New Lower Bounds for Permutation Codes Using Linear Block Codes

Micheli, Giacomo; Neri, Alessandro

doi:10.1109/tit.2019.2957354

Cited by 8 publications

(6 citation statements)

References 23 publications

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“…Another thing that is worth mentioning is that optimal permutation codes are far from being optimal in the sense of Corollary 4.5. We remark that, similar bounds to those of Theorem 6.8 are proven for a different type of permutation codes in [27]. Question 6.9.…”

Section: Permutation Codessupporting

confidence: 70%

Submodule codes as spherical codes in buildings

Stanojkovski¹

2022

Preprint

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In this paper, we give a generalization of subspace codes by means of codes of modules over finite commutative chain rings. We define the new class of Sperner codes and use results from extremal combinatorics to prove the optimality of such codes in different cases. Moreover, we explain the connection with Bruhat-Tits buildings and show how our codes are the buildings' analogue of spherical codes in the Euclidean sense.

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Section: Permutation Codessupporting

confidence: 70%

Submodule codes as spherical codes in buildings

Stanojkovski¹

2022

Preprint

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“…The basics of spatial modulation of code structures for Gaussian channels and their classification are considered in [7,8]. Determining the lower limits for permutation codes when using them in multidimensional signals for transmission along continuous channels with damping is given in [9,10]. Certain promising directions for the construction of new types of code structures, including Gaussian signals, are described in [11][12][13].…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Analyzing the code structures of multidimensional signals for a continuous information transmission channel

Беркман

Turovsky

Kyrpach

et al. 2021

EEJET

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One of the directions to improve the efficiency of modern telecommunication systems is the transition to the use of multidimensional signals for continuous channels of information transmission. As a result of studies carried out in recent years, it has been established that it is possible to ensure high quality of information transmission in continuous channels by combining demodulation and decoding operations into a single procedure that involves the construction of a code construct for a multidimensional signal. This paper considers issues related to estimating the possibility to improve the efficiency of continuous information transmission channel by changing the signal distance of the code structure. It has been established that the code structures of such types as a hierarchical code construct of signals, a hierarchical code construct of signals with Euclidean metric, a reversible code construct of signals, a reversible code construct of signals with Euclidean metric have the potential, when used, to increase the speed of information transmission along a continuous channel. With a signal distance reduced by 10 percent or larger, it could increase by two times or faster. The estimation of the effect of reducing a signal distance on the efficiency of certain types of code structures was carried out. It has been established that the hierarchical reversible code construct, compared to the hierarchical code construct, provides a win of up to two or more times in the speed of information transmission with a halved signal distance. Implementing the modulation procedure has no fundamental difficulties, on the condition that for each code of the code construct the encoding procedure is known when using binary codes. The results reported here make it possible to build an acceptably complex demodulation procedure according to the specified types of code structures

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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%

“…Yang et al [23] used PRFs to compute, for example, an improved lower bound for M (19,14). Ferraguti and Micheli [11] enumerated all PRFs of degree 3.…”

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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New Lower Bounds for Permutation Codes Using Linear Block Codes

Cited by 8 publications

References 23 publications

Submodule codes as spherical codes in buildings

Submodule codes as spherical codes in buildings

Analyzing the code structures of multidimensional signals for a continuous information transmission channel

New lower bounds for permutation arrays using contraction

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