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On one-lee weight and two-lee weight $ \mathbb{Z}_2\mathbb{Z}_4[u] $ additive codes and their constructions

  • * Corresponding author: Huazhang Wu

    * Corresponding author: Huazhang Wu 

The research is supported by the Open Fund Research of Fund of Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, Anhui University

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  • This paper mainly study $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes. A Gray map from $ \mathbb{Z}_{2}^{\alpha}\times\mathbb{Z}_{4}^{\beta}[u] $ to $ \mathbb{Z}_{4}^{\alpha+2\beta} $ is defined, and we prove that is a weight preserving and distance preserving map. A MacWilliams-type identity between the Lee weight enumerator of a $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive code and its dual is proved. Some properties of one-weight $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes and two-weight projective $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes are discussed. As main results, some construction methods for one-weight and two-weight $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes are studied, meanwhile several examples are presented to illustrate the methods.

    Mathematics Subject Classification: Primary: 94B05, 94B15, 94B60.

    Citation:

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  • Table 1.  One-weight $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes

    Cases Weight Remark
    $ w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})=w_{L} ((2 +u)\mathbf{c})\neq0 $ $ 8k_{1} $ $ a+4a_{7}=2k_{1} $
    $ w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L} ((2+u)\mathbf{c})=0 $ $ 4k_{3} $ $ a+4a_{7}=2k_{3} $
    $ w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})=0 $ $ 4k_{2} $ $ a+4a_{7}=2k_{2} $
    $ w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})=w_{L} ((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})=0 $ $ 4k_{1} $ $ a+4a_{7}=2k_{1} $
    $ w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L} ((2 +u)\mathbf{c})=0 $ / /
    $ w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})=w_{L} ((2 +u)\mathbf{c})=0 $ / /
    $ w_{L}(\mathbf{c})=w_{L} ((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})=0 $ / /
    $ w_{L}(\mathbf{c})\neq0,w_{L} ((2 +u)\mathbf{c})=w_{L}(2\mathbf{c})=w_{L}(u\mathbf{c})=0 $ $ a+4a_{7} $ /
     | Show Table
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    Table 2.  Two-weight $\mathbb{Z}_{2}\mathbb{Z}_{4}[u]$-additive codes

    Cases $m_{1}$ $m_{2}$ Remark
    $w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})\neq0$ $4(k_{1}+k_{3})$ $8k_{1}$ $a=2k_{3}-4a_{7},k_{1}\neq k_{3}$
    $w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})\neq0$ $4(k_{1}+k_{2})$ $8k_{1}$ $a=2k_{2}-4a_{7},k_{1}\neq k_{2}$
    $w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})\neq0$ $4(k_{1}+k_{2})$ $8k_{2}$ $a=2k_{1}-4a_{7},k_{1}\neq k_{2}$
    $w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(\mathbf{c})\neq0$ $a+6k_{1}+4a_{7}$ $8k_{1}$ $a+4a_{7}\neq2k_{1}$
    $w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0$ $4(k_{1}+k_{2})$ $8k_{2}$ $a=4k_{2}-2k_{1}-4a_{7},k_{1}\neq k_{2}$
    $w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0$ $4(k_{1}+k_{2})$ $8k_{1}$ $a=4k_{1}-2k_{2}-4a_{7},k_{1}\neq k_{2}$
    $w_{L}(\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L}(u\mathbf{c})\neq0$ $4(k_{1}+k_{3})$ $8k_{1}$ $a=4k_{1}-2k_{3}-4a_{7},k_{1}\neq k_{3}$
    $w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})=0$ / / /
    $w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})=0$ / / /
    $w_{L}(u\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}(\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})=0$ $a+2k_{3}+4a_{7}$ $4k_{3}$ $a+4a_{7}\neq2k_{3}$
    $w_{L}(\mathbf{c})=w_{L}(u\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})=0$ / / /
    $w_{L}(\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})=0$ / / /
    $w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(\mathbf{c})\neq0,w_{L}(2\mathbf{c})=0$ $a+2k_{2}+4a_{7}$ $4k_{2}$ $a+4a_{7}\neq2k_{2}$
    $w_{L}(\mathbf{c})=w_{L}(2\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(u\mathbf{c})=0$ / / /
    $w_{L}(\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})=0$ / / /
    $w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(\mathbf{c})\neq0,w_{L}(u\mathbf{c})=0$ $a+2k_{1}+4a_{7}$ $4k_{1}$ $a+4a_{7}\neq2k_{1}$
    $w_{L}(\mathbf{c})\neq0,w_{L}(u\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L}((2 +u)\mathbf{c})=0$ / / /
    $w_{L}(\mathbf{c})\neq0,w_{L}(2\mathbf{c})\neq0,w_{L}(u\mathbf{c})=w_{L}((2 +u)\mathbf{c})=0$ / / /
    $w_{L}(\mathbf{c})\neq0,w_{L}((2 +u)\mathbf{c})\neq0,w_{L}(2\mathbf{c})=w_{L}(u\mathbf{c})=0$ / / /
     | Show Table
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    Table 3.  Code parameters comparison

    Examples Length of $\Phi(\mathcal{C})$ Size of $\Phi(\mathcal{C})$ Lee weight of $\Phi(\mathcal{C})$ Lee weight in Database in http://www.Z4codes.info/ Remark
    Ex. 5.3 (i) 8 4 8 8/10 As good as in Database
    Ex. 5.3 (ii) 10 2 12 / New value
    Ex. 5.5 (i) 32 4 32 32/42 As good as in Database
    Ex. 5.5 (ii) 30 4 32 30/40 Better than Database
    Ex. 5.5 (iii) 62 4 64 82
    Ex. 5.7 36 8 32 / New value
    Ex. 6.2 (i) 9 4 6 and 12 9/12 Optimal as per Database
    Ex. 6.2 (ii) 16 4 8 and 16 16/21 Optimal as per Database
    Ex. 6.4 (i) 18 4 16 and 20 18/24 Improves on Database
    Ex. 6.4 (ii) 11 4 12 and 13 11/14 Improves on Database
    Ex. 6.6 46 8 36 and 48 / New value
     | Show Table
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