# American Institute of Mathematical Sciences

May  2020, 14(2): 379-395. 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  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.

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
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 [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.

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##### 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.
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 $\boldsymbol{r_{a}}\leq \boldsymbol{s_{b}}$ $2$ $2$ $1$ $1$ $3$ $2$ $8$ $6$ $4$ $3$ $2$ $2$ $5$ $3$ $36$ $26$ $6$ $4$ $1$ $1$ $7$ $4$ $6$ $6$ $8$ $4$ $292$ $197$ $9$ $4$ $4338$ $2709$ $10$ $5$ $24$ $24$ $11$ $5$ $325$ $292$ $12$ $6$ $6$ $6$ $13$ $6$ $?$ $28$
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 $(w,1,1,0,0)$ $(w,0,0,1,1)$ $(w,0,1,1,0)$ $(w,0,1,1,0)$
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 $(w,1,0,0,1)$ $(w,1,1,0,0)$ $(w,1,0,1,0)$ $(w,0,1,1,1)$
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
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
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
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
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
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
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
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
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
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
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 $2$ $2$ $1$ $1$ $-$ $3$ $2$ $2$ $2$ $-$ $4$ $3$ $1$ $1$ $-$ $5$ $3$ $4$ $2$ $2$ $6$ $4$ $1$ $1$ $-$ $7$ $4$ $1$ $1$ $-$ $8$ $4$ $102$ $10$ $92$ $9$ $4$ $2083$ $15$ $2068$ $10$ $5$ $3$ $3$ $-$ $11$ $5$ $52$ $13$ $39$ $12$ $6$ $2$ $2$ $-$ $13$ $6$ $2$ $2$ $-$
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 $(w,1)$ $(w,1)$
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 $(w,1,0)$ $(w,0,1)$ $(w,1,1)$ $(w,1,1)$
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 $(w,1,1,0)$ $(w,0,1,1)$
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 $(w,1,1,0,0)$ $(w,0,0,1,1)$ $(w,0,1,1,0)$ $(w,0,1,1,0)$
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 $(w,1,0,0,1)$ $(w,1,1,0,0)$ $(w,1,0,1,0)$ $(w,0,1,1,1)$
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 $(w,0,1,1,1,0)$ $(w,0,1,1,1,0)$
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 $(w,1,0,1,1,0,0)$ $(w,0,0,1,1,0,1)$
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 $(w,1,1,0,1,0,0,0)$ $(w,0,0,0,1,0,1,1)$ $(w,1,0,1,1,0,0,0)$ $(w,0,0,0,1,1,0,1)$ $(w,0,1,1,1,0,0,0)$ $(w,0,0,0,1,1,1,0)$ $(w,1,1,0,0,1,0,0)$ $(w,0,0,1,0,0,1,1)$ $(w,1,0,1,0,1,0,0)$ $(w,0,0,1,0,1,0,1)$ $(w,0,1,1,0,1,0,0)$ $(w,0,0,1,0,1,1,0)$ $(w,1,1,0,1,1,0,0)$ $(w,0,0,1,1,0,1,1)$ $(w,0,0,1,1,1,0,0)$ $(w,0,0,1,1,1,0,0)$ $(w,1,0,1,1,1,0,0)$ $(w,0,0,1,1,1,0,1)$ $(w,1,1,0,1,0,1,0)$ $(w,0,1,0,1,0,1,1)$
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 $(w,0,1,1,0,1,0,0)$ $(w,0,0,1,0,1,1,1)$ $(w,0,1,1,0,1,0,0)$ $(w,0,0,1,1,1,1,0)$ $(w,0,1,1,0,1,0,0)$ $(w,1,1,1,0,0,0,1)$ $(w,0,1,1,0,1,0,0)$ $(w,0,1,1,1,0,1,0)$ $(w,0,1,1,0,1,0,0)$ $(w,0,1,0,0,1,1,1)$ $(w,1,0,0,1,0,1,0)$ $(w,0,1,0,1,0,1,1)$ $(w,1,0,0,1,0,1,0)$ $(w,1,1,0,0,1,1,0)$ $(w,1,0,0,1,0,0,1)$ $(w,1,0,1,0,1,1,0)$ $(w,0,1,0,1,0,0,1)$ $(w,1,1,0,1,0,1,0)$ $(w,0,1,0,1,0,0,1)$ $(w,1,1,0,0,0,1,1)$ $(w,0,0,1,0,0,1,1)$ $(w,1,1,0,1,1,0,0)$ $(w,0,1,1,0,0,0,1)$ $(w,1,0,0,1,1,1,1)$ $(w,0,1,0,0,1,0,1)$ $(w,1,1,0,0,0,1,1)$ $(w,0,1,0,0,1,0,1)$ $(w,1,1,1,0,1,0,1)$ $(w,0,1,0,0,1,0,1)$ $(w,1,1,1,0,1,1,1)$ $(w,1,0,1,1,0,0,0)$ $(w,0,1,1,1,0,0,0)$ $(w,1,0,1,1,0,0,0)$ $(w,0,0,1,1,1,0,0)$ $(w,1,0,1,1,0,0,0)$ $(w,1,1,0,0,1,1,0)$ $(w,1,0,1,1,0,0,0)$ $(w,0,1,1,1,0,0,1)$ $(w,1,0,1,1,0,0,0)$ $(w,1,0,0,1,1,0,1)$ $(w,1,0,1,1,0,0,0)$ $(w,0,0,1,1,1,0,1)$ $(w,1,0,1,1,0,0,0)$ $(w,1,1,1,0,1,1,0)$ $(w,0,0,1,1,0,1,0)$ $(w,0,1,0,1,1,0,1)$ $(w,0,0,1,0,1,0,1)$ $(w,1,1,0,0,1,0,0)$ $(w,0,0,1,0,1,0,1)$ $(w,1,0,1,1,1,0,0)$ $(w,0,0,1,0,1,0,1)$ $(w,1,1,0,1,1,0,0)$ $(w,0,0,1,0,1,0,1)$ $(w,1,0,1,1,1,1,1)$ $(w,1,1,0,0,0,1,0)$ $(w,1,1,0,1,0,0,0)$ $(w,1,1,0,0,0,1,0)$ $(w,1,1,0,0,1,0,1)$ $(w,1,1,0,0,0,1,0)$ $(w,1,1,0,1,0,1,1)$ $(w,1,1,0,0,0,1,0)$ $(w,1,1,0,1,1,1,0)$ $(w,0,0,0,1,1,0,1)$ $(w,1,1,0,1,0,0,0)$ $(w,0,0,0,1,1,0,1)$ $(w,1,0,1,1,1,0,0)$ $(w,0,0,0,1,1,0,1)$ $(w,1,1,0,1,0,1,0)$ $(w,1,1,0,1,0,0,0)$ $(w,1,0,0,0,0,1,1)$ $(w,1,1,0,1,0,0,0)$ $(w,1,1,0,0,1,1,0)$ $(w,1,1,0,1,0,0,0)$ $(w,1,1,0,0,0,1,1)$ $(w,1,1,0,1,0,0,0)$ $(w,1,0,1,1,1,0,1)$ $(w,1,0,0,0,1,1,0)$ $(w,0,1,1,1,0,0,0)$ $(w,0,1,1,1,0,0,0)$ $(w,0,1,0,1,1,0,1)$ $(w,0,1,1,1,0,0,0)$ $(w,0,1,0,1,1,1,0)$ $(w,1,0,1,0,0,0,1)$ $(w,1,1,0,0,0,0,1)$ $(w,1,0,1,0,0,0,1)$ $(w,1,0,0,0,1,1,1)$ $(w,1,0,1,0,0,0,1)$ $(w,1,1,0,0,0,1,1)$ $(w,1,0,1,0,0,0,1)$ $(w,1,0,0,1,1,0,1)$ $(w,1,0,1,0,0,0,1)$ $(w,1,0,0,1,1,1,1)$
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 $(w,1,1,0,0,1,0,0)$ $(w,0,1,0,1,1,0,1)$ $(w,0,0,1,1,1,0,0)$ $(w,1,0,1,0,1,0,0)$ $(w,0,0,1,0,1,1,0)$ $(w,1,0,1,1,1,0,0)$ $(w,0,0,1,0,1,1,0)$ $(w,0,1,1,1,1,0,1)$ $(w,0,0,1,0,1,1,0)$ $(w,1,1,1,1,0,1,1)$ $(w,1,0,0,0,1,0,1)$ $(w,1,1,1,0,0,0,1)$ $(w,1,0,0,0,1,0,1)$ $(w,1,1,1,0,1,0,0)$ $(w,1,0,0,0,0,1,1)$ $(w,1,1,0,0,1,1,0)$ $(w,1,0,0,0,0,1,1)$ $(w,1,1,0,1,1,0,0)$ $(w,1,0,0,0,0,1,1)$ $(w,1,1,0,1,0,0,1)$ $(w,1,0,1,0,1,0,0)$ $(w,0,1,1,0,1,0,1)$ $(w,1,0,0,1,1,0,0)$ $(w,0,1,0,1,1,1,0)$ $(w,1,0,0,1,1,0,0)$ $(w,1,1,1,0,1,1,0)$ $(w,1,0,0,1,1,0,0)$ $(w,1,1,0,1,1,1,0)$ $(w,0,0,0,1,0,1,1)$ $(w,1,1,1,0,1,1,0)$ $(w,0,0,0,1,0,1,1)$ $(w,1,1,0,1,1,1,1)$ $(w,0,0,0,1,1,1,0)$ $(w,0,1,1,1,0,1,0)$ $(w,0,0,0,1,1,1,0)$ $(w,0,1,1,1,0,1,1)$ $(w,1,1,0,0,0,0,1)$ $(w,1,0,1,1,1,0,0)$ $(w,1,1,0,0,0,0,1)$ $(w,1,0,1,1,1,0,1)$ $(w,0,1,0,1,1,0,0)$ $(w,0,0,1,1,0,1,1)$ $(w,0,1,0,0,1,1,0)$ $(w,1,1,1,1,0,0,1)$ $(w,0,1,0,1,1,0,1)$ $(w,0,0,1,1,0,1,1)$ $(w,0,0,1,0,1,1,1)$ $(w,1,0,1,1,1,0,0)$ $(w,0,0,1,0,1,1,1)$ $(w,1,0,1,0,1,1,0)$ $(w,0,0,1,1,1,1,0)$ $(w,0,1,1,0,1,0,1)$ $(w,0,0,1,1,1,1,0)$ $(w,0,1,1,0,1,1,1)$ $(w,0,0,1,1,1,1,0)$ $(w,0,1,1,1,1,0,1)$ $(w,0,1,1,0,1,0,1)$ $(w,1,1,1,1,0,1,0)$ $(w,0,0,1,1,0,1,1)$ $(w,0,1,1,1,0,1,1)$ $(w,0,0,1,1,0,1,1)$ $(w,1,0,1,1,1,1,1)$ $(w,0,1,0,1,1,1,0)$ $(w,0,1,1,0,1,1,1)$ $(w,1,0,1,0,0,1,1)$ $(w,0,1,1,1,1,1,0)$ $(w,1,0,1,1,1,0,0)$ $(w,1,1,0,0,1,1,0)$ $(w,1,1,0,0,1,1,0)$ $(w,1,1,0,1,0,1,1)$ $(w,1,0,1,1,0,1,0)$ $(w,1,1,1,0,1,1,0)$ $(w,1,0,1,1,0,1,0)$ $(w,1,1,1,0,0,1,1)$ $(w,1,1,0,0,0,1,1)$ $(w,1,1,0,1,0,1,1)$ $(w,1,1,0,0,0,1,1)$ $(w,1,1,0,1,1,1,0)$ $(w,1,1,0,0,1,0,1)$ $(w,1,1,0,0,1,0,1)$ $(w,1,1,0,0,1,0,1)$ $(w,1,1,1,0,0,1,1)$ $(w,1,0,0,1,1,0,1)$ $(w,1,0,0,1,1,0,1)$ $(w,1,0,0,1,1,0,1)$ $(w,1,1,1,1,0,1,1)$ $(w,1,0,1,1,1,0,1)$ $(w,0,1,0,1,1,1,1)$ $(w,0,1,1,1,0,1,1)$ $(w,1,0,1,0,1,1,1)$ $(w,0,1,1,0,1,1,1)$ $(w,1,1,1,0,1,0,1)$
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 $(w,1,1,0,1,0,0,0,0)$ $(w,0,0,0,0,1,0,1,1)$ $(w,0,1,1,1,0,0,0,0)$ $(w,0,0,0,0,1,1,1,0)$ $(w,1,1,1,0,1,0,0,0)$ $(w,0,0,0,1,0,1,1,1)$ $(w,1,0,0,1,1,0,0,0)$ $(w,0,0,0,1,1,0,0,1)$ $(w,0,1,0,1,1,0,0,0)$ $(w,0,0,0,1,1,0,1,0)$ $(w,0,0,1,1,1,0,0,0)$ $(w,0,0,0,1,1,1,0,0)$ $(w,1,1,1,0,0,1,0,0)$ $(w,0,0,1,0,0,1,1,1)$ $(w,1,0,0,1,1,1,0,0)$ $(w,0,0,1,1,1,0,0,1)$ $(w,0,0,1,1,1,1,0,0)$ $(w,0,0,1,1,1,1,0,0)$ $(w,0,1,1,1,1,1,0,0)$ $(w,0,0,1,1,1,1,1,0)$ $(w,1,1,0,1,0,0,1,0)$ $(w,0,1,0,0,1,0,1,1)$ $(w,1,1,1,1,0,0,1,0)$ $(w,0,1,0,0,1,1,1,1)$ $(w,1,1,0,1,1,0,1,0)$ $(w,0,1,0,1,1,0,1,1)$ $(w,1,1,0,1,0,1,1,0)$ $(w,0,1,1,0,1,0,1,1)$ $(w,1,1,1,1,0,1,1,0)$ $(w,0,1,1,0,1,1,1,1)$
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 $(w,1,0,1,1,0,1,0,0,0)$ $(w,0,0,0,1,0,1,1,0,1)$ $(w,1,0,1,0,0,1,1,0,0)$ $(w,0,0,1,1,0,0,1,0,1)$ $(w,0,1,1,0,1,1,1,0,0)$ $(w,0,0,1,1,1,0,1,1,0)$
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 $(w,0,0,1,0,0,1,1,0,1,0)$ $(w,0,1,0,1,1,0,0,1,0,0)$ $(w,0,1,0,0,0,0,0,1,1,1)$ $(w,1,1,1,0,0,0,0,0,1,0)$ $(w,1,0,1,0,1,1,0,0,0,0)$ $(w,0,0,0,0,1,1,0,1,0,1)$ $(w,0,1,1,0,1,0,0,0,1,0)$ $(w,0,1,0,0,0,1,0,1,1,0)$ $(w,0,0,0,1,1,0,1,1,0,0)$ $(w,0,0,1,1,0,1,1,0,0,0)$ $(w,0,0,1,0,0,1,0,0,1,1)$ $(w,1,1,0,0,1,0,0,1,0,0)$ $(w,1,0,1,1,1,0,0,0,0,0)$ $(w,0,0,0,0,0,1,1,1,0,1)$ $(w,0,1,1,1,0,0,1,0,0,1)$ $(w,1,0,0,1,0,0,1,1,1,0)$ $(w,1,1,0,1,1,0,0,1,0,0)$ $(w,0,0,1,0,0,1,1,0,1,1)$ $(w,1,1,0,0,1,0,1,1,0,0)$ $(w,0,0,1,1,0,1,0,0,1,1)$ $(w,1,0,1,1,0,0,0,1,1,0)$ $(w,0,1,1,0,0,0,1,1,0,1)$ $(w,0,0,1,1,0,0,1,0,1,1)$ $(w,1,1,0,1,0,0,1,1,0,0)$ $(w,0,1,0,0,0,1,1,1,0,1)$ $(w,1,0,1,1,1,0,0,0,1,0)$
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 $(w,0,1,0,0,0,0,0,1,1,1)$ $(w,1,1,1,0,1,0,1,1,0,1)$ $(w,0,1,1,1,0,0,1,0,0,0)$ $(w,1,0,1,0,1,0,0,0,1,1)$ $(w,1,1,0,0,0,1,0,1,0,0)$ $(w,1,0,1,0,1,0,0,0,1,1)$ $(w,1,1,0,1,0,0,0,0,1,0)$ $(w,1,1,0,0,0,0,1,0,1,1)$ $(w,0,1,1,0,1,0,0,0,1,0)$ $(w,1,1,0,1,1,1,1,0,0,0)$ $(w,1,0,0,0,0,1,0,0,1,1)$ $(w,1,1,1,1,0,1,1,1,1,0)$ $(w,0,0,0,0,1,0,1,1,0,1)$ $(w,1,1,1,1,0,1,0,0,0,0)$ $(w,0,0,0,1,1,0,0,1,1,0)$ $(w,1,1,1,0,0,1,1,0,0,0)$ $(w,0,0,1,0,1,0,0,1,0,1)$ $(w,1,0,1,0,1,1,0,1,0,0)$ $(w,0,1,0,0,0,1,1,0,1,0)$ $(w,0,1,1,0,1,0,0,1,1,0)$ $(w,0,0,1,0,0,0,1,1,1,0)$ $(w,1,1,0,1,1,0,1,0,1,0)$ $(w,1,0,1,0,0,1,0,1,0,0)$ $(w,0,0,1,0,1,1,0,1,0,1)$ $(w,1,1,0,0,0,0,1,0,0,1)$ $(w,1,1,1,0,1,1,1,1,0,0)$ $(w,1,1,0,1,0,1,0,0,0,0)$ $(w,0,0,0,0,1,1,1,0,1,1)$ $(w,1,1,0,0,0,0,0,1,1,0)$ $(w,0,1,1,0,0,1,0,1,1,0)$ $(w,0,0,1,1,0,0,0,1,1,0)$ $(w,1,0,1,1,1,1,1,0,1,1)$ $(w,0,1,0,0,1,1,0,1,0,0)$ $(w,0,1,0,0,1,1,1,1,1,1)$ $(w,0,1,0,0,1,1,1,0,0,0)$ $(w,0,0,0,1,1,1,0,0,1,1)$ $(w,0,0,0,1,0,1,1,0,0,1)$ $(w,1,0,0,1,1,0,1,1,0,0)$ $(w,0,0,0,0,1,0,1,0,1,1)$ $(w,1,1,0,1,1,1,0,0,0,0)$ $(w,1,0,0,0,1,1,0,0,0,1)$ $(w,1,1,0,0,1,1,0,0,0,0)$ $(w,1,0,0,0,1,1,0,0,0,1)$ $(w,1,1,0,0,1,1,0,0,0,1)$ $(w,1,1,0,0,1,1,0,0,0,0)$ $(w,1,0,0,0,1,1,0,0,1,1)$ $(w,0,0,1,0,1,1,0,1,0,0)$ $(w,1,0,1,0,1,1,0,1,0,0)$ $(w,1,0,1,0,0,1,0,0,1,0)$ $(w,1,0,1,0,0,1,0,0,1,0)$ $(w,1,0,1,1,0,0,0,1,0,0)$ $(w,1,1,1,1,0,1,0,0,1,1)$ $(w,1,0,1,1,0,0,0,1,1,0)$ $(w,1,0,1,1,1,1,0,1,1,0)$ $(w,0,1,1,0,0,1,0,1,0,1)$ $(w,1,1,1,1,1,0,0,1,1,0)$ $(w,0,0,1,1,1,0,1,1,0,0)$ $(w,1,0,1,1,0,1,1,1,0,0)$ $(w,1,0,0,1,1,0,0,0,1,1)$ $(w,0,1,1,0,1,0,1,0,1,0)$ $(w,0,0,0,1,1,0,1,1,1,0)$ $(w,1,0,0,1,1,0,1,1,0,0)$ $(w,0,1,0,0,1,0,1,1,1,0)$ $(w,0,1,0,1,1,1,1,0,0,1)$ $(w,1,0,0,1,1,0,0,1,0,1)$ $(w,0,1,0,1,1,1,1,1,0,1)$ $(w,0,0,0,1,1,1,0,1,0,1)$ $(w,1,0,1,0,1,0,1,1,0,0)$ $(w,0,0,0,1,0,1,1,0,1,1)$ $(w,1,1,0,1,1,0,1,0,0,1)$ $(w,0,1,1,1,1,0,0,0,1,0)$ $(w,1,1,0,0,0,1,1,1,1,0)$ $(w,0,1,1,0,0,0,1,1,0,1)$ $(w,0,0,1,1,0,1,1,1,0,1)$ $(w,1,1,1,0,1,1,0,0,0,1)$ $(w,0,1,0,1,1,1,1,1,0,0)$ $(w,0,1,0,1,1,1,1,0,0,1)$ $(w,1,0,1,1,1,0,1,1,1,1)$
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 $(w,0,0,1,0,1,1,1,0,1,0,0)$ $(w,0,0,1,0,1,1,1,0,1,0,0)$ $(w,0,1,1,0,1,1,1,1,0,1,0)$ $(w,0,1,0,1,1,1,1,0,1,1,0)$
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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