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On some classes of codes with a few weights

  • * Corresponding author: Zhimin Sun

    * Corresponding author: Zhimin Sun
This work was supported by The National Science Foundation of China (No. 11171366)
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  • 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.

    Mathematics Subject Classification: 94B05.

    Citation:

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

    WeightMultiplicity
    01
    $\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

    WeightMultiplicity
    01
    $\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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