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A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes

The first author was supported by CAPES (Brazil). The work of the second author was partially supported by Spanish grants AICO/2017/128 of the Generalitat Valenciana and VIGROB-287 of the Universitat d'Alacant. The third and fourth authors were supported by NSERC (Canada). The first, third and fourth authors acknowledge support from FAPESP SPRINT grant 2016/50476-0

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  • In this paper we construct $ \mathbb{F}_2 $-linear codes over $ \mathbb{F}_{2}^{b} $ with length $ n $ and dimension $ n-r $ where $ n = rb $. These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime $ p $, let $ n = p-1 $ and utilize a partition of $ \mathbb{Z}_n $. Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to $ 0 $ as $ n $ grows (for a fixed $ r $). When $ r = 2 $ we prove that these codes are always maximum distance separable. For higher $ r $ some of them retain this property.

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

    Citation:

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  • Table 1.  Index array constructed in Section 3

    $0 $ $1 $ $2 $ $\cdots$ $p-3 $ $p-2 $
    $D_0 $ $ D_0+1 $ $ D_0+2 $ $\cdots$ $ D_0+p-3 $ $D_0+p-2 $
    $ D_1 $ $ D_1+1 $ $ D_1+2 $ $\cdots$ $ D_1+p-3 $ $D_1+p-2$
    $\vdots$ $\vdots$ $ \vdots $ $ $ \vdots $ $ \vdots $
    $ D_{u-1} $ $D_{u-1}+1$ $D_{u-1}+2 $ $\cdots$ $ D_{u-1}+p-3 $ $D_{u-1}+p-2 $
    $D_{u+1}$ $ D_{u+1}+1 $ $D_{u+1}+2$ $\cdots$ $ D_{u+1}+p-3 $ $D_{u+1}+p-2 $
    $\vdots$ $\vdots$ $ \vdots $ $ $ \vdots $ $ \vdots $
    $ D_{b-1} $ $ D_{b-1}+1 $ $D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $ $D_{b-1}+p-2 $
     | Show Table
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    Table 2.  Index array representing the parity-check matrix when $ r = 2 $

    $0 $ $1 $ $2 $ $\cdots $ $p-3 $ $p-2 $
    $ D_1 $ $ D_1+1 $ $ D_1+2 $ $ \cdots $ $ D_1+p-3 $ $D_1+p-2$
    $ D_2 $ $ D_2+1 $ $ D_2+2 $ $ \cdots $ $ D_2+p-3 $ $D_2+p-2 $
    $ \vdots $ $ \vdots $ $ \vdots $ $ $ \vdots $ $ \vdots $
    $ D_{b-1} $ $ D_{b-1}+1 $ $D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $ $D_{b-1}+p-2 $
     | Show Table
    DownLoad: CSV

    Table 3.  Index array representing the generator matrix computed from the index array in Table 2

    $B $ $ B+(b-1) $ $ B+2(b-1) $ $ B+3(b-1) $ $ B+4(b-1) $ $ \cdots $ $ B+(n-1)(b-1) $
    $ 0 $ $ b-1 $ $ 2(b-1) $ $ 3(b-1) $ $ 4(b-1) $ $ \cdots $ $(n-1)(b-1) $
    $ 1 $ $ b $ $ 2(b-1)+1 $ $ 3(b-1)+1 $ $ 4(b-1)+1 $ $ \cdots $ $(n-1)(b-1)+1 $
    $ 2 $ $ b+1 $ $ 2(b-1)+2 $ $ 3(b-1)+2 $ $ 4(b-1)+2 $ $ \cdots $ $(n-1)(b-1)+2 $
    $ \vdots $ $ \vdots $ $ \vdots $ $ \cdots $ $ \vdots $ $ $\vdots $
    $ b-2 $ $ 2b-3 $ $ 3(b-1)-1 $ $ 4(b-1)-1 $ $ 5(b-1)-1 $ $ \ldots $ $n(b-1)-1 $
     | Show Table
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    Table 4.  Index array obtained by subtracting $ 1 $ modulo $ n $ to every set in the array given in Table 1

    $p-2 $ $0 $ $1 $ $2 $ $\cdots $ $p-3 $
    $ D_0+p-2 $ $D_0 $ $D_0+1 $ $ D_0+2 % $ $ \cdots $ $ D_0+p-3 $
    $ D_1+p-2 $ $D_1 $ $D_1 +1 $ $ D_1+2 % $ $ \cdots $ $ D_1+p-3$
    $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ $ \vdots $
    $ D_{u-1}+p-2 $ $D_{u-1} $ $D_{u-1}+1 $ $ D_{u-1}+2 % $ $ \cdots $ $ D_{u-1}+p-3 $
    $ D_{u+1}+p-2 $ $D_{u+1} $ $D_{u+1} +1 $ $ D_{u+1}+2 % $ $ \cdots $ $ D_{u+1}+p-3 $
    $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ $ \vdots $
    $ D_{b-1}+p-2 $ $D_{b-1} $ $D_{b-1} +1 $ $ D_{b-1}+2 $ $ \cdots $ $ D_{b-1}+p-3 $
     | Show Table
    DownLoad: CSV

    Table 5.  MDS property of $ C(p, r, \alpha) $ for different values of $ p $ and $ r $

    $p $
    5 7 11 13 17 19 23 29 31 37 41 43
    $ r $ 2
    3 ×
    4 × × ×
    5 × × ×
    6 × × × × ×
    7 × ×
    8 × ×
    9 × ×
    10 × ×
    11 ×
    12 ×
    13
    14 × ×
    15 ×
     | Show Table
    DownLoad: CSV
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