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

Gerardo Vega; Jesús E. Cuén-Ramos

doi:10.3934/amc.2020059

Article Contents

2020, Volume 14, Issue 3: 525-533. Doi: 10.3934/amc.2020059

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

Gerardo Vega^1, , and
Jesús E. Cuén-Ramos^2,†,

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
^* Corresponding author: Gerardo Vega

^† Ph.D. student.

Received: November 2018

Revised: October 2019

Early access: January 2020

Published: August 2020

Partially supported by PAPIIT-UNAM IN109818

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 94B15, 11T71.

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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 $

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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 $

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Table 3.

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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References

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