    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 $2p^n$

 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.
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:
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##### References:
  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  L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Inform. Control, 20 (1972), 158-175. Google Scholar  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  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  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  F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. Google Scholar  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  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  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  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  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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