February  2015, 9(1): 23-36. doi: 10.3934/amc.2015.9.23

Some new classes of cyclic codes with three or six weights

1. 

Department of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China

2. 

Department of Informatics, University of Bergen, N-5020 Bergen, Norway

3. 

CIPSI, Department of Electrical Engineering and Computer Science, University of Stavanger, 4036 Stavanger, Norway

Received  December 2013 Revised  July 2014 Published  February 2015

In this paper, a class of three-weight cyclic codes over prime fields $\mathbb{F}_p$ of odd order whose duals have two zeros, and a class of six-weight cyclic codes whose duals have three zeros are presented. The weight distributions of these cyclic codes are derived.
Citation: Yongbo Xia, Tor Helleseth, Chunlei Li. Some new classes of cyclic codes with three or six weights. Advances in Mathematics of Communications, 2015, 9 (1) : 23-36. doi: 10.3934/amc.2015.9.23
References:
[1]

C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102. doi: 10.1109/TIT.2005.847722.  Google Scholar

[2]

S. T. Choi, J. Y. Kim and J. S. No, On the cross-correlation of a p-ary m-sequence and its decimated sequences by $d = \frac{p^n+1}{p^k+1} + \frac{p^n-1}{2}$, IEICE Trans. Comm., B96 (2013), 2190-2197. Google Scholar

[3]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  Google Scholar

[4]

C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946. doi: 10.1109/TIT.2013.2281205.  Google Scholar

[5]

K. Feng and J. Luo, Value distribution of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inf. Theory, 53 (2007), 3035-3041. doi: 10.1109/TIT.2007.903153.  Google Scholar

[6]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409. doi: 10.1016/j.ffa.2007.03.003.  Google Scholar

[7]

Z. Hu, X. Li, D. Mills, E. N. Müller, W. Sun, W. Willems, Y. Yang and Z. Zhang, On the crosscorrelation of sequences with the decimation factor $d = \frac{p^n+1}{p+1} - \frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 12 (2001), 255-263. doi: 10.1007/s002000100073.  Google Scholar

[8]

C. Li, N. Li, T. Helleseth and C. Ding, On the weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721. doi: 10.1109/TIT.2014.2329694.  Google Scholar

[9]

R. Lidl and H. Niederreiter, Finite fields, in Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Amsterdam, 1983.  Google Scholar

[10]

Y. Liu, H. Yan and C. Liu, A class of six-weight cyclic codes and their weight distribution,, Des. Codes Cryptogr., ().  doi: 10.1007/s10623-014-9984-y.  Google Scholar

[11]

J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344. doi: 10.1109/TIT.2008.2006424.  Google Scholar

[12]

C. Ma, L. Zeng, Y. Liu, D. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402. doi: 10.1109/TIT.2010.2090272.  Google Scholar

[13]

E. N. Müller, On the crosscorrelation of sequences over $GF(p)$ with short periods, IEEE Trans. Inf. Theory, 45 (1999), 289-295. doi: 10.1109/18.746820.  Google Scholar

[14]

G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965), 170-179.  Google Scholar

[15]

B. Wang, C. Tang, Y. Qi, Y. Yang and M. Xu, The weight distributions of cyclic codes and elliptic curves, IEEE Trans. Inf. Theory, 58 (2012), 7253-7259. doi: 10.1109/TIT.2012.2210386.  Google Scholar

[16]

Y. Xia, X. Zeng and L. Hu, Further crosscorrelation properties of sequences with the decimation factor $d = \frac{p^n+1}{p+1} - \frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 21 (2010), 329-342. doi: 10.1007/s00200-010-0128-y.  Google Scholar

[17]

Z. Zhou and C. Ding, Seven families of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126. Google Scholar

[18]

Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93. doi: 10.1016/j.ffa.2013.08.005.  Google Scholar

[19]

Z. Zhou, C. Ding, J. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682. doi: 10.1109/TIT.2013.2267722.  Google Scholar

show all references

References:
[1]

C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102. doi: 10.1109/TIT.2005.847722.  Google Scholar

[2]

S. T. Choi, J. Y. Kim and J. S. No, On the cross-correlation of a p-ary m-sequence and its decimated sequences by $d = \frac{p^n+1}{p^k+1} + \frac{p^n-1}{2}$, IEICE Trans. Comm., B96 (2013), 2190-2197. Google Scholar

[3]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  Google Scholar

[4]

C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946. doi: 10.1109/TIT.2013.2281205.  Google Scholar

[5]

K. Feng and J. Luo, Value distribution of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inf. Theory, 53 (2007), 3035-3041. doi: 10.1109/TIT.2007.903153.  Google Scholar

[6]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409. doi: 10.1016/j.ffa.2007.03.003.  Google Scholar

[7]

Z. Hu, X. Li, D. Mills, E. N. Müller, W. Sun, W. Willems, Y. Yang and Z. Zhang, On the crosscorrelation of sequences with the decimation factor $d = \frac{p^n+1}{p+1} - \frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 12 (2001), 255-263. doi: 10.1007/s002000100073.  Google Scholar

[8]

C. Li, N. Li, T. Helleseth and C. Ding, On the weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721. doi: 10.1109/TIT.2014.2329694.  Google Scholar

[9]

R. Lidl and H. Niederreiter, Finite fields, in Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Amsterdam, 1983.  Google Scholar

[10]

Y. Liu, H. Yan and C. Liu, A class of six-weight cyclic codes and their weight distribution,, Des. Codes Cryptogr., ().  doi: 10.1007/s10623-014-9984-y.  Google Scholar

[11]

J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344. doi: 10.1109/TIT.2008.2006424.  Google Scholar

[12]

C. Ma, L. Zeng, Y. Liu, D. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402. doi: 10.1109/TIT.2010.2090272.  Google Scholar

[13]

E. N. Müller, On the crosscorrelation of sequences over $GF(p)$ with short periods, IEEE Trans. Inf. Theory, 45 (1999), 289-295. doi: 10.1109/18.746820.  Google Scholar

[14]

G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965), 170-179.  Google Scholar

[15]

B. Wang, C. Tang, Y. Qi, Y. Yang and M. Xu, The weight distributions of cyclic codes and elliptic curves, IEEE Trans. Inf. Theory, 58 (2012), 7253-7259. doi: 10.1109/TIT.2012.2210386.  Google Scholar

[16]

Y. Xia, X. Zeng and L. Hu, Further crosscorrelation properties of sequences with the decimation factor $d = \frac{p^n+1}{p+1} - \frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 21 (2010), 329-342. doi: 10.1007/s00200-010-0128-y.  Google Scholar

[17]

Z. Zhou and C. Ding, Seven families of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126. Google Scholar

[18]

Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93. doi: 10.1016/j.ffa.2013.08.005.  Google Scholar

[19]

Z. Zhou, C. Ding, J. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682. doi: 10.1109/TIT.2013.2267722.  Google Scholar

[1]

Gerardo Vega, Jesús E. Cuén-Ramos. The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes. Advances in Mathematics of Communications, 2020, 14 (3) : 525-533. doi: 10.3934/amc.2020059

[2]

Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023

[3]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002

[4]

Lanqiang Li, Shixin Zhu, Li Liu. The weight distribution of a class of p-ary cyclic codes and their applications. Advances in Mathematics of Communications, 2019, 13 (1) : 137-156. doi: 10.3934/amc.2019008

[5]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032

[6]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433

[7]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017

[8]

Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395

[9]

Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409

[10]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013

[11]

Yuk L. Yung, Cameron Taketa, Ross Cheung, Run-Lie Shia. Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 229-248. doi: 10.3934/dcdsb.2010.13.229

[12]

Pankaj Kumar, Monika Sangwan, Suresh Kumar Arora. The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$. Advances in Mathematics of Communications, 2015, 9 (3) : 277-289. doi: 10.3934/amc.2015.9.277

[13]

Chengju Li, Qin Yue, Ziling Heng. Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates. Advances in Mathematics of Communications, 2015, 9 (3) : 341-352. doi: 10.3934/amc.2015.9.341

[14]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[15]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95

[16]

Mehmet Duran Toksari, Emel Kizilkaya Aydogan, Berrin Atalay, Saziye Sari. Some scheduling problems with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021044

[17]

Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai. Asymptotic distribution of values of isotropic here quadratic forms at S-integral points. Journal of Modern Dynamics, 2017, 11: 501-550. doi: 10.3934/jmd.2017020

[18]

Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110

[19]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221

[20]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]