    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, doi: 10.3934/amc.2020059
##### References:
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show all references

##### References:
  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  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  T. Kløve, The weight distribution for a class of ICCs, Discrete Mathematics, 20 (1977), 87-90.   Google Scholar  R. Lidl and H. Niederreiter, Finite Fields, Second edition. Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. Google Scholar  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  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  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  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  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  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  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  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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