American Institute of Mathematical Sciences

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May  2020, 14(2): 379-395. doi: 10.3934/amc.2020026

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

Murat Şahİn 1, and  Hayrullah Özİmamoğlu 2,,

1. 

Department of Mathematics, Faculty of Science, Ankara University, Ankara, 06100, Turkey
2. 

Department of Mathematics, Faculty of Arts and Sciences, Nevşehir Hacı Bektaş Veli University, Nevşehir, 50300, Turkey

* Corresponding author: Hayrullah Özimamoğlu

Received  October 2018 Revised  March 2019 Published  May 2020 Early access  September 2019

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.

Keywords: Additive codes, additive circulant codes, codes over $ \mathbb{F}_{4} $, optimal codes, classification.
Mathematics Subject Classification: Primary: 94B60; Secondary: 94B05.
Citation: Murat Şahİn, Hayrullah Özİmamoğlu. Additive Toeplitz codes over $ \mathbb{F}_{4} $. Advances in Mathematics of Communications, 2020, 14 (2) : 379-395. doi: 10.3934/amc.2020026
References:
[1]

A. R. Calderbank, E. M. Rains, P. 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. Gaborit, W. C. Huffman, J. 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. 

show all references

References:
[1]

A. R. Calderbank, E. M. Rains, P. 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. Gaborit, W. C. Huffman, J. 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. 

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 $
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$ \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 $
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) $
Table Options

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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) $
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) $
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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) $
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
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$ \boldsymbol{a_{2}} $ $ \boldsymbol{b_{2}} $ Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{2}}} $ $ (w,1) $ Circulant
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
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$ \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
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
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$ \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
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
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$ \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
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
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$ \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
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
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$ \boldsymbol{a_{3}} $ $ \boldsymbol{b_{3}} $ Circulant/Non-Circulant
$ \boldsymbol{u_{N_{3}}} $ $ \boldsymbol{u_{N_{3}}} $ Non-Circulant
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
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$ \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
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
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$ \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
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
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$ \boldsymbol{a_{11}} $ $ \boldsymbol{b_{11}} $ Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{11}}} $ $ \boldsymbol{u_{Q_{11}}} $ Non-Circulant
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
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$ \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
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 $ $ - $
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$ \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 $ $ - $
Table A.1.  The Generator Vector of Optimal Additive Circulant Code for $ n = 2 $
Upper Generator Vector Lower Generator Vector
$ (w,1) $ $ (w,1) $
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Upper Generator Vector Lower Generator Vector
$ (w,1) $ $ (w,1) $
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) $
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Upper Generator Vectors Lower Generator Vectors
$ (w,1,0) $ $ (w,0,1) $
$ (w,1,1) $ $ (w,1,1) $
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) $
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Upper Generator Vector Lower Generator Vector
$ (w,1,1,0) $ $ (w,0,1,1) $
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) $
Table Options

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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) $
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) $
Table Options

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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) $
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) $
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Upper Generator Vector Lower Generator Vector
$ (w,0,1,1,1,0) $ $ (w,0,1,1,1,0) $
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) $
Table Options

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Upper Generator Vector Lower Generator Vector
$ (w,1,0,1,1,0,0) $ $ (w,0,0,1,1,0,1) $
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) $
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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) $
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) $
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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) $
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) $
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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) $
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) $
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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) $
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) $
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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) $
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) $
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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) $
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) $
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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) $
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) $
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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) $
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) $
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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) $
[1]

W. Cary Huffman. Additive cyclic codes over $\mathbb F_4$. Advances in Mathematics of Communications, 2008, 2 (3) : 309-343. doi: 10.3934/amc.2008.2.309
[2]

W. Cary Huffman. Additive cyclic codes over $\mathbb F_4$. Advances in Mathematics of Communications, 2007, 1 (4) : 427-459. doi: 10.3934/amc.2007.1.427
[3]

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[4]

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[5]

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[6]

W. Cary Huffman. Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order. Advances in Mathematics of Communications, 2007, 1 (3) : 357-398. doi: 10.3934/amc.2007.1.357
[7]

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[10]

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[11]

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[12]

Habibul Islam, Om Prakash, Patrick Solé. $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes. Advances in Mathematics of Communications, 2021, 15 (4) : 737-755. doi: 10.3934/amc.2020094
[13]

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[14]

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[15]

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[16]

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[17]

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