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:
[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. DingY. LiuC. 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. MaL. W. ZengY. LiuD. 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. YangM. S. XiongC. 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

show all references

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. DingY. LiuC. 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. MaL. W. ZengY. LiuD. 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. YangM. S. XiongC. 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

Table 1.   
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 $
Table 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 $
Table 3.   
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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