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A decoding algorithm for 2D convolutional codes over the erasure channel

  • * Corresponding author: Julia Lieb

    * Corresponding author: Julia Lieb 
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  • Two-dimensional (2D) convolutional codes are a generalization of (1D) convolutional codes, which are suitable for transmission over an erasure channel. In this paper, we present a decoding algorithm for 2D convolutional codes over such a channel by reducing the decoding process to several decoding steps applied to 1D convolutional codes. Moreover, we provide constructions of 2D convolutional codes that are specially tailored to our decoding algorithm.

    Mathematics Subject Classification: Primary: 94B10; Secondary: 94B35.

    Citation:

    \begin{equation} \\ \end{equation}
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  • $ \hat{v}_{ij} $ $ j=0 $ $ j=1 $ $ j=2 $ $ j=3 $ $ j=4 $
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    $ \ast $ $ v_{01,2} $ $ \ast $ $ \ast $ $ \ast $
    $ \ast $ $ v_{01,3} $ $ \ast $ $ v_{03,3} $ $ \ast $
    $ i=1 $ $ \ast $ $ \ast $ $ v_{12,1} $ $ \ast $ $ v_{14,1} $
    $ \ast $ $ \ast $ $ v_{12,2} $ $ \ast $ $ v_{14,2} $
    $ \ast $ $ \ast $ $ v_{12,3} $ $ v_{13,3} $ $ v_{14,3} $
    $ i=2 $ $ v_{20,1} $ $ \ast $ $ \ast $ $ \ast $ $ v_{24,1} $
    $ v_{20,2} $ $ \ast $ $ \ast $ $ \ast $ $ v_{24,2} $
    $ v_{20,3} $ $ v_{21,3} $ $ \ast $ $ v_{23,3} $ $ v_{24,3} $
    $ i=3 $ $ \ast $ $ v_{31,1} $ $ \ast $ $ \ast $ $ v_{34,1} $
    $ \ast $ $ v_{31,2} $ $ \ast $ $ \ast $ $ v_{34,2} $
    $ \ast $ $ v_{31,3} $ $ \ast $ $ v_{33,3} $ $ v_{34,3} $
    $ i=4 $ $ \ast $ $ v_{41,1} $ $ \ast $ $ \ast $ $ v_{44,1} $
    $ \ast $ $ v_{41,2} $ $ \ast $ $ \ast $ $ v_{44,2} $
    $ \ast $ $ v_{41,3} $ $ \ast $ $ v_{43,3} $ $ v_{44,3} $
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    $ \hat{v}_{ij} $ $ j=0 $ $ j=1 $ $ j=2 $ $ j=3 $ $ j=4 $ $ j=5 $ $ j=6 $
    $ i=0 $ $ v_{00,1} $ $ \ast $ $ v_{02,1} $ $ \ast $ $ v_{04,1} $ $ v_{05,1} $ $ v_{06,1} $
    $ v_{00,2} $ $ \ast $ $ v_{02,2} $ $ \ast $ $ \ast $ $ v_{05,2} $ $ v_{06,2} $
    $ i=1 $ $ v_{10,1} $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{15,1} $ $ v_{16,1} $
    $ v_{10,2} $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{15,2} $ $ v_{16,2} $
    $ i=2 $ $ \ast $ $ v_{21,1} $ $ v_{22,1} $ $ \ast $ $ \ast $ $ v_{25,1} $ $ v_{26,1} $
    $ \ast $ $ v_{21,2} $ $ v_{22,2} $ $ \ast $ $ \ast $ $ v_{25,2} $ $ v_{26,2} $
    $ i=3 $ $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{34,1} $ $ v_{35,1} $ $ \ast $
    $ \ast $ $ \ast $ $ \ast $ $ \ast $ $ v_{34,2} $ $ v_{35,2} $ $ \ast $
    $ i=4 $ $ v_{40,1} $ $ v_{41,1} $ $ v_{42,1} $ $ \ast $ $ \ast $ $ v_{45,1} $ $ v_{46,1} $
    $ v_{40,2} $ $ v_{41,2} $ $ v_{42,2} $ $ \ast $ $ \ast $ $ v_{45,2} $ $ v_{46,2} $
    $ i=5 $ $ v_{50,1} $ $ v_{51,1} $ $ v_{52,1} $ $ \ast $ $ \ast $ $ v_{55,1} $ $ v_{56,1} $
    $ v_{50,2} $ $ v_{51,2} $ $ v_{52,2} $ $ \ast $ $ \ast $ $ v_{55,2} $ $ v_{56,2} $
    $ i=6 $ $ v_{60,1} $ $ v_{61,1} $ $ v_{62,1} $ $ \ast $ $ \ast $ $ v_{65,1} $ $ v_{66,1} $
    $ v_{60,2} $ $ v_{61,2} $ $ v_{62,2} $ $ \ast $ $ \ast $ $ v_{65,2} $ $ v_{66,2} $
     | Show Table
    DownLoad: CSV
    $ \hat{v}_{ij} $ $ j=0 $ $ \cdots $ $ j=\deg_{z_1}(v(z_1,z_2)) $
    $ i=0 $ $ \ast $ $ \cdots $ $ \ast $
    $ \vdots $ $ \vdots $ $ \vdots $
    $ i=(L_1+1)(n-k) $ $ \ast $ $ \cdots $ $ \ast $
    $ i=(L_1+1)(n-k)+1 $ $ v_{(L_1+1)(n-k)+1,0} $ $ \cdots $ $ v_{(L_1+1)(n-k)+1,\deg_{z_1}(v(z_1,z_2))} $
    $ \vdots $ $ \vdots $ $ \vdots $
    $ i=(L_1+1)n $ $ v_{(L_1+1)n,0} $ $ \cdots $ $ v_{(L_1+1)n,\deg_{z_1}(v(z_1,z_2))} $
     | Show Table
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