Parallel Partition and Extension

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

doi:10.1109/ciss.2017.7926083

Cited by 5 publications

(3 citation statements)

References 11 publications

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“…The column on the left shows the PAs M 1 , M 2 and M 2 . M 1 is the cyclic subgroup of AGL (1,4), and M 2 and M 3 are two of its cosets. The blue symbols are the symbols of Q i that occupy positions in P i , for i ∈ 1, 2.…”

Section: Previous Results On Partition and Extensionmentioning

confidence: 99%

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Constructing permutation arrays using partition and extension

Bereg

Mojica

Morales

et al. 2019

Des. Codes Cryptogr.

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We give new lower bounds for M (n, d), for various positive integers n and d with n > d, where M (n, d) is the largest number of permutations on n symbols with pairwise Hamming distance at least d. Large sets of permutations on n symbols with pairwise Hamming distance d are needed for constructing error correcting permutation codes, which have been proposed for power-line communications. Our technique, partition and extension, is universally applicable to constructing such sets for all n and all d, d < n. We describe three new techniques, sequential partition and extension, parallel partition and extension, and a modified Kronecker product operation, which extend the applicability of partition and extension in different ways. We describe how partition and extension gives improved lower bounds for M (n, n − 1) using mutually orthogonal Latin squares (MOLS). We present efficient algorithms for computing new partitions: an iterative greedy algorithm and an algorithm based on integer linear programming. These algorithms yield partitions of positions (or symbols) used as input to our partition and extension techniques. We report many new lower bounds for M (n, d) found using these techniques for n up to 600.

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Section: Previous Results On Partition and Extensionmentioning

confidence: 99%

“…Recently, we described a new technique, called partition and extension [4,5] and we illustrated how to use this technique to improve several lower bounds for M (n, n−1) over those given by MOLS. Partition and extension operates on permutation arrays that can be decomposed into subsets with certain properties.…”

Section: Introductionmentioning

confidence: 99%

Constructing permutation arrays using partition and extension

Bereg

Mojica

Morales

et al. 2019

Des. Codes Cryptogr.

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“…To illustrate G-cycles on coefficients, consider GF (2 4 ). The elements of GF (2 4 ) are partitioned into 6 disjoint equivalence classes (i.e., G-cycles on coefficients) by the G-map, namely the equivalence classes [0], [1], [2], [4], [6], and [8]. To see this, observe that where mod 15 arithmetic is used in the exponents.…”

Section: The G-mapmentioning

confidence: 99%

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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Constructing permutation arrays from groups

Bereg

Levy

Sudborough

2017

Des. Codes Cryptogr.

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Let M (n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M (n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1, q) and P GL(2, q) with Frobenius maps to obtain new, improved lower bounds for M (n, d). We give new randomized algorithms. We give better lower bounds for M (n, d) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower bound for M (n, n − 2) for all n ≡ 2 (mod 3) such that n + 1 is a prime power.

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Parallel Partition and Extension

Cited by 5 publications

References 11 publications

Constructing permutation arrays using partition and extension

Constructing permutation arrays using partition and extension

Equivalence Relations for Computing Permutation Polynomials

Constructing permutation arrays from groups

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