\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Additive Toeplitz codes over $ \mathbb{F}_{4} $

  • * Corresponding author: Hayrullah Özimamoğlu

    * Corresponding author: Hayrullah Özimamoğlu
Abstract Full Text(HTML) Figure(0) / Table(30) Related Papers Cited by
  • In this paper, we introduce additive Toeplitz codes over $ \mathbb{F}_{4} $. The additive Toeplitz codes are a generalization of additive circulant codes over $ \mathbb{F}_{4} $. We find many optimal additive Toeplitz codes (OATC) over $ \mathbb{F}_{4} $. These optimal codes also contain optimal non-circulant codes, so we find new additive codes in this manner. We provide some theorems to partially classify OATC. Then, we give a new algorithm that fully classifies OATC by combining these theorems with Gaborit's algorithm. We classify OATC over $ \mathbb{F}_{4} $ of length up to $ 13 $. We obtain $ 2 $ inequivalent optimal additive toeplitz codes (IOATC) that are non-circulant codes of length $ 5 $, $ 92 $ of length $ 8 $, $ 2068 $ of length $ 9 $, and $ 39 $ of length $ 11 $. Moreover, we improve an idea related to quadratic residue codes to construct optimal and near-optimal additive Toeplitz codes over $ \mathbb{F}_{4} $ of length prime $ p $. We obtain many optimal and near-optimal additive Toeplitz codes for some primes $ p $ from this construction.

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

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Table 2.1.  Number of OATC

    $ \boldsymbol{n} $ $ \boldsymbol{d_{max}} $ $ \boldsymbol{\#} $ All OATC $ \boldsymbol{\#} $ OATC with $ \boldsymbol{r_{a}}\leq \boldsymbol{s_{b}} $
    $ 2 $ $ 2 $ $ 1 $ $ 1 $
    $ 3 $ $ 2 $ $ 8 $ $ 6 $
    $ 4 $ $ 3 $ $ 2 $ $ 2 $
    $ 5 $ $ 3 $ $ 36 $ $ 26 $
    $ 6 $ $ 4 $ $ 1 $ $ 1 $
    $ 7 $ $ 4 $ $ 6 $ $ 6 $
    $ 8 $ $ 4 $ $ 292 $ $ 197 $
    $ 9 $ $ 4 $ $ 4338 $ $ 2709 $
    $ 10 $ $ 5 $ $ 24 $ $ 24 $
    $ 11 $ $ 5 $ $ 325 $ $ 292 $
    $ 12 $ $ 6 $ $ 6 $ $ 6 $
    $ 13 $ $ 6 $ $ ? $ $ 28 $
     | Show Table
    DownLoad: CSV

    Table 2.2.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 5 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,1,0,0) $ $ (w,0,0,1,1) $
    $ (w,0,1,1,0) $ $ (w,0,1,1,0) $
     | Show Table
    DownLoad: CSV

    Table 2.3.  The Generator Vectors of Optimal Additive Non-Circulant Codes for $ n = 5 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,0,0,1) $ $ (w,1,1,0,0) $
    $ (w,1,0,1,0) $ $ (w,0,1,1,1) $
     | Show Table
    DownLoad: CSV

    Table 2.4.  The Generator Vectors of Optimal Additive Toeplitz Code for $ p = 2 $

    $ \boldsymbol{a_{2}} $ $ \boldsymbol{b_{2}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{2}}} $ $ (w,1) $ Circulant
     | Show Table
    DownLoad: CSV

    Table 2.5.  The Generator Vectors of Optimal Additive Toeplitz Codes for $ p = 3 $

    $ \boldsymbol{a_{3}} $ $ \boldsymbol{b_{3}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{3}}} $ $ (w,1,0) $ Non-Circulant
    $ \boldsymbol{u_{Q_{3}}} $ $ (w,0,1) $ Circulant
    $ \boldsymbol{u_{Q_{3}}} $ $ (w,1,1) $ Non-Circulant
    $ \boldsymbol{u_{N_{3}}} $ $ (w,1,0) $ Circulant
    $ \boldsymbol{u_{N_{3}}} $ $ (w,1,1) $ Non-Circulant
     | Show Table
    DownLoad: CSV

    Table 2.6.  The Generator Vectors of Optimal Additive Toeplitz Codes for $ p = 5 $

    $ \boldsymbol{a_{5}} $ $ \boldsymbol{b_{5}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{5}}} $ $ (w,1,0,0,1) $ Circulant
    $ \boldsymbol{u_{Q_{5}}} $ $ (w,1,1,0,0) $ Non-Circulant
    $ \boldsymbol{u_{Q_{5}}} $ $ (w,1,1,1,0) $ Non-Circulant
    $ \boldsymbol{u_{Q_{5}}} $ $ (w,1,0,1,1) $ Non-Circulant
    $ \boldsymbol{u_{Q_{5}}} $ $ (w,1,1,0,1) $ Non-Circulant
    $ \boldsymbol{u_{N_{5}}} $ $ (w,0,1,1,0) $ Circulant
    $ \boldsymbol{u_{N_{5}}} $ $ (w,1,1,1,0) $ Non-Circulant
    $ \boldsymbol{u_{N_{5}}} $ $ (w,0,1,1,1) $ Non-Circulant
     | Show Table
    DownLoad: CSV

    Table 2.7.  The Generator Vectors of Optimal Additive Toeplitz Codes for $ p = 11 $

    $ \boldsymbol{a_{11}} $ $ \boldsymbol{b_{11}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{11}}} $ $ (w,0,1,0,0,0,1,1,1,0,1) $ Circulant
    $ \boldsymbol{u_{N_{11}}} $ $ (w,1,0,1,1,1,0,0,0,1,0) $ Circulant
     | Show Table
    DownLoad: CSV

    Table 2.8.  The Generator Vectors of Near-Optimal Additive Toeplitz Codes for $ p = 2 $

    $ \boldsymbol{a_{2}} $ $ \boldsymbol{b_{2}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{2}}} $ $ \boldsymbol{u_{N_{2}}} $ Non-Circulant
    $ \boldsymbol{u_{N_{2}}} $ $ \boldsymbol{u_{Q_{2}}} $ Non-Circulant
    $ \boldsymbol{u_{N_{2}}} $ $ \boldsymbol{u_{N_{2}}} $ Circulant
     | Show Table
    DownLoad: CSV

    Table 2.9.  The Generator Vectors of Near-Optimal Additive Toeplitz Code for $ p = 3 $

    $ \boldsymbol{a_{3}} $ $ \boldsymbol{b_{3}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{N_{3}}} $ $ \boldsymbol{u_{N_{3}}} $ Non-Circulant
     | Show Table
    DownLoad: CSV

    Table 2.10.  The Generator Vectors of Near-Optimal Additive Toeplitz Codes for $ p = 5 $

    $ \boldsymbol{a_{5}} $ $ \boldsymbol{b_{5}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{5}}} $ $ \boldsymbol{u_{N_{5}}} $ Non-Circulant
    $ \boldsymbol{u_{N_{5}}} $ $ \boldsymbol{u_{Q_{5}}} $ Non-Circulant
     | Show Table
    DownLoad: CSV

    Table 2.11.  The Generator Vectors of Near-Optimal Additive Toeplitz Codes for $ p = 7 $

    $ \boldsymbol{a_{7}} $ $ \boldsymbol{b_{7}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{7}}} $ $ \boldsymbol{u_{Q_{7}}} $ Non-Circulant
    $ \boldsymbol{u_{Q_{7}}} $ $ \boldsymbol{u_{N_{7}}} $ Circulant
    $ \boldsymbol{u_{N_{7}}} $ $ \boldsymbol{u_{Q_{7}}} $ Circulant
     | Show Table
    DownLoad: CSV

    Table 2.12.  The Generator Vectors of Near-Optimal Additive Toeplitz Code for $ p = 11 $

    $ \boldsymbol{a_{11}} $ $ \boldsymbol{b_{11}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{11}}} $ $ \boldsymbol{u_{Q_{11}}} $ Non-Circulant
     | Show Table
    DownLoad: CSV

    Table 2.13.  The Generator Vectors of Near-Optimal Additive Toeplitz Codes for $ p = 13 $

    $ \boldsymbol{a_{13}} $ $ \boldsymbol{b_{13}} $ Circulant/Non-Circulant
    $ \boldsymbol{u_{Q_{13}}} $ $ \boldsymbol{u_{Q_{13}}} $ Circulant
    $ \boldsymbol{u_{N_{13}}} $ $ \boldsymbol{u_{N_{13}}} $ Circulant
     | Show Table
    DownLoad: CSV

    Table 3.1.  Number of Inequivalent Optimal Additive Circulant and Non-Circulant Codes

    $ \boldsymbol{n} $ $ \boldsymbol{d_{max}} $ $ \boldsymbol{\#} $ All Toeplitz Codes $ \boldsymbol{\#} $ Circulant Codes $ \boldsymbol{\#} $ Non-Circulant Codes
    $ 2 $ $ 2 $ $ 1 $ $ 1 $ $ - $
    $ 3 $ $ 2 $ $ 2 $ $ 2 $ $ - $
    $ 4 $ $ 3 $ $ 1 $ $ 1 $ $ - $
    $ 5 $ $ 3 $ $ 4 $ $ 2 $ $ 2 $
    $ 6 $ $ 4 $ $ 1 $ $ 1 $ $ - $
    $ 7 $ $ 4 $ $ 1 $ $ 1 $ $ - $
    $ 8 $ $ 4 $ $ 102 $ $ 10 $ $ 92 $
    $ 9 $ $ 4 $ $ 2083 $ $ 15 $ $ 2068 $
    $ 10 $ $ 5 $ $ 3 $ $ 3 $ $ - $
    $ 11 $ $ 5 $ $ 52 $ $ 13 $ $ 39 $
    $ 12 $ $ 6 $ $ 2 $ $ 2 $ $ - $
    $ 13 $ $ 6 $ $ 2 $ $ 2 $ $ - $
     | Show Table
    DownLoad: CSV

    Table A.1.  The Generator Vector of Optimal Additive Circulant Code for $ n = 2 $

    Upper Generator Vector Lower Generator Vector
    $ (w,1) $ $ (w,1) $
     | Show Table
    DownLoad: CSV

    Table A.2.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 3 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,0) $ $ (w,0,1) $
    $ (w,1,1) $ $ (w,1,1) $
     | Show Table
    DownLoad: CSV

    Table A.3.  The Generator Vector of Optimal Additive Circulant Code for $ n = 4 $

    Upper Generator Vector Lower Generator Vector
    $ (w,1,1,0) $ $ (w,0,1,1) $
     | Show Table
    DownLoad: CSV

    Table A.4.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 5 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,1,0,0) $ $ (w,0,0,1,1) $
    $ (w,0,1,1,0) $ $ (w,0,1,1,0) $
     | Show Table
    DownLoad: CSV

    Table A.5.  The Generator Vectors of Optimal Additive Non-Circulant Codes for $ n = 5 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,0,0,1) $ $ (w,1,1,0,0) $
    $ (w,1,0,1,0) $ $ (w,0,1,1,1) $
     | Show Table
    DownLoad: CSV

    Table A.6.  The Generator Vector of Optimal Additive Circulant Code for $ n = 6 $

    Upper Generator Vector Lower Generator Vector
    $ (w,0,1,1,1,0) $ $ (w,0,1,1,1,0) $
     | Show Table
    DownLoad: CSV

    Table A.7.  The Generator Vector of Optimal Additive Circulant Code for $ n = 7 $

    Upper Generator Vector Lower Generator Vector
    $ (w,1,0,1,1,0,0) $ $ (w,0,0,1,1,0,1) $
     | Show Table
    DownLoad: CSV

    Table A.8.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 8 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,1,0,1,0,0,0) $ $ (w,0,0,0,1,0,1,1) $
    $ (w,1,0,1,1,0,0,0) $ $ (w,0,0,0,1,1,0,1) $
    $ (w,0,1,1,1,0,0,0) $ $ (w,0,0,0,1,1,1,0) $
    $ (w,1,1,0,0,1,0,0) $ $ (w,0,0,1,0,0,1,1) $
    $ (w,1,0,1,0,1,0,0) $ $ (w,0,0,1,0,1,0,1) $
    $ (w,0,1,1,0,1,0,0) $ $ (w,0,0,1,0,1,1,0) $
    $ (w,1,1,0,1,1,0,0) $ $ (w,0,0,1,1,0,1,1) $
    $ (w,0,0,1,1,1,0,0) $ $ (w,0,0,1,1,1,0,0) $
    $ (w,1,0,1,1,1,0,0) $ $ (w,0,0,1,1,1,0,1) $
    $ (w,1,1,0,1,0,1,0) $ $ (w,0,1,0,1,0,1,1) $
     | Show Table
    DownLoad: CSV

    Table A.9.  The Generator Vectors of Optimal Additive Non-Circulant Codes for $ n = 8 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,0,1,1,0,1,0,0) $ $ (w,0,0,1,0,1,1,1) $
    $ (w,0,1,1,0,1,0,0) $ $ (w,0,0,1,1,1,1,0) $
    $ (w,0,1,1,0,1,0,0) $ $ (w,1,1,1,0,0,0,1) $
    $ (w,0,1,1,0,1,0,0) $ $ (w,0,1,1,1,0,1,0) $
    $ (w,0,1,1,0,1,0,0) $ $ (w,0,1,0,0,1,1,1) $
    $ (w,1,0,0,1,0,1,0) $ $ (w,0,1,0,1,0,1,1) $
    $ (w,1,0,0,1,0,1,0) $ $ (w,1,1,0,0,1,1,0) $
    $ (w,1,0,0,1,0,0,1) $ $ (w,1,0,1,0,1,1,0) $
    $ (w,0,1,0,1,0,0,1) $ $ (w,1,1,0,1,0,1,0) $
    $ (w,0,1,0,1,0,0,1) $ $ (w,1,1,0,0,0,1,1) $
    $ (w,0,0,1,0,0,1,1) $ $ (w,1,1,0,1,1,0,0) $
    $ (w,0,1,1,0,0,0,1) $ $ (w,1,0,0,1,1,1,1) $
    $ (w,0,1,0,0,1,0,1) $ $ (w,1,1,0,0,0,1,1) $
    $ (w,0,1,0,0,1,0,1) $ $ (w,1,1,1,0,1,0,1) $
    $ (w,0,1,0,0,1,0,1) $ $ (w,1,1,1,0,1,1,1) $
    $ (w,1,0,1,1,0,0,0) $ $ (w,0,1,1,1,0,0,0) $
    $ (w,1,0,1,1,0,0,0) $ $ (w,0,0,1,1,1,0,0) $
    $ (w,1,0,1,1,0,0,0) $ $ (w,1,1,0,0,1,1,0) $
    $ (w,1,0,1,1,0,0,0) $ $ (w,0,1,1,1,0,0,1) $
    $ (w,1,0,1,1,0,0,0) $ $ (w,1,0,0,1,1,0,1) $
    $ (w,1,0,1,1,0,0,0) $ $ (w,0,0,1,1,1,0,1) $
    $ (w,1,0,1,1,0,0,0) $ $ (w,1,1,1,0,1,1,0) $
    $ (w,0,0,1,1,0,1,0) $ $ (w,0,1,0,1,1,0,1) $
    $ (w,0,0,1,0,1,0,1) $ $ (w,1,1,0,0,1,0,0) $
    $ (w,0,0,1,0,1,0,1) $ $ (w,1,0,1,1,1,0,0) $
    $ (w,0,0,1,0,1,0,1) $ $ (w,1,1,0,1,1,0,0) $
    $ (w,0,0,1,0,1,0,1) $ $ (w,1,0,1,1,1,1,1) $
    $ (w,1,1,0,0,0,1,0) $ $ (w,1,1,0,1,0,0,0) $
    $ (w,1,1,0,0,0,1,0) $ $ (w,1,1,0,0,1,0,1) $
    $ (w,1,1,0,0,0,1,0) $ $ (w,1,1,0,1,0,1,1) $
    $ (w,1,1,0,0,0,1,0) $ $ (w,1,1,0,1,1,1,0) $
    $ (w,0,0,0,1,1,0,1) $ $ (w,1,1,0,1,0,0,0) $
    $ (w,0,0,0,1,1,0,1) $ $ (w,1,0,1,1,1,0,0) $
    $ (w,0,0,0,1,1,0,1) $ $ (w,1,1,0,1,0,1,0) $
    $ (w,1,1,0,1,0,0,0) $ $ (w,1,0,0,0,0,1,1) $
    $ (w,1,1,0,1,0,0,0) $ $ (w,1,1,0,0,1,1,0) $
    $ (w,1,1,0,1,0,0,0) $ $ (w,1,1,0,0,0,1,1) $
    $ (w,1,1,0,1,0,0,0) $ $ (w,1,0,1,1,1,0,1) $
    $ (w,1,0,0,0,1,1,0) $ $ (w,0,1,1,1,0,0,0) $
    $ (w,0,1,1,1,0,0,0) $ $ (w,0,1,0,1,1,0,1) $
    $ (w,0,1,1,1,0,0,0) $ $ (w,0,1,0,1,1,1,0) $
    $ (w,1,0,1,0,0,0,1) $ $ (w,1,1,0,0,0,0,1) $
    $ (w,1,0,1,0,0,0,1) $ $ (w,1,0,0,0,1,1,1) $
    $ (w,1,0,1,0,0,0,1) $ $ (w,1,1,0,0,0,1,1) $
    $ (w,1,0,1,0,0,0,1) $ $ (w,1,0,0,1,1,0,1) $
    $ (w,1,0,1,0,0,0,1) $ $ (w,1,0,0,1,1,1,1) $
     | Show Table
    DownLoad: CSV

    Table A.10.  The Generator Vectors of Optimal Additive Non-Circulant Codes for $ n = 8 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,1,0,0,1,0,0) $ $ (w,0,1,0,1,1,0,1) $
    $ (w,0,0,1,1,1,0,0) $ $ (w,1,0,1,0,1,0,0) $
    $ (w,0,0,1,0,1,1,0) $ $ (w,1,0,1,1,1,0,0) $
    $ (w,0,0,1,0,1,1,0) $ $ (w,0,1,1,1,1,0,1) $
    $ (w,0,0,1,0,1,1,0) $ $ (w,1,1,1,1,0,1,1) $
    $ (w,1,0,0,0,1,0,1) $ $ (w,1,1,1,0,0,0,1) $
    $ (w,1,0,0,0,1,0,1) $ $ (w,1,1,1,0,1,0,0) $
    $ (w,1,0,0,0,0,1,1) $ $ (w,1,1,0,0,1,1,0) $
    $ (w,1,0,0,0,0,1,1) $ $ (w,1,1,0,1,1,0,0) $
    $ (w,1,0,0,0,0,1,1) $ $ (w,1,1,0,1,0,0,1) $
    $ (w,1,0,1,0,1,0,0) $ $ (w,0,1,1,0,1,0,1) $
    $ (w,1,0,0,1,1,0,0) $ $ (w,0,1,0,1,1,1,0) $
    $ (w,1,0,0,1,1,0,0) $ $ (w,1,1,1,0,1,1,0) $
    $ (w,1,0,0,1,1,0,0) $ $ (w,1,1,0,1,1,1,0) $
    $ (w,0,0,0,1,0,1,1) $ $ (w,1,1,1,0,1,1,0) $
    $ (w,0,0,0,1,0,1,1) $ $ (w,1,1,0,1,1,1,1) $
    $ (w,0,0,0,1,1,1,0) $ $ (w,0,1,1,1,0,1,0) $
    $ (w,0,0,0,1,1,1,0) $ $ (w,0,1,1,1,0,1,1) $
    $ (w,1,1,0,0,0,0,1) $ $ (w,1,0,1,1,1,0,0) $
    $ (w,1,1,0,0,0,0,1) $ $ (w,1,0,1,1,1,0,1) $
    $ (w,0,1,0,1,1,0,0) $ $ (w,0,0,1,1,0,1,1) $
    $ (w,0,1,0,0,1,1,0) $ $ (w,1,1,1,1,0,0,1) $
    $ (w,0,1,0,1,1,0,1) $ $ (w,0,0,1,1,0,1,1) $
    $ (w,0,0,1,0,1,1,1) $ $ (w,1,0,1,1,1,0,0) $
    $ (w,0,0,1,0,1,1,1) $ $ (w,1,0,1,0,1,1,0) $
    $ (w,0,0,1,1,1,1,0) $ $ (w,0,1,1,0,1,0,1) $
    $ (w,0,0,1,1,1,1,0) $ $ (w,0,1,1,0,1,1,1) $
    $ (w,0,0,1,1,1,1,0) $ $ (w,0,1,1,1,1,0,1) $
    $ (w,0,1,1,0,1,0,1) $ $ (w,1,1,1,1,0,1,0) $
    $ (w,0,0,1,1,0,1,1) $ $ (w,0,1,1,1,0,1,1) $
    $ (w,0,0,1,1,0,1,1) $ $ (w,1,0,1,1,1,1,1) $
    $ (w,0,1,0,1,1,1,0) $ $ (w,0,1,1,0,1,1,1) $
    $ (w,1,0,1,0,0,1,1) $ $ (w,0,1,1,1,1,1,0) $
    $ (w,1,0,1,1,1,0,0) $ $ (w,1,1,0,0,1,1,0) $
    $ (w,1,1,0,0,1,1,0) $ $ (w,1,1,0,1,0,1,1) $
    $ (w,1,0,1,1,0,1,0) $ $ (w,1,1,1,0,1,1,0) $
    $ (w,1,0,1,1,0,1,0) $ $ (w,1,1,1,0,0,1,1) $
    $ (w,1,1,0,0,0,1,1) $ $ (w,1,1,0,1,0,1,1) $
    $ (w,1,1,0,0,0,1,1) $ $ (w,1,1,0,1,1,1,0) $
    $ (w,1,1,0,0,1,0,1) $ $ (w,1,1,0,0,1,0,1) $
    $ (w,1,1,0,0,1,0,1) $ $ (w,1,1,1,0,0,1,1) $
    $ (w,1,0,0,1,1,0,1) $ $ (w,1,0,0,1,1,0,1) $
    $ (w,1,0,0,1,1,0,1) $ $ (w,1,1,1,1,0,1,1) $
    $ (w,1,0,1,1,1,0,1) $ $ (w,0,1,0,1,1,1,1) $
    $ (w,0,1,1,1,0,1,1) $ $ (w,1,0,1,0,1,1,1) $
    $ (w,0,1,1,0,1,1,1) $ $ (w,1,1,1,0,1,0,1) $
     | Show Table
    DownLoad: CSV

    Table A.11.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 9 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,1,0,1,0,0,0,0) $ $ (w,0,0,0,0,1,0,1,1) $
    $ (w,0,1,1,1,0,0,0,0) $ $ (w,0,0,0,0,1,1,1,0) $
    $ (w,1,1,1,0,1,0,0,0) $ $ (w,0,0,0,1,0,1,1,1) $
    $ (w,1,0,0,1,1,0,0,0) $ $ (w,0,0,0,1,1,0,0,1) $
    $ (w,0,1,0,1,1,0,0,0) $ $ (w,0,0,0,1,1,0,1,0) $
    $ (w,0,0,1,1,1,0,0,0) $ $ (w,0,0,0,1,1,1,0,0) $
    $ (w,1,1,1,0,0,1,0,0) $ $ (w,0,0,1,0,0,1,1,1) $
    $ (w,1,0,0,1,1,1,0,0) $ $ (w,0,0,1,1,1,0,0,1) $
    $ (w,0,0,1,1,1,1,0,0) $ $ (w,0,0,1,1,1,1,0,0) $
    $ (w,0,1,1,1,1,1,0,0) $ $ (w,0,0,1,1,1,1,1,0) $
    $ (w,1,1,0,1,0,0,1,0) $ $ (w,0,1,0,0,1,0,1,1) $
    $ (w,1,1,1,1,0,0,1,0) $ $ (w,0,1,0,0,1,1,1,1) $
    $ (w,1,1,0,1,1,0,1,0) $ $ (w,0,1,0,1,1,0,1,1) $
    $ (w,1,1,0,1,0,1,1,0) $ $ (w,0,1,1,0,1,0,1,1) $
    $ (w,1,1,1,1,0,1,1,0) $ $ (w,0,1,1,0,1,1,1,1) $
     | Show Table
    DownLoad: CSV

    Table A.12.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 10 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,0,1,1,0,1,0,0,0) $ $ (w,0,0,0,1,0,1,1,0,1) $
    $ (w,1,0,1,0,0,1,1,0,0) $ $ (w,0,0,1,1,0,0,1,0,1) $
    $ (w,0,1,1,0,1,1,1,0,0) $ $ (w,0,0,1,1,1,0,1,1,0) $
     | Show Table
    DownLoad: CSV

    Table A.13.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 11 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,0,0,1,0,0,1,1,0,1,0) $ $ (w,0,1,0,1,1,0,0,1,0,0) $
    $ (w,0,1,0,0,0,0,0,1,1,1) $ $ (w,1,1,1,0,0,0,0,0,1,0) $
    $ (w,1,0,1,0,1,1,0,0,0,0) $ $ (w,0,0,0,0,1,1,0,1,0,1) $
    $ (w,0,1,1,0,1,0,0,0,1,0) $ $ (w,0,1,0,0,0,1,0,1,1,0) $
    $ (w,0,0,0,1,1,0,1,1,0,0) $ $ (w,0,0,1,1,0,1,1,0,0,0) $
    $ (w,0,0,1,0,0,1,0,0,1,1) $ $ (w,1,1,0,0,1,0,0,1,0,0) $
    $ (w,1,0,1,1,1,0,0,0,0,0) $ $ (w,0,0,0,0,0,1,1,1,0,1) $
    $ (w,0,1,1,1,0,0,1,0,0,1) $ $ (w,1,0,0,1,0,0,1,1,1,0) $
    $ (w,1,1,0,1,1,0,0,1,0,0) $ $ (w,0,0,1,0,0,1,1,0,1,1) $
    $ (w,1,1,0,0,1,0,1,1,0,0) $ $ (w,0,0,1,1,0,1,0,0,1,1) $
    $ (w,1,0,1,1,0,0,0,1,1,0) $ $ (w,0,1,1,0,0,0,1,1,0,1) $
    $ (w,0,0,1,1,0,0,1,0,1,1) $ $ (w,1,1,0,1,0,0,1,1,0,0) $
    $ (w,0,1,0,0,0,1,1,1,0,1) $ $ (w,1,0,1,1,1,0,0,0,1,0) $
     | Show Table
    DownLoad: CSV

    Table A.14.  The Generator Vectors of Optimal Additive Non-Circulant Codes for $ n = 11 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,0,1,0,0,0,0,0,1,1,1) $ $ (w,1,1,1,0,1,0,1,1,0,1) $
    $ (w,0,1,1,1,0,0,1,0,0,0) $ $ (w,1,0,1,0,1,0,0,0,1,1) $
    $ (w,1,1,0,0,0,1,0,1,0,0) $ $ (w,1,0,1,0,1,0,0,0,1,1) $
    $ (w,1,1,0,1,0,0,0,0,1,0) $ $ (w,1,1,0,0,0,0,1,0,1,1) $
    $ (w,0,1,1,0,1,0,0,0,1,0) $ $ (w,1,1,0,1,1,1,1,0,0,0) $
    $ (w,1,0,0,0,0,1,0,0,1,1) $ $ (w,1,1,1,1,0,1,1,1,1,0) $
    $ (w,0,0,0,0,1,0,1,1,0,1) $ $ (w,1,1,1,1,0,1,0,0,0,0) $
    $ (w,0,0,0,1,1,0,0,1,1,0) $ $ (w,1,1,1,0,0,1,1,0,0,0) $
    $ (w,0,0,1,0,1,0,0,1,0,1) $ $ (w,1,0,1,0,1,1,0,1,0,0) $
    $ (w,0,1,0,0,0,1,1,0,1,0) $ $ (w,0,1,1,0,1,0,0,1,1,0) $
    $ (w,0,0,1,0,0,0,1,1,1,0) $ $ (w,1,1,0,1,1,0,1,0,1,0) $
    $ (w,1,0,1,0,0,1,0,1,0,0) $ $ (w,0,0,1,0,1,1,0,1,0,1) $
    $ (w,1,1,0,0,0,0,1,0,0,1) $ $ (w,1,1,1,0,1,1,1,1,0,0) $
    $ (w,1,1,0,1,0,1,0,0,0,0) $ $ (w,0,0,0,0,1,1,1,0,1,1) $
    $ (w,1,1,0,0,0,0,0,1,1,0) $ $ (w,0,1,1,0,0,1,0,1,1,0) $
    $ (w,0,0,1,1,0,0,0,1,1,0) $ $ (w,1,0,1,1,1,1,1,0,1,1) $
    $ (w,0,1,0,0,1,1,0,1,0,0) $ $ (w,0,1,0,0,1,1,1,1,1,1) $
    $ (w,0,1,0,0,1,1,1,0,0,0) $ $ (w,0,0,0,1,1,1,0,0,1,1) $
    $ (w,0,0,0,1,0,1,1,0,0,1) $ $ (w,1,0,0,1,1,0,1,1,0,0) $
    $ (w,0,0,0,0,1,0,1,0,1,1) $ $ (w,1,1,0,1,1,1,0,0,0,0) $
    $ (w,1,0,0,0,1,1,0,0,0,1) $ $ (w,1,1,0,0,1,1,0,0,0,0) $
    $ (w,1,0,0,0,1,1,0,0,0,1) $ $ (w,1,1,0,0,1,1,0,0,0,1) $
    $ (w,1,1,0,0,1,1,0,0,0,0) $ $ (w,1,0,0,0,1,1,0,0,1,1) $
    $ (w,0,0,1,0,1,1,0,1,0,0) $ $ (w,1,0,1,0,1,1,0,1,0,0) $
    $ (w,1,0,1,0,0,1,0,0,1,0) $ $ (w,1,0,1,0,0,1,0,0,1,0) $
    $ (w,1,0,1,1,0,0,0,1,0,0) $ $ (w,1,1,1,1,0,1,0,0,1,1) $
    $ (w,1,0,1,1,0,0,0,1,1,0) $ $ (w,1,0,1,1,1,1,0,1,1,0) $
    $ (w,0,1,1,0,0,1,0,1,0,1) $ $ (w,1,1,1,1,1,0,0,1,1,0) $
    $ (w,0,0,1,1,1,0,1,1,0,0) $ $ (w,1,0,1,1,0,1,1,1,0,0) $
    $ (w,1,0,0,1,1,0,0,0,1,1) $ $ (w,0,1,1,0,1,0,1,0,1,0) $
    $ (w,0,0,0,1,1,0,1,1,1,0) $ $ (w,1,0,0,1,1,0,1,1,0,0) $
    $ (w,0,1,0,0,1,0,1,1,1,0) $ $ (w,0,1,0,1,1,1,1,0,0,1) $
    $ (w,1,0,0,1,1,0,0,1,0,1) $ $ (w,0,1,0,1,1,1,1,1,0,1) $
    $ (w,0,0,0,1,1,1,0,1,0,1) $ $ (w,1,0,1,0,1,0,1,1,0,0) $
    $ (w,0,0,0,1,0,1,1,0,1,1) $ $ (w,1,1,0,1,1,0,1,0,0,1) $
    $ (w,0,1,1,1,1,0,0,0,1,0) $ $ (w,1,1,0,0,0,1,1,1,1,0) $
    $ (w,0,1,1,0,0,0,1,1,0,1) $ $ (w,0,0,1,1,0,1,1,1,0,1) $
    $ (w,1,1,1,0,1,1,0,0,0,1) $ $ (w,0,1,0,1,1,1,1,1,0,0) $
    $ (w,0,1,0,1,1,1,1,0,0,1) $ $ (w,1,0,1,1,1,0,1,1,1,1) $
     | Show Table
    DownLoad: CSV

    Table A.15.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 12 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,0,0,1,0,1,1,1,0,1,0,0) $ $ (w,0,0,1,0,1,1,1,0,1,0,0) $
    $ (w,0,1,1,0,1,1,1,1,0,1,0) $ $ (w,0,1,0,1,1,1,1,0,1,1,0) $
     | Show Table
    DownLoad: CSV

    Table A.16.  The Generator Vectors of Optimal Additive Circulant Codes for $ n = 13 $

    Upper Generator Vectors Lower Generator Vectors
    $ (w,1,0,1,0,0,1,1,1,0,0,0,0) $ $ (w,0,0,0,0,1,1,1,0,0,1,0,1) $
    $ (w,1,1,1,0,1,1,1,1,1,0,1,0) $ $ (w,0,1,0,1,1,1,1,1,0,1,1,1) $
     | Show Table
    DownLoad: CSV
  • [1] A. R. CalderbankE. M. RainsP. M. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.
    [2] J. Cannon, W. Bosma, C. Fieker and A. Steel, Handbook of Magma Functions, Version 2.19, Sydney, 2013.
    [3] L. E. Danielsen and M. G. Parker, Directed graph representation of half-rate additive codes over GF(4), Des. Codes Cryptogr., 59 (2011), 119-130.  doi: 10.1007/s10623-010-9469-6.
    [4] L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over GF(4) of length up to 12, J. Combin. Theory Ser. A, 113 (2006), 1351-1367.  doi: 10.1016/j.jcta.2005.12.004.
    [5] P. GaboritW. C. HuffmanJ. L. Kim and V. Pless, On additive GF(4) codes, DIMACS Workshop Codes Assoc. Schemes, DIMACS Ser. Discr. Math. Theoret. Comp. Sci., Amer. Math. Soc., 56 (2001), 135-149. 
    [6] T. A. Gulliver and J.-L. Kim, Circulant based extremal additive self-dual codes over GF(4), IEEE Trans. on Inform. Theory, 50 (2004), 359-366.  doi: 10.1109/TIT.2003.822616.
    [7] G. Höhn, Self-dual codes over the Kleinian four group, Math. Ann., 327 (2003), 227-255.  doi: 10.1007/s00208-003-0440-y.
    [8] P. R. J. Östergard, Classifying subspaces of Hamming spaces, Des. Codes Cryptogr., 27 (2002), 297-305.  doi: 10.1023/A:1019903407222.
    [9] V. S. Pless and W. C. Huffman, Handbook of Coding Theory, North-Holland, Amsterdam, 1998.
    [10] Z. Varbanov, Some new results for additive self-dual codes over GF(4), Serdica J. Comput., 1 (2007), 213-227. 
  • 加载中

Tables(30)

SHARE

Article Metrics

HTML views(1656) PDF downloads(408) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return