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August  2015, 9(3): 277-289. doi: 10.3934/amc.2015.9.277

The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$

Pankaj Kumar 1, , Monika Sangwan 1, and  Suresh Kumar Arora 2,

1. 

Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar, Pin-125001, India, India
2. 

Department of Mathematics, M. D. University, Rohtak, Pin-124001, India

Received  October 2013 Published  July 2015

In this paper, an algorithm is given for computing the weight distributions of all irreducible cyclic codes of dimension $p^jd$ generated by $x^{p^j}-1$, where $p$ is an odd prime, $j\geq 0 $ and $d > 1$. Then the weight distributions of all irreducible cyclic codes of length $p^n$ and $ 2p^n $ over $F_q$, where $n$ is a positive integer, $p$, $q$ are distinct odd primes and $q$ is primitive root modulo $ p^n$, are obtained. The weight distributions of all the irreducible cyclic codes of length $3^{n+1}$ over $F_5$ are also determined explicitly.
Keywords: weight distribution., Irreducible cyclic codes.
Mathematics Subject Classification: Primary: 20C05, 94B05, 94B6.
Citation: 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
References:
[1]

S. K. Arora and M. Pruthi, Minimal cyclic codes of length $2p^n$, Finite Fields Appl., 5 (1999), 177-187. doi: 10.1006/ffta.1998.0238.  Google Scholar

[2]

L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Inform. Control, 20 (1972), 158-175.  Google Scholar

[3]

C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960. doi: 10.1109/TIT.2008.2011511.  Google Scholar

[4]

R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes, Finite Fields Appl., 11 (2005), 89-110. doi: 10.1016/j.ffa.2004.06.002.  Google Scholar

[5]

F. J. MacWilliams and J. Seery, The weight distributions of some minimal cyclic codes, IEEE Trans. Inf. Theory, 27 (1981), 796-806. doi: 10.1109/TIT.1981.1056420.  Google Scholar

[6]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. Google Scholar

[7]

M. J. Moisio and K. O. Väänänen, Two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes, IEEE Trans. Inf. Theory, 45 (1999), 1244-1249. doi: 10.1109/18.761277.  Google Scholar

[8]

M. Pruthi and S. K. Arora, Minimal cyclic codes of prime power length, Finite Fields Appl., 3 (1997), 99-113. doi: 10.1006/ffta.1996.0156.  Google Scholar

[9]

A. Sharma and G. K. Bakshi, The weight distributions of some irreducible cyclic codes, Finite Fields Appl., 18 (2012), 144-159. doi: 10.1016/j.ffa.2011.07.002.  Google Scholar

[10]

A. Sharma, G. K. Bakshi and M. Raka, The weight distributions of irreducible cyclic codes of length $2^m$, Finite Fields Appl., 13 (2007), 1086-1095. doi: 10.1016/j.ffa.2007.07.004.  Google Scholar

[11]

M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, J. Number Theory, 55 (1995), 145-159. doi: 10.1006/jnth.1995.1133.  Google Scholar

show all references

References:
[1]

S. K. Arora and M. Pruthi, Minimal cyclic codes of length $2p^n$, Finite Fields Appl., 5 (1999), 177-187. doi: 10.1006/ffta.1998.0238.  Google Scholar

[2]

L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Inform. Control, 20 (1972), 158-175.  Google Scholar

[3]

C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960. doi: 10.1109/TIT.2008.2011511.  Google Scholar

[4]

R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes, Finite Fields Appl., 11 (2005), 89-110. doi: 10.1016/j.ffa.2004.06.002.  Google Scholar

[5]

F. J. MacWilliams and J. Seery, The weight distributions of some minimal cyclic codes, IEEE Trans. Inf. Theory, 27 (1981), 796-806. doi: 10.1109/TIT.1981.1056420.  Google Scholar

[6]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. Google Scholar

[7]

M. J. Moisio and K. O. Väänänen, Two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes, IEEE Trans. Inf. Theory, 45 (1999), 1244-1249. doi: 10.1109/18.761277.  Google Scholar

[8]

M. Pruthi and S. K. Arora, Minimal cyclic codes of prime power length, Finite Fields Appl., 3 (1997), 99-113. doi: 10.1006/ffta.1996.0156.  Google Scholar

[9]

A. Sharma and G. K. Bakshi, The weight distributions of some irreducible cyclic codes, Finite Fields Appl., 18 (2012), 144-159. doi: 10.1016/j.ffa.2011.07.002.  Google Scholar

[10]

A. Sharma, G. K. Bakshi and M. Raka, The weight distributions of irreducible cyclic codes of length $2^m$, Finite Fields Appl., 13 (2007), 1086-1095. doi: 10.1016/j.ffa.2007.07.004.  Google Scholar

[11]

M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, J. Number Theory, 55 (1995), 145-159. doi: 10.1006/jnth.1995.1133.  Google Scholar

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