Article Contents

2018, Volume 12, Issue 2: 415-428. Doi: 10.3934/amc.2018025

This issue Previous Article New families of strictly optimal frequency hopping sequence sets Next Article

On some classes of codes with a few weights

Yiwei Liu^1, and
Zihui Liu^2, ,

1.
School of Computer Science & Technology, Beijing Institute of Technology, Beijing 100081, China
2.
Department of Mathematics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

^* Corresponding author: Zhimin Sun
^* Corresponding author: Zhimin Sun

Received: June 2017

Revised: February 2018

Published: May 2018

This work was supported by The National Science Foundation of China (No. 11171366)

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Abstract

We generalize the code constructed recently by Wang et al, and obtain many classes of codes with a few weights. The weight distribution of these codes is completely determined, and the minimum distance of the duals of these codes is determined. We also show that some subclasses of the duals of these codes are optimal. Furthermore, some parameters of the generalized Hamming weight of these codes are calculated in certain cases.

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Mathematics Subject Classification: 94B05.

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Table 1. The weight distribution of the code $\mathcal{C}_{A}$ in Theorem 5

Weight	Multiplicity
0	1
$\frac{q+\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$	$\frac{q-1}{2}-\frac{q-1}{2r^{m}}S(a)$
$\frac{q-\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$	$\frac{q-1}{2}+\frac{q-1}{2r^{m}}S(a)$

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Table 2. The weight distribution of the code $\mathcal{C}_{A}$ in Theorem 10

Weight	Multiplicity
0	1
$\frac{q}{4}$	$\frac{q-3}{2}+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\frac{q-1}{2r^{m}}S(a)$
$\frac{q+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}2\sqrt{q}}{4}$	$\frac{q-1}{2}-(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\frac{q-1}{2r^{m}}S(a)$
$\frac{q+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\sqrt{q}}{4}-\frac{q+\sqrt{q}}{4r^{m}}S(a)$	1

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