An Improvement on the Gilbert–Varshamov Bound for Permutation Codes

Gao, Fei; Yang, Yiting; Ge, Gennian

doi:10.1109/tit.2013.2237945

Cited by 22 publications

(19 citation statements)

References 18 publications

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“…(It should be noted that one can often get far more than N GV (n, d) permutations in a PA, because the assumption that the spheres are disjoint is not required. In fact, this combinatorial lower bound has recently been improved [11].) Of course, such sets need not be disjoint and one can choose a larger set of permutations by eliminating this condition.…”

Section: Resultsmentioning

confidence: 99%

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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Section: Resultsmentioning

confidence: 99%

Constructing permutation arrays from groups

Bereg

Levy

Sudborough

2017

Des. Codes Cryptogr.

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“…Permutation codes have been of great interest recently due to their applications (for example in powerline communications [2,3]) and for their intrinsecal combinatorial interest [8,9,10,15,18]. Let us now briefly explain what permutation codes are.…”

Section: Introductionmentioning

confidence: 99%

New Lower Bounds for Permutation Codes Using Linear Block Codes

Micheli

Neri

2020

IEEE Trans. Inform. Theory

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In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an [n, k, d]q linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree n, having minimum distance at least d and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code.

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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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An Improvement on the Gilbert–Varshamov Bound for Permutation Codes

Cited by 22 publications

References 18 publications

Constructing permutation arrays from groups

Constructing permutation arrays from groups

New Lower Bounds for Permutation Codes Using Linear Block Codes

New lower bounds for permutation arrays using contraction

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