nach oben

Applicable Algebra in Engineering, Communication and Computing

Erschienen in:

06.05.2022 | Original Paper

Nonexistence of perfect permutation codes under the \(\ell _{\infty }\)-metric

verfasst von: Xiang Wang, Wenjuan Yin, Fang-Wei Fu

Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 3/2024

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Permutation codes are studied due to their numerous applications in various applications, such as power line communications, block ciphers, and coding for storage. In this paper, we study perfect permutation codes in \(S_n\), the set of all permutations on n elements, under the \(\ell _{\infty }\)-metric. We present some nonexistence results on perfect t-error-correcting permutation codes in \(S_n\) under the \(\ell _{\infty }\)-metric for some t and n. More precisely, we prove that there does not exist a perfect t-error-correcting code in \(S_n\) under the \(\ell _{\infty }\)-metric for t and n satisfying \(1\le t\le 3, 2t+1\le n\) or for t and n satisfying \(R_{2t+1}(n)=0,1,2t\), where \(0\le R_d(n)< d\) is the residue when dividing the positive integer n by the positive integer d.

Vorheriger Artikel A class of three-weight linear codes over finite fields of odd characteristic

Nächster Artikel Construction of quasi self-dual codes over a commutative non-unital ring of order 4

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vinck, A.J.H.: Coded modulation for power line communications. AE Int. J. Electron. Commun. 54, 45–49 (2011)

Colbourn, C.J., Kløve, T., Ling, A.C.H.: Permutation arrays for powerline communication and mutually orthogonal Latin squares. IEEE Trans. Inf. Theory 50, 1289–1291 (2004)MathSciNetCrossRef

Pavlidou, N., Vinck, A.J.H., Yazdani, J., Honary, B.: Power line communications: State of the art and future trends. IEEE Commun. Mag. 41, 34–40 (2003)CrossRef

de la Torre D.R., Colbourn C.J., Ling A.C.H.: An application of permutation arrays to block ciphers. Congr Numer, 5–7 (2000)

Jiang, A., Schwartz, M., Bruck, J.: Correcting charge-constrained errors in the rank-modulation scheme. IEEE Trans. Inf. Theory 56, 2112–2120 (2010)MathSciNetCrossRef

Barg, A., Mazumdar, A.: Codes in permutations and error correction for rank modulation. IEEE Trans. Inf. Theory 56, 3158–3165 (2010)MathSciNetCrossRef

Wang, X., Zhang, Y.W., Yang, Y.T., Ge, G.N.: New bounds of permutation codes under Hamming metric and Kendall’s \(\tau\)-metric. Des. Codes Cryptogr. 85, 533–545 (2017)MathSciNetCrossRef

Farnoud, F., Skachek, V., Milenkovic, O.: Error-correction in flash memories via codes in the Ulam metric. IEEE Trans. Inf. Theory 59, 3003–3020 (2013)MathSciNetCrossRef

Kong J., Hagiwara M.: Nonexistence of perfect permutation codes in the Ulam metric. in Proc. IEEE Int. Symp. Inf. Theory and its Applications (ISITA), 691-695 (2016)

10.

Wang, X., Fu, F.-W.: On the snake-in-the-box codes for rank modulation under Kendall’s \(\tau\)-metric. Des. Codes Cryptogr. 83, 455–465 (2017)MathSciNetCrossRef

11.

Kløve, T., Lin, T.T., Tsai, S.C., Tzeng, W.G.: Permutation arrays under the Chebyshev distance. IEEE Trans. Inf. Theory 56, 2611–2617 (2010)MathSciNetCrossRef

12.

Wang, X., Fu, F.-W.: Snake-in-the-box codes under the \(\ell _{\infty }\)-metric for rank modulation. Des. Codes Cryptogr. 88, 487–503 (2020)MathSciNetCrossRef

13.

Yehezkeally, Y., Schwartz, M.: Snake-in-the-box codes for rank modulation. IEEE Trans. Inf. Theory 58, 5471–5483 (2012)MathSciNetCrossRef

14.

Haykin, S.: Communication Systems, 4th edn. Wiley, Hoboken (2001)

15.

Tamo, I., Schwartz, M.: Correcting limited-magnitude errors in the rank-modulation scheme. IEEE Trans. Inf. Theory 56, 2551–2560 (2010)MathSciNetCrossRef

16.

Tamo, I., Schwartz, M.: On the labeling problem of permutation group codes for the infinity metric. IEEE Trans. Inf. Theory 58, 6595–6604 (2012)MathSciNetCrossRef

17.

Schwartz, M., Vontobel, P.O.: Improved lower bounds on the size of balls over permutations with the infinity metric. IEEE Trans. Inf. Theory 63, 6227–6239 (2017)MathSciNetCrossRef

18.

Kløve, T.: Lower bounds on the size of spheres of permutations under the Chebychev distance. Des. Codes Cryptogr. 59, 183–191 (2010)MathSciNetCrossRef

19.

Buzaglo, S., Etzion, T.: Bounds on the size of permutation codes with the Kendall \(\tau\)-metric. IEEE Trans. Inf. Theory 61, 3241–3250 (2015)MathSciNetCrossRef

20.

Wang, X., Wang, Y.J., Yin, W.J., Fu, F.-W.: Nonexistence of perfect permutation codes under the Kendall \(\tau\)-metric. Des. Codes Cryptogr. 89, 2511–2531 (2021)MathSciNetCrossRef

21.

Minc, H.: Upper bounds for permanents of \((0,1)\)-matrices. Bull. Am. Math. Soc. 69, 789–791 (1963)MathSciNetCrossRef

22.

Bregman, L.M.: Some properties of nonnegative matrices and their permanents. Soviet Math. Dokl. 14, 945–949 (1973)

23.

Kløve, T.: Spheres of permutations under the infinity norm-permutations with limited displacement, p. 376. University of Bergen, Bergen, Norway, Tech, Rep (2008)

Titel: Nonexistence of perfect permutation codes under the -metric
verfasst von: Xiang Wang
Wenjuan Yin
Fang-Wei Fu
Publikationsdatum: 06.05.2022
Verlag: Springer Berlin Heidelberg
Erschienen in: Applicable Algebra in Engineering, Communication and Computing / Ausgabe 3/2024
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI: https://doi.org/10.1007/s00200-022-00556-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 3/2024

A class of three-weight linear codes over finite fields of odd characteristic

RLWE and PLWE over cyclotomic fields are not equivalent

δ-dual codes over finite commutative semi-simple rings

Construction of quasi self-dual codes over a commutative non-unital ring of order 4

Several classes of permutation polynomials with trace functions over

New constructions of optimal frequency hopping sequences

Premium Partner