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Two-weight and three-weight linear codes constructed from Weil sums

  • * Corresponding author: Shudi Yang

    * Corresponding author: Shudi Yang

The work is partially supported by National Natural Science Foundation of China under Grant 12071247 and by Research and Innovation Fund for Graduate Dissertations of Qufu Normal University in 2021 under Grant LWCXS202133

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  • Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.

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

    Citation:

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  • Table 1.  The weight distribution of $ C_{D_1} $

    weight multiplicity
    0 1
    $ q^{2m-2}\left(q-1 \right) -q^{3s-2} $ $ \left(q^m-q^{m-1}+q^s-q^{s-1}\right) \left(q-1\right) $
    $ q^{2m-2}\left( q-1\right) $ $ q^m\left(q^m-q+1 \right)-1 $
    $ \left( q^{2m-2}+q^{3s-2}\right)\left(q-1 \right) $ $ \left(q^{m-1}-q^s+q^{s-1}\right) \left(q-1\right) $
     | Show Table
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    Table 2.  The weight distribution of $ C_{D_2} $ when $ q\equiv 1\pmod{4} $

    weight multiplicity
    0 1
    $ \; \qquad q^{2m-2}\left( q-1\right) $ $ \left( q^m-1\right)\left( q^{m-1}+1\right) \qquad \; $
    $ \; \qquad q^{m-1}\left( q^{m-1}+1\right) \left(q-1 \right) $ $ q^{m-1}\left( q^m-1\right) \left( q-1\right) \qquad \; $
     | Show Table
    DownLoad: CSV

    Table 3.  The weight distribution of $ C_{D_2} $ when $ q\equiv 3\pmod{4} $

    weight multiplicity
    0 1
    $ \;q^{m-1}\left( q^{m-1}+(-1)^s\right) \left(q-1 \right) $ $ q^{m-1}\left( q^m-(-1)^s\right) \left( q-1\right) \; $
    $ \;q^{2m-2}\left( q-1\right) $ $ \left( q^m-(-1)^s\right)\left( q^{m-1}+(-1)^s\right) \; $
     | Show Table
    DownLoad: CSV

    Table 4.  The weight distribution of $ C_{D_2} $ when $ q $ is even and $ q>2 $

    weight multiplicity
    0 1
    $ \left( q^{2m-2}-q^{3s-2}\right)\left(q-1 \right) $ $ \dfrac{1}{2}\left(q^m+q^s \right) \left(q-1\right) $
    $ q^{2m-2}\left( q-1\right) $ $ q^m\left(q^m-q+1 \right)-1 $
    $ \left( q^{2m-2}+q^{3s-2}\right) \left(q-1 \right) $ $ \dfrac{1}{2}\left(q^m-q^s \right) \left(q-1\right) $
     | Show Table
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