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Ternary Primitive LCD BCH codes

This paper was supported by the National Natural Science Foundation of China (No.61772015), the Foundation of Science and Technology on Information Assurance Laboratory (No.KJ-17-010) and the Foundation of Jinling Institute of Technology (No.JIT-B-202016, No.JIT-FHXM-2020). Y. Wu was sponsored by NUPTSF (No. NY220137). (Corresponding author:Yansheng Wu.)

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  • Absolute coset leaders were first proposed by the authors which have advantages in constructing binary LCD BCH codes. As a continue work, in this paper we focus on ternary linear codes. Firstly, we find the largest, second largest, and third largest absolute coset leaders of ternary primitive BCH codes. Secondly, we present three classes of ternary primitive BCH codes and determine their weight distributions. Finally, we obtain some LCD BCH codes and calculate some weight distributions. However, the calculation of weight distributions of two of these codes is equivalent to that of Kloosterman sums.

    Mathematics Subject Classification: Primary: 11T23, 94B05; Secondary: 11L05.


    \begin{equation} \\ \end{equation}
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  • Weight Frequency
    0 1
    $ \frac{2}3\cdot(3^m-3^{\frac m2}) $ $ \frac{3^m-1}{2} $
    $ \frac{2}3\cdot(3^m+3^{\frac m2}) $ $ \frac{3^m-1}{2} $
     | Show Table
    DownLoad: CSV
    Weight Frequency
    0 1
    $ \frac{2}3 \cdot(3^m-3^{\frac{m}2}) $ $ \frac{3^m-1}2 $
    $ \frac{2}3 \cdot(3^m+3^{\frac{m}2}) $ $ \frac{3^m-1}2 $
    $ \frac{1}3(2\cdot3^m+3^\frac{m}2)-1 $ $ 3^m-1 $
    $ \frac{1}3(2\cdot3^m-3^\frac{m}2)-1 $ $ 3^m-1 $
    $ 3^m-1 $ 2
     | Show Table
    DownLoad: CSV
    Weight Frequency
    0 1
    $ \frac12\cdot(3^m-1) $ 12
    $ \frac34\cdot(3^m-1) $ 8
    $ 3^m-1 $ 6
     | Show Table
    DownLoad: CSV
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