# American Institute of Mathematical Sciences

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:
 [1] S. K. Arora and M. Pruthi, Minimal cyclic codes of length $2p^n$,, Finite Fields Appl., 5 (1999), 177. doi: 10.1006/ffta.1998.0238. [2] L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes,, Inform. Control, 20 (1972), 158. [3] C. Ding, The weight distribution of some irreducible cyclic codes,, IEEE Trans. Inf. Theory, 55 (2009), 955. doi: 10.1109/TIT.2008.2011511. [4] R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes,, Finite Fields Appl., 11 (2005), 89. doi: 10.1016/j.ffa.2004.06.002. [5] F. J. MacWilliams and J. Seery, The weight distributions of some minimal cyclic codes,, IEEE Trans. Inf. Theory, 27 (1981), 796. doi: 10.1109/TIT.1981.1056420. [6] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North Holland, (1977). [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. doi: 10.1109/18.761277. [8] M. Pruthi and S. K. Arora, Minimal cyclic codes of prime power length,, Finite Fields Appl., 3 (1997), 99. doi: 10.1006/ffta.1996.0156. [9] A. Sharma and G. K. Bakshi, The weight distributions of some irreducible cyclic codes,, Finite Fields Appl., 18 (2012), 144. doi: 10.1016/j.ffa.2011.07.002. [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. doi: 10.1016/j.ffa.2007.07.004. [11] M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes,, J. Number Theory, 55 (1995), 145. doi: 10.1006/jnth.1995.1133.

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##### References:
 [1] S. K. Arora and M. Pruthi, Minimal cyclic codes of length $2p^n$,, Finite Fields Appl., 5 (1999), 177. doi: 10.1006/ffta.1998.0238. [2] L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes,, Inform. Control, 20 (1972), 158. [3] C. Ding, The weight distribution of some irreducible cyclic codes,, IEEE Trans. Inf. Theory, 55 (2009), 955. doi: 10.1109/TIT.2008.2011511. [4] R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes,, Finite Fields Appl., 11 (2005), 89. doi: 10.1016/j.ffa.2004.06.002. [5] F. J. MacWilliams and J. Seery, The weight distributions of some minimal cyclic codes,, IEEE Trans. Inf. Theory, 27 (1981), 796. doi: 10.1109/TIT.1981.1056420. [6] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North Holland, (1977). [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. doi: 10.1109/18.761277. [8] M. Pruthi and S. K. Arora, Minimal cyclic codes of prime power length,, Finite Fields Appl., 3 (1997), 99. doi: 10.1006/ffta.1996.0156. [9] A. Sharma and G. K. Bakshi, The weight distributions of some irreducible cyclic codes,, Finite Fields Appl., 18 (2012), 144. doi: 10.1016/j.ffa.2011.07.002. [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. doi: 10.1016/j.ffa.2007.07.004. [11] M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes,, J. Number Theory, 55 (1995), 145. doi: 10.1006/jnth.1995.1133.
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