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Complete weight enumerators of a class of linear codes over finite fields

  • *Corresponding author: Xiangli Kong

    *Corresponding author: Xiangli Kong 

The work is partially supported by the National Natural Science Foundation of China (11701317, 11801303, 11571380) and the Natural Science Foundation of Shandong Province of China (ZR2016AM04). This work is also partially supported by Guangzhou Science and Technology Program (201607010144) and the Natural Science Foundation of Jiangsu Higher Education Institutions of China (17KJB110018)

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  • We investigate a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. These codes have at most three weights and some of them are almost optimal so that they are suitable for applications in secret sharing schemes. This is a supplement of the results raised by Wang et al. (2017) and Kong et al. (2019).

    Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 11T23.

    Citation:

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  • Table 1.  Weight distribution of $ \mathcal{C} _{D_0} $ for odd $ e $

    Weight $ w $ Frequency $ A_w $
    $ 0 $ $ 1 $
    $ (p-1) p^{e-2} $ $ p^{e-1} -1 $
    $ (p-1) (p^{e-2}- p^{-2} \varepsilon_0 \eta_e(a) \sqrt{p^*} ) $ $ \dfrac{p-1}{2p} {\big( {{ q+ \varepsilon_0 \eta_e(a) \sqrt{p^*}}} \big) } $
    $ (p-1) (p^{e-2}+ p^{-2} \varepsilon_0 \eta_e(a) \sqrt{p^*} ) $ $ \dfrac{p-1}{2p} {\big( {{ q- \varepsilon_0 \eta_e(a) \sqrt{p^*}}} \big) } $
     | Show Table
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    Table 2.  Weight distribution of $ \mathcal{C} _{D_0} $ for even $ e $

    Weight $ w $ Frequency $ A_w $
    $ 0 $ $ 1 $
    $ (p-1) p^{e-2} $ $ p^{e-1} + (p-1)p^{-1} \varepsilon_0 \eta_e(a)-1 $
    $ (p-1) (p^{e-2}+ p^{-1} \varepsilon_0 \eta_e(a) ) $ $ (p-1) {\big( {{p^{e-1}- p^{-1} \varepsilon_0 \eta_e(a)}} \big) } $
     | Show Table
    DownLoad: CSV

    Table 3.  Weight distribution of $ \mathcal{C} _{D_c} $ for odd $ e $ and $ c \neq 0 $

    Weight $ w $ Frequency $ A_w $
    $ 0 $ $ 1 $
    $ (p-1) p^{e-2} $ $ p^{e-1} -1 $
    $ n_c-p^{e-2} + p^{-2}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) $ $ \frac{p-1}{2} n_c $
    $ n_c-p^{e-2} - p^{-2}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) $ $ \frac{p-1}{2}{\big( {{p^{e-1} - p^{-1}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) }} \big) } $
     | Show Table
    DownLoad: CSV

    Table 4.  Weight distribution of $ \mathcal{C} _{D_c} $ for even $ e $ and $ c\neq 0 $

    Weight $ w $ Frequency $ A_w $
    $ 0 $ $ 1 $
    $ (p-1) p^{e-2} $ $ \frac{p+1}{2}p^{e-1}+\frac{p-1}{2} p^{-1}\varepsilon_0 \eta_e(a)-1 $
    $ (p-1)p^{e-2} -2 p^{-1}\varepsilon_0 \eta_e(a) $ $ \frac{p-1}{2}n_c $
     | Show Table
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