nach oben

Journal of Applied Mathematics and Computing

Erschienen in:

05.02.2016 | Original Research

A class of binary cyclic codes and sequence families

verfasst von: Hua Liang, Wenbing Chen, Yuansheng Tang

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2017

Einloggen

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

search-config

KI-gestützte Suche

Aus

Abstract

For two odd integers l, k with \(0<l<k\) and \(\gcd (l,k)=1\), let \(m=2k\) and \(d=\frac{2^{lk}+1}{2^l+1}+\frac{2(2^m-1)}{3}\). In this paper, we determine the value distribution of the exponential sum \(\sum _{x\in \mathbb {F}_{2^m}}(-1)^{\mathrm {Tr}_1^m(ax+bx^d)}\). As applications, the weight distribution of a class of binary cyclic codes is settled. Second, we determine the correlation distribution among sequences in a sequence family.

Vorheriger Artikel Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions

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

Ding, C.: The weight distribution of some irreducible cyclic codes. IEEE Trans. Inf. Theory 55(3), 955–960 (2009)MathSciNetCrossRef

Ding, C., Yang, J.: Hamming weights in irreducible cyclic codes. Discret. Math 313(4), 434–446 (2013)MathSciNetCrossRefMATH

van der Vlugt, M.: Hasse\(-\)Davenport curves, gauss sums, and weight distributions of irreducible cyclic codes. J. Number Theory 55(2), 145–159 (1995)MathSciNetCrossRefMATH

Ding, C., Liu, Y., Ma, C., Zeng, L.: The weight distributions of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 57(12), 8000–8006 (2011)MathSciNetCrossRef

Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014)MathSciNetCrossRefMATH

Feng, K., Luo, J.: Value distributions of exponential sums from perfect nonlinear functions and their applications. IEEE Trans. Inf. Theory 53(9), 3035–3041 (2007)MathSciNetCrossRefMATH

Li, C., Yue, Q., Li, F.: Hamming weights of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 60(7), 3895–3902 (2014)MathSciNetCrossRef

Li, S., Feng, T., Ge, G.: On the weight distribution of cyclic codes with Niho exponents. IEEE Trans. Inf. Theory 60(7), 3903–3912 (2014)MathSciNetCrossRef

Luo, J., Feng, K.: On the weight distribution of two classes of cyclic codes. IEEE Trans. Inf. Theory 54(5), 5332–5344 (2008)MathSciNetCrossRefMATH

10.

Luo, J., Feng, K.: Cyclic codes and sequences from generalized Coulter-Matthews function. IEEE Trans. Inf. Theory 54(12), 5345–5353 (2008)MathSciNetCrossRefMATH

11.

Zhou, Z., Ding, C., Luo, J., Zhang, A.: A family of five-weight cyclic codes and their weight enumerators. IEEE Trans. Inf. Theory 59(10), 6674–6682 (2013)MathSciNetCrossRef

12.

Delsarte, P.: On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inf. Theory 21(5), 575–576 (1975)MathSciNetCrossRefMATH

13.

Choi, S.T., Lim, T., No, J.S., Chung, H.: On the cross-correlation of a \(p\)-ary m-sequence of period \(p^{2m}-1\) and its decimated sequence by \(\frac{(p^{m}+1)^{2}}{2(p+1)}\). IEEE Trans. Inf. Theory 58(3), 1873–1879 (2012)MathSciNetCrossRef

14.

Luo, J.: Binary sequences with three-valued cross correlations of different lengths, arXiv:1503.05650vl, (2015)

15.

Gong, G.: New designs for signal sets with low cross correlation, balance property, and large linear span: GF(p) case. IEEE Trans. Inf. Theory 48(11), 2847–2867 (2002)MathSciNetCrossRefMATH

16.

Seo, E.Y., Kim, Y.S., No, J.S., Shin, D.J.: Cross-correlation distribution of p-ary m-sequence of period \(p^{4k}-1\) and its decimated sequences by \((\frac{p^{2k}+1}{2})^{2}\). IEEE Trans. Inf. Theory 54(7), 3140–3149 (2008)

17.

Ness, G.J., Helleseth, T., Kholosha, A.: On the correlation distribution of the Coulter-Matthews decimation. IEEE Trans. Inf. Theory 52(5), 2241–2247 (2006)MathSciNetCrossRefMATH

18.

Muller, E.N.: On the crosscorrelation of sequences over \(GF(p)\) with short periods. IEEE Trans. Inf. Theory 45(1), 289–295 (1999)MathSciNetCrossRefMATH

19.

Yu, N.Y., Gong, G.: A new binary sequence family with low correlation and large size. IEEE Trans. Inf. Theory 52(4), 1624–1636 (2006)MathSciNetCrossRefMATH

20.

Liang, H., Tang, Y.: The cross correlation distribution of a \(p\)-ary \(m\)-sequence of period \(p^m-1\) and its decimated sequences by \((p^k+1)(p^m+1)/4\). Finite Fields Appl. 31, 137–161 (2015)MathSciNetCrossRefMATH

21.

No, J.S., Kumar, P.V.: A new family of binary pseudorandom sequences having optimal periodic correlation properties and larger linear span. IEEE Trans. Inf. Theory 35(2), 371–379 (1989)CrossRefMATH

22.

Lidl, R., Niederreiter, H.: Finite Fields, Encyclopedia of Mathmatics, vol. 20. Cambridge University Press, Cambridge (1983)MATH

23.

Moisio, M.: A note on evaluations of some exponential sums. Acta Arith. 93, 117–119 (2000)MathSciNetMATH

Titel: A class of binary cyclic codes and sequence families
verfasst von: Hua Liang
Wenbing Chen
Yuansheng Tang
Publikationsdatum: 05.02.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Applied Mathematics and Computing / Ausgabe 1-2/2017
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-016-0993-z

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 1-2/2017

Solvability for fractional p-Laplacian differential equations with multipoint boundary conditions at resonance on infinite interval

Iterative algorithms for least-squares solutions of a quaternion matrix equation

Stability of nonlocal fractional Langevin differential equations involving fractional integrals

On some classes of linear codes over and their covering radii

Dynamics of an oasis-vegetation degradation model with impulsive irrigation and diffusion in arid area

Existence of periodic solutions of a continuous flow bioreactor model with impulsive control in microorganisms