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The punctured codes of two classes of cyclic codes with few weights

  • *Corresponding author: Xiaoqiang Wang

    *Corresponding author: Xiaoqiang Wang

This work was partially supported by National Natural Science Foundation of China under Grant 11971156 and 12001175, and National Key Research and Development Program of China under Grant 2021YFA1000600

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  • The puncturing technique is one of important tools for constructing new codes from old ones. In recent years, many classes of linear codes with interesting parameters have been obtained with this technique. Based on quadratic Gauss sums, the puncturing technique and cyclotomic classes, we investigate two classes of punctured codes of cyclic codes over finite fields. The weight distribution and the parameters of the duals of these codes are studied. The parameters of these codes are new.

    Mathematics Subject Classification: Primary: 94B05; Secondary: 94B15.


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  • Table 1.  The weight distribution of $ \mathcal{C}_{D_0} $

    Weight Multiplicity
    $ 0 $ $ 1 $
    $ (p-1)p^{m-2} $ $ 2p^{2m-2}-2p^{2m-3}+p^{m-2}-1 $
    $ (p-1)p^{m-2}\pm p^{\frac{m}{2}-1} $ $ \begin{array}{c}\frac{1}{2}\left(p^{\frac{m}{2}-1}-p^{\frac{m}{2}-2}\right)\Big(p^{\frac{3m}{2}}-p^{\frac{3m}{2}-1}+p^{\frac{3m}{2}-2}\mp p^{m+1}\pm p^m \\ \mp p^{m-2}- p^{\frac{m}{2}}\pm1\Big)\end{array} $
    $ (p-1)p^{m-2}\pm (p-1)p^{\frac{m}{2}-1} $ $ \frac{1}{2}\left(p^{m-4}\mp(p-1)p^{\frac{m}{2}-3}\right)\left(p^m-p\mp(p-1)p^{\frac{m}{2}}\right) $
     | Show Table
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    Table 2.  The weight distribution of $ \mathcal{C}_{D_\epsilon} $ for $ \epsilon\in {\mathrm{GF}}(p)^* $

    Weight Multiplicity
    $ 0 $ $ 1 $
    $ (p-1)p^{m-2} $ $ 2p^{2m-1}-2p^{2m-2}+p^{m-1}-p $
    $ (p-1)p^{m-2}\pm p^{\frac{m}{2}-1} $ $ \frac{1}{2}(p^{m-1}-p^{m-2})(p^{m+1}-p^m+p^{m-1}\pm p^{\frac{m}{2}}\mp p^{\frac{m}{2}-1}-1) $
    $ (p-1)p^{m-2}\pm (p-1)p^{\frac{m}{2}-1} $ $ \frac{1}{2}\left(p^{2m-3}-p^{m-2}\mp(p-1)p^{\frac{3m}{2}-3}\right) $
    $ p^{m-1} $ $ p-1 $
     | Show Table
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    Table 3.  The weight distribution of $ \mathcal{C}_D $

    Weight Multiplicity
    $ 0 $ $ 1 $
    $ \frac{h(t-1)p^{r-s}}{t} $ $ t^2(p^{r}-1) $
    $ \frac{hp^{r-s}}{2} $ $ 2(p^{r}-1) $
    $ \frac{h(2t-1)p^{r-s}}{2t} $ $ 2t(p^{r}-t)(p^{r}-1) $
    $ hp^{r-s} $ $ (p^{r}-1)(p^{2r}+p^{r}-2tp^{r}+t^2-1) $
     | Show Table
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