doi: 10.3934/amc.2020026

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

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  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.

Citation: Murat Şahİn, Hayrullah Özİmamoğlu. Additive Toeplitz codes over $ \mathbb{F}_{4} $. Advances in Mathematics of Communications, doi: 10.3934/amc.2020026
References:
[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. Google Scholar

[2]

J. Cannon, W. Bosma, C. Fieker and A. Steel, Handbook of Magma Functions, Version 2.19, Sydney, 2013.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

G. Höhn, Self-dual codes over the Kleinian four group, Math. Ann., 327 (2003), 227-255. doi: 10.1007/s00208-003-0440-y. Google Scholar

[8]

P. R. J. Östergard, Classifying subspaces of Hamming spaces, Des. Codes Cryptogr., 27 (2002), 297-305. doi: 10.1023/A:1019903407222. Google Scholar

[9]

V. S. Pless and W. C. Huffman, Handbook of Coding Theory, North-Holland, Amsterdam, 1998.Google Scholar

[10]

Z. Varbanov, Some new results for additive self-dual codes over GF(4), Serdica J. Comput., 1 (2007), 213-227. Google Scholar

show all references

References:
[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. Google Scholar

[2]

J. Cannon, W. Bosma, C. Fieker and A. Steel, Handbook of Magma Functions, Version 2.19, Sydney, 2013.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

G. Höhn, Self-dual codes over the Kleinian four group, Math. Ann., 327 (2003), 227-255. doi: 10.1007/s00208-003-0440-y. Google Scholar

[8]

P. R. J. Östergard, Classifying subspaces of Hamming spaces, Des. Codes Cryptogr., 27 (2002), 297-305. doi: 10.1023/A:1019903407222. Google Scholar

[9]

V. S. Pless and W. C. Huffman, Handbook of Coding Theory, North-Holland, Amsterdam, 1998.Google Scholar

[10]

Z. Varbanov, Some new results for additive self-dual codes over GF(4), Serdica J. Comput., 1 (2007), 213-227. Google Scholar

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 $
$ \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) $
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) $
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
$ \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
$ \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
$ \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
$ \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
$ \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
$ \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
$ \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
$ \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
$ \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
$ \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 $ $ - $
$ \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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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) $
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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