# American Institute of Mathematical Sciences

August  2020, 14(3): 525-533. doi: 10.3934/amc.2020059

## The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes

 1 Dirección General de Cómputo y de Tecnologías de Información y Comunicación, Universidad Nacional Autónoma de México, 04510 Ciudad de México, Mexico 2 Posgrado en Ciencias Matemáticas, Universidad Nacional Autónoma de México, 20059 Ciudad de México, Mexico

* Corresponding author: Gerardo Vega

Ph.D. student.

Received  November 2018 Revised  October 2019 Published  January 2020

Fund Project: Partially supported by PAPIIT-UNAM IN109818

The calculation of the weight distribution for some reducible cyclic codes can be reduced down to the corresponding one of a particular kind of irreducible cyclic codes. This reduction is achieved by means of a known identity (see [3,Theorem 1.1]). In fact, as will be shown here, the weight distribution of some families of reducible cyclic codes, recently reported in several works ([2,5,7,11,12]), and that of others not previously reported, can be obtained almost directly by means of this identity.

Citation: 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
##### References:
 [1] C. S. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inform. Theory, 55 (2009), 955-960.  doi: 10.1109/TIT.2008.2011511.  Google Scholar [2] C. S. Ding, Y. Liu, C. L. Ma and L. W. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, 57 (2011), 8000-8006.  doi: 10.1109/TIT.2011.2165314.  Google Scholar [3] T. Kløve, The weight distribution for a class of ICCs, Discrete Mathematics, 20 (1977), 87-90.   Google Scholar [4] R. Lidl and H. Niederreiter, Finite Fields, Second edition. Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.  Google Scholar [5] C. L. Ma, L. W. Zeng, Y. Liu, D. G. Feng and C. S. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inform. Theory, 57 (2011), 397-402.  doi: 10.1109/TIT.2010.2090272.  Google Scholar [6] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [7] G. Vega and L. B. Morales, A general description for the weight distribution of some reducible cyclic codes, IEEE Trans. Inform. Theory, 59 (2013), 5994-6001.  doi: 10.1109/TIT.2013.2263195.  Google Scholar [8] G. Vega, A critical review and some remarks about one- and two-weight irreducible cyclic codes, Finite Fields and Their Appl., 33 (2015), 1-13.  doi: 10.1016/j.ffa.2014.11.001.  Google Scholar [9] G. Vega, An improved method for determining the weight distribution of a family of $q$-ary cyclic codes, Applicable Algebra in Engineering, Communication and Computing, 28 (2017), 527-533.  doi: 10.1007/s00200-017-0318-y.  Google Scholar [10] J. Wolfmann, Are 2-weight projective cyclic codes irreducible?, IEEE Trans. Inform. Theory, 51 (2005), 733-737.  doi: 10.1109/TIT.2004.840882.  Google Scholar [11] J. Yang, M. S. Xiong, C. S. Ding and J. Q. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory, 59 (2013), 5985-5993.  doi: 10.1109/TIT.2013.2266731.  Google Scholar [12] L. Yu and H. W. Liu, The weight distribution of a family of $p$-ary cyclic codes, Des. Codes Cryptogr., 78 (2016), 731-745.  doi: 10.1007/s10623-014-0029-3.  Google Scholar

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##### References:
 [1] C. S. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inform. Theory, 55 (2009), 955-960.  doi: 10.1109/TIT.2008.2011511.  Google Scholar [2] C. S. Ding, Y. Liu, C. L. Ma and L. W. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, 57 (2011), 8000-8006.  doi: 10.1109/TIT.2011.2165314.  Google Scholar [3] T. Kløve, The weight distribution for a class of ICCs, Discrete Mathematics, 20 (1977), 87-90.   Google Scholar [4] R. Lidl and H. Niederreiter, Finite Fields, Second edition. Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.  Google Scholar [5] C. L. Ma, L. W. Zeng, Y. Liu, D. G. Feng and C. S. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inform. Theory, 57 (2011), 397-402.  doi: 10.1109/TIT.2010.2090272.  Google Scholar [6] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [7] G. Vega and L. B. Morales, A general description for the weight distribution of some reducible cyclic codes, IEEE Trans. Inform. Theory, 59 (2013), 5994-6001.  doi: 10.1109/TIT.2013.2263195.  Google Scholar [8] G. Vega, A critical review and some remarks about one- and two-weight irreducible cyclic codes, Finite Fields and Their Appl., 33 (2015), 1-13.  doi: 10.1016/j.ffa.2014.11.001.  Google Scholar [9] G. Vega, An improved method for determining the weight distribution of a family of $q$-ary cyclic codes, Applicable Algebra in Engineering, Communication and Computing, 28 (2017), 527-533.  doi: 10.1007/s00200-017-0318-y.  Google Scholar [10] J. Wolfmann, Are 2-weight projective cyclic codes irreducible?, IEEE Trans. Inform. Theory, 51 (2005), 733-737.  doi: 10.1109/TIT.2004.840882.  Google Scholar [11] J. Yang, M. S. Xiong, C. S. Ding and J. Q. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory, 59 (2013), 5985-5993.  doi: 10.1109/TIT.2013.2266731.  Google Scholar [12] L. Yu and H. W. Liu, The weight distribution of a family of $p$-ary cyclic codes, Des. Codes Cryptogr., 78 (2016), 731-745.  doi: 10.1007/s10623-014-0029-3.  Google Scholar
 Weight Frequency 0 1 $\frac{h}{q} (q^k+q^{k/2})$ $(N_1-1)^2(\frac{q^k-1}{N_1})^2$ $\frac{h}{2q} (2q^k-(N_1-2)q^{k/2})$ $2(N_1-1)(\frac{q^k-1}{N_1})^2$ $\frac{h}{2q} (q^k-(N_1-1)q^{k/2})$ $2\frac{q^k-1}{N_1}$ $\frac{h}{2q} (q^k+q^{k/2})$ $2(N_1-1)\frac{q^k-1}{N_1}$ $\frac{h}{q} (q^k-(N_1-1)q^{k/2})$ $(\frac{q^k-1}{N_1})^2$
 Weight Frequency 0 1 $\frac{h}{q} (q^k+q^{k/2})$ $(N_1-1)^2(\frac{q^k-1}{N_1})^2$ $\frac{h}{2q} (2q^k-(N_1-2)q^{k/2})$ $2(N_1-1)(\frac{q^k-1}{N_1})^2$ $\frac{h}{2q} (q^k-(N_1-1)q^{k/2})$ $2\frac{q^k-1}{N_1}$ $\frac{h}{2q} (q^k+q^{k/2})$ $2(N_1-1)\frac{q^k-1}{N_1}$ $\frac{h}{q} (q^k-(N_1-1)q^{k/2})$ $(\frac{q^k-1}{N_1})^2$
 Weight Frequency 0 1 $\frac{h}{q} (q^k-(-1)^{\alpha}q^{k/2})$ $(N_1-1)^2(\frac{q^k-1}{N_1})^2$ $\frac{h}{2q} (2q^k-(-1)^{\alpha+1}(N_1-2)q^{k/2})$ $2(N_1-1)(\frac{q^k-1}{N_1})^2$ $\frac{h}{2q} (q^k-(-1)^{\alpha+1}(N_1-1)q^{k/2})$ $2\frac{q^k-1}{N_1}$ $\frac{h}{2q} (q^k-(-1)^{\alpha}q^{k/2})$ $2(N_1-1)\frac{q^k-1}{N_1}$ $\frac{h}{q} (q^k-(-1)^{\alpha+1}(N_1-1)q^{k/2})$ $(\frac{q^k-1}{N_1})^2$
 Weight Frequency 0 1 $\frac{h}{q} (q^k-(-1)^{\alpha}q^{k/2})$ $(N_1-1)^2(\frac{q^k-1}{N_1})^2$ $\frac{h}{2q} (2q^k-(-1)^{\alpha+1}(N_1-2)q^{k/2})$ $2(N_1-1)(\frac{q^k-1}{N_1})^2$ $\frac{h}{2q} (q^k-(-1)^{\alpha+1}(N_1-1)q^{k/2})$ $2\frac{q^k-1}{N_1}$ $\frac{h}{2q} (q^k-(-1)^{\alpha}q^{k/2})$ $2(N_1-1)\frac{q^k-1}{N_1}$ $\frac{h}{q} (q^k-(-1)^{\alpha+1}(N_1-1)q^{k/2})$ $(\frac{q^k-1}{N_1})^2$
 Weight  Frequency $(0\leq u\leq e)$  $\frac{(q-1)q^k}{\delta eq}u$ $\binom{e}{u}(q^k-1)^u$
 Weight  Frequency $(0\leq u\leq e)$  $\frac{(q-1)q^k}{\delta eq}u$ $\binom{e}{u}(q^k-1)^u$
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