based Block Code for Short Packet Communication Systems

Faure

Shcherba

Makhynko³

et al. 2022

Sensors

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This paper presents an approach to the construction of block error-correcting code for data transmission systems with short packets. The need for this is driven by the necessity of information interaction between objects of machine-type communication network with a dynamically changing structure and unique system of commands or alerts for each network object. The codewords of a code are permutations with a given minimum pairwise Hamming distance. The purpose of the study is to develop a statistical method for constructing a code, in contrast to known algebraic methods, and to investigate the code size. An algorithm for generating codewords has been developed. It can be implemented both by enumeration of the full set of permutations, and by enumeration of a given number of randomly selected permutations. We have experimentally determined the dependencies of the average and the maximum values of the code size on the size of a subset of permutations used for constructing the code. A technique for computing approximation quadratic polynomials for the determined code size dependencies has been developed. These polynomials and their corresponding curves estimate the size of a code generated from a subset of random permutations of such a size that a statistically significant experiment cannot be performed. The results of implementing the developed technique for constructing a code based on permutations of lengths 7 and 11 have been presented. The prediction relative error of the code size did not exceed the value of 0.72% for permutation length 11, code distance 9, random permutation subset size 50,000, and permutation statistical study range limited by 5040.

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“…Permutation arrays (PAs) with large Hamming distance have been the subject of many recent papers with applications in the design of error correcting codes. New lower bounds for the size of such permutation arrays are given, for example, in [1,2,3,4,5,6,7,12,15,14,19,20,22].…”

Section: Introductionmentioning

confidence: 99%

New lower bounds for permutation arrays using contraction

Bereg

Miller

Mojica

et al. 2019

Des. Codes Cryptogr.

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We consider rational functions of the form V (x)/U (x), where both V (x) and U (x) are polynomials over the finite field F q . Polynomials that permute the elements of a field, called permutation polynomials (P P s), have been the subject of research for decades. Let P 1 (F q ) denote Z q ∪ {∞}. If the rational function, V (x)/U (x), permutes the elements of P 1 (F q ), it is called a permutation rational function (PRF). Let N d (q) denote the number of PPs of degree d over F q , and let N v,u (q) denote the number of PRFs with a numerator of degree v and a denominator of degree u. It follows that N d,0 (q) = N d (q), so PRFs are a generalization of PPs. The number of monic degree 3 PRFs is known [11]. We develop efficient computational techniques for N v,u (q), and use them to show N 4,3 (q) = (q + 1)q 2 (q − 1) 2 /3, for all prime powers q ≤ 307, N 5,4 (q) > (q + 1)q 3 (q − 1) 2 /2, for all prime powers q ≤ 97, and N 4,4 (p) = (p + 1)p 2 (p − 1) 3 /3, for all primes p ≤ 47. We conjecture that these formulas are, in fact, true for all prime powers q. Let M (n, D) denote the maximum number of permutations on n symbols with pairwise Hamming distance D. Computing improved lower bounds for M (n, D) is the subject of much current research with applications in error correcting codes. Using PRFs, we obtain significantly improved lower bounds on M (q, q − d) and M (q + 1, q − d), for d ∈ {5, 7, 9}.

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Constructing permutation arrays using partition and extension

Cited by 7 publications

References 19 publications

Equivalence Relations for Computing Permutation Polynomials

Equivalence Relations for Computing Permutation Polynomials

Permutation-Based Block Code for Short Packet Communication Systems

New lower bounds for permutation arrays using contraction

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