American Institute of Mathematical Sciences

May  2018, 12(2): 415-428. doi: 10.3934/amc.2018025

On some classes of codes with a few weights

 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

Received  June 2017 Revised  February 2018 Published  March 2018

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

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.

Citation: Yiwei Liu, Zihui Liu. On some classes of codes with a few weights. Advances in Mathematics of Communications, 2018, 12 (2) : 415-428. doi: 10.3934/amc.2018025
References:

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References:
The weight distribution of the code $\mathcal{C}_{A}$ in Theorem 5
 Weight Multiplicity 0 1 $\frac{q+\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$ $\frac{q-1}{2}-\frac{q-1}{2r^{m}}S(a)$ $\frac{q-\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$ $\frac{q-1}{2}+\frac{q-1}{2r^{m}}S(a)$
 Weight Multiplicity 0 1 $\frac{q+\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$ $\frac{q-1}{2}-\frac{q-1}{2r^{m}}S(a)$ $\frac{q-\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$ $\frac{q-1}{2}+\frac{q-1}{2r^{m}}S(a)$
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
 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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