Kronecker product and tiling of permutation arrays for hamming distances

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

doi:10.1109/isit.2017.8006919

Cited by 5 publications

(8 citation statements)

References 19 publications

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“…Define the modified Kronecker product [3] of PAs X and Y , denoted by (X ⊗ Y ), to be the PA on lm symbols defined by (X ⊗ Y ) = l i=1 (X ⊗ Y ) i . This is illustrated in Figure 1.…”

Section: Partition and Extension Of Modified Kronecker Productmentioning

confidence: 99%

“…. , A (m) , such that for all i (1 ≤ i ≤ m), hd(A (i) ) = n. A detailed discussion of block decomposition appears in [3], along with several examples using AGL(1, q) and P GL(2, q), where q is a prime or a prime power. We use block decompositions of PAs and the modified Kronecker product to produce new PAs, which in some cases give new lower bounds for M (n+1, n).…”

Section: Partition and Extension Of Modified Kronecker Productmentioning

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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“…Define the modified Kronecker product [3] of PAs X and Y , denoted by (X ⊗ Y ), to be the PA on lm symbols defined by (X ⊗ Y ) = l i=1 (X ⊗ Y ) i . This is illustrated in Figure 1.…”

Section: Partition and Extension Of Modified Kronecker Productmentioning

confidence: 99%

Section: Partition and Extension Of Modified Kronecker Productmentioning

confidence: 99%

Constructing permutation arrays using partition and extension

Bereg

Mojica

Morales

et al. 2019

Des. Codes Cryptogr.

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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“…nPP Equivalence Classes for q = 37, degree 7, and primitive polynomial x + 13. class number of nPPs x 9 + 3x 5 + 10x 24 x 9 + 2x 7 + 3x 5 + 4x3 24 x 9 + 2x 7 + 21x 5 + 26x 3 + 36x 48 TOTAL 96…”

mentioning

confidence: 99%

“…class a number of nPPs x 9 -1 x 9 + 2ax 5 + 3a 2 x {1,2,3} 30 x 9 + x 6 + 6x 3 -40 x 9 + x 7 + 4a 9 x 5 + 36a 19 x 3 + 33a −1 x {1,2} 40 x 9 + x 7 + 14a 25 x 5 + 28a −2 x 3 + 5a 4 x {1,2} 40 x 9 + ax 7 + 19a −8 x 6 + 5a 11 x 5 + 3a −7 x 4 + +16a 14 x 3 + 15a 7 x 2 + 39a −10 x {1,2} 80 x 9 + 2x 7 + 15a 16 x 5 + 2a −3 x 3 + 31a 22 x {1,2} 40 x 9 + 2x 7 + 33ax 5 + 18a…”

mentioning

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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Kronecker product and tiling of permutation arrays for hamming distances

Cited by 5 publications

References 19 publications

Constructing permutation arrays using partition and extension

Constructing permutation arrays using partition and extension

Permutation-Based Block Code for Short Packet Communication Systems

Equivalence Relations for Computing Permutation Polynomials

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