American Institute of Mathematical Sciences

Advanced
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  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, 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. 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. 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. 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. 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. 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. 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. 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 $
Table Options

Download as excel

$ \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

Download as excel

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) $
Table Options

Download as excel

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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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 $ $ - $
Table Options

Download as excel

$ \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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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

Download as excel

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

Download as excel

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) $
Table Options

Download as excel

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

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
Table Options

Download as excel

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]

Ken Saito. Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs. Advances in Mathematics of Communications, 2019, 13 (2) : 213-220. doi: 10.3934/amc.2019014
[4]

Evangeline P. Bautista, Philippe Gaborit, Jon-Lark Kim, Judy L. Walker. s-extremal additive $\mathbb F_4$ codes. Advances in Mathematics of Communications, 2007, 1 (1) : 111-130. doi: 10.3934/amc.2007.1.111
[5]

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
[6]

Helena Rifà-Pous, Josep Rifà, Lorena Ronquillo. $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography. Advances in Mathematics of Communications, 2011, 5 (3) : 425-433. doi: 10.3934/amc.2011.5.425
[7]

Makoto Araya, Masaaki Harada, Hiroki Ito, Ken Saito. On the classification of $\mathbb{Z}_4$-codes. Advances in Mathematics of Communications, 2017, 11 (4) : 747-756. doi: 10.3934/amc.2017054
[8]

Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038
[9]

Lars Eirik Danielsen. Graph-based classification of self-dual additive codes over finite fields. Advances in Mathematics of Communications, 2009, 3 (4) : 329-348. doi: 10.3934/amc.2009.3.329
[10]

T. Aaron Gulliver, Masaaki Harada, Hiroki Miyabayashi. Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$. Advances in Mathematics of Communications, 2007, 1 (2) : 223-238. doi: 10.3934/amc.2007.1.223
[11]

Thomas Feulner. Canonization of linear codes over $\mathbb Z$4. Advances in Mathematics of Communications, 2011, 5 (2) : 245-266. doi: 10.3934/amc.2011.5.245
[12]

Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[13]

Karim Samei, Saadoun Mahmoudi. Singleton bounds for R-additive codes. Advances in Mathematics of Communications, 2018, 12 (1) : 107-114. doi: 10.3934/amc.2018006
[14]

T. Aaron Gulliver, Masaaki Harada. On the performance of optimal double circulant even codes. Advances in Mathematics of Communications, 2017, 11 (4) : 767-775. doi: 10.3934/amc.2017056
[15]

Joaquim Borges, Cristina Fernández-Córdoba, Roger Ten-Valls. On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 169-179. doi: 10.3934/amc.2018011
[16]

Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020029
[17]

Fatmanur Gursoy, Irfan Siap, Bahattin Yildiz. Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$. Advances in Mathematics of Communications, 2014, 8 (3) : 313-322. doi: 10.3934/amc.2014.8.313
[18]

Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043
[19]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[20]

Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (17)
  • HTML views (87)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences