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

Murat Şahİn; Hayrullah Özİmamoğlu

doi:10.3934/amc.2020026

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:

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

Citation:

Table 2.1. Number of OATC

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

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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) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,1,0,0,1) $	$ (w,1,1,0,0) $
$ (w,1,0,1,0) $	$ (w,0,1,1,1) $

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

$ \boldsymbol{a_{2}} $	$ \boldsymbol{b_{2}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{2}}} $	$ (w,1) $	Circulant

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

$ \boldsymbol{a_{3}} $	$ \boldsymbol{b_{3}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{3}}} $	$ (w,1,0) $	Non-Circulant
$ \boldsymbol{u_{Q_{3}}} $	$ (w,0,1) $	Circulant
$ \boldsymbol{u_{Q_{3}}} $	$ (w,1,1) $	Non-Circulant
$ \boldsymbol{u_{N_{3}}} $	$ (w,1,0) $	Circulant
$ \boldsymbol{u_{N_{3}}} $	$ (w,1,1) $	Non-Circulant

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

$ \boldsymbol{a_{5}} $	$ \boldsymbol{b_{5}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{5}}} $	$ (w,1,0,0,1) $	Circulant
$ \boldsymbol{u_{Q_{5}}} $	$ (w,1,1,0,0) $	Non-Circulant
$ \boldsymbol{u_{Q_{5}}} $	$ (w,1,1,1,0) $	Non-Circulant
$ \boldsymbol{u_{Q_{5}}} $	$ (w,1,0,1,1) $	Non-Circulant
$ \boldsymbol{u_{Q_{5}}} $	$ (w,1,1,0,1) $	Non-Circulant
$ \boldsymbol{u_{N_{5}}} $	$ (w,0,1,1,0) $	Circulant
$ \boldsymbol{u_{N_{5}}} $	$ (w,1,1,1,0) $	Non-Circulant
$ \boldsymbol{u_{N_{5}}} $	$ (w,0,1,1,1) $	Non-Circulant

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

$ \boldsymbol{a_{11}} $	$ \boldsymbol{b_{11}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{11}}} $	$ (w,0,1,0,0,0,1,1,1,0,1) $	Circulant
$ \boldsymbol{u_{N_{11}}} $	$ (w,1,0,1,1,1,0,0,0,1,0) $	Circulant

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

$ \boldsymbol{a_{2}} $	$ \boldsymbol{b_{2}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{2}}} $	$ \boldsymbol{u_{N_{2}}} $	Non-Circulant
$ \boldsymbol{u_{N_{2}}} $	$ \boldsymbol{u_{Q_{2}}} $	Non-Circulant
$ \boldsymbol{u_{N_{2}}} $	$ \boldsymbol{u_{N_{2}}} $	Circulant

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

$ \boldsymbol{a_{3}} $	$ \boldsymbol{b_{3}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{N_{3}}} $	$ \boldsymbol{u_{N_{3}}} $	Non-Circulant

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

$ \boldsymbol{a_{5}} $	$ \boldsymbol{b_{5}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{5}}} $	$ \boldsymbol{u_{N_{5}}} $	Non-Circulant
$ \boldsymbol{u_{N_{5}}} $	$ \boldsymbol{u_{Q_{5}}} $	Non-Circulant

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

$ \boldsymbol{a_{7}} $	$ \boldsymbol{b_{7}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{7}}} $	$ \boldsymbol{u_{Q_{7}}} $	Non-Circulant
$ \boldsymbol{u_{Q_{7}}} $	$ \boldsymbol{u_{N_{7}}} $	Circulant
$ \boldsymbol{u_{N_{7}}} $	$ \boldsymbol{u_{Q_{7}}} $	Circulant

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

$ \boldsymbol{a_{11}} $	$ \boldsymbol{b_{11}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{11}}} $	$ \boldsymbol{u_{Q_{11}}} $	Non-Circulant

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

$ \boldsymbol{a_{13}} $	$ \boldsymbol{b_{13}} $	Circulant/Non-Circulant
$ \boldsymbol{u_{Q_{13}}} $	$ \boldsymbol{u_{Q_{13}}} $	Circulant
$ \boldsymbol{u_{N_{13}}} $	$ \boldsymbol{u_{N_{13}}} $	Circulant

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

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

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

Upper Generator Vector	Lower Generator Vector
$ (w,1) $	$ (w,1) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,1,0) $	$ (w,0,1) $
$ (w,1,1) $	$ (w,1,1) $

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

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

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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) $

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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) $

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

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

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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) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,1,1,0,1,0,0,0) $	$ (w,0,0,0,1,0,1,1) $
$ (w,1,0,1,1,0,0,0) $	$ (w,0,0,0,1,1,0,1) $
$ (w,0,1,1,1,0,0,0) $	$ (w,0,0,0,1,1,1,0) $
$ (w,1,1,0,0,1,0,0) $	$ (w,0,0,1,0,0,1,1) $
$ (w,1,0,1,0,1,0,0) $	$ (w,0,0,1,0,1,0,1) $
$ (w,0,1,1,0,1,0,0) $	$ (w,0,0,1,0,1,1,0) $
$ (w,1,1,0,1,1,0,0) $	$ (w,0,0,1,1,0,1,1) $
$ (w,0,0,1,1,1,0,0) $	$ (w,0,0,1,1,1,0,0) $
$ (w,1,0,1,1,1,0,0) $	$ (w,0,0,1,1,1,0,1) $
$ (w,1,1,0,1,0,1,0) $	$ (w,0,1,0,1,0,1,1) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,0,1,1,0,1,0,0) $	$ (w,0,0,1,0,1,1,1) $
$ (w,0,1,1,0,1,0,0) $	$ (w,0,0,1,1,1,1,0) $
$ (w,0,1,1,0,1,0,0) $	$ (w,1,1,1,0,0,0,1) $
$ (w,0,1,1,0,1,0,0) $	$ (w,0,1,1,1,0,1,0) $
$ (w,0,1,1,0,1,0,0) $	$ (w,0,1,0,0,1,1,1) $
$ (w,1,0,0,1,0,1,0) $	$ (w,0,1,0,1,0,1,1) $
$ (w,1,0,0,1,0,1,0) $	$ (w,1,1,0,0,1,1,0) $
$ (w,1,0,0,1,0,0,1) $	$ (w,1,0,1,0,1,1,0) $
$ (w,0,1,0,1,0,0,1) $	$ (w,1,1,0,1,0,1,0) $
$ (w,0,1,0,1,0,0,1) $	$ (w,1,1,0,0,0,1,1) $
$ (w,0,0,1,0,0,1,1) $	$ (w,1,1,0,1,1,0,0) $
$ (w,0,1,1,0,0,0,1) $	$ (w,1,0,0,1,1,1,1) $
$ (w,0,1,0,0,1,0,1) $	$ (w,1,1,0,0,0,1,1) $
$ (w,0,1,0,0,1,0,1) $	$ (w,1,1,1,0,1,0,1) $
$ (w,0,1,0,0,1,0,1) $	$ (w,1,1,1,0,1,1,1) $
$ (w,1,0,1,1,0,0,0) $	$ (w,0,1,1,1,0,0,0) $
$ (w,1,0,1,1,0,0,0) $	$ (w,0,0,1,1,1,0,0) $
$ (w,1,0,1,1,0,0,0) $	$ (w,1,1,0,0,1,1,0) $
$ (w,1,0,1,1,0,0,0) $	$ (w,0,1,1,1,0,0,1) $
$ (w,1,0,1,1,0,0,0) $	$ (w,1,0,0,1,1,0,1) $
$ (w,1,0,1,1,0,0,0) $	$ (w,0,0,1,1,1,0,1) $
$ (w,1,0,1,1,0,0,0) $	$ (w,1,1,1,0,1,1,0) $
$ (w,0,0,1,1,0,1,0) $	$ (w,0,1,0,1,1,0,1) $
$ (w,0,0,1,0,1,0,1) $	$ (w,1,1,0,0,1,0,0) $
$ (w,0,0,1,0,1,0,1) $	$ (w,1,0,1,1,1,0,0) $
$ (w,0,0,1,0,1,0,1) $	$ (w,1,1,0,1,1,0,0) $
$ (w,0,0,1,0,1,0,1) $	$ (w,1,0,1,1,1,1,1) $
$ (w,1,1,0,0,0,1,0) $	$ (w,1,1,0,1,0,0,0) $
$ (w,1,1,0,0,0,1,0) $	$ (w,1,1,0,0,1,0,1) $
$ (w,1,1,0,0,0,1,0) $	$ (w,1,1,0,1,0,1,1) $
$ (w,1,1,0,0,0,1,0) $	$ (w,1,1,0,1,1,1,0) $
$ (w,0,0,0,1,1,0,1) $	$ (w,1,1,0,1,0,0,0) $
$ (w,0,0,0,1,1,0,1) $	$ (w,1,0,1,1,1,0,0) $
$ (w,0,0,0,1,1,0,1) $	$ (w,1,1,0,1,0,1,0) $
$ (w,1,1,0,1,0,0,0) $	$ (w,1,0,0,0,0,1,1) $
$ (w,1,1,0,1,0,0,0) $	$ (w,1,1,0,0,1,1,0) $
$ (w,1,1,0,1,0,0,0) $	$ (w,1,1,0,0,0,1,1) $
$ (w,1,1,0,1,0,0,0) $	$ (w,1,0,1,1,1,0,1) $
$ (w,1,0,0,0,1,1,0) $	$ (w,0,1,1,1,0,0,0) $
$ (w,0,1,1,1,0,0,0) $	$ (w,0,1,0,1,1,0,1) $
$ (w,0,1,1,1,0,0,0) $	$ (w,0,1,0,1,1,1,0) $
$ (w,1,0,1,0,0,0,1) $	$ (w,1,1,0,0,0,0,1) $
$ (w,1,0,1,0,0,0,1) $	$ (w,1,0,0,0,1,1,1) $
$ (w,1,0,1,0,0,0,1) $	$ (w,1,1,0,0,0,1,1) $
$ (w,1,0,1,0,0,0,1) $	$ (w,1,0,0,1,1,0,1) $
$ (w,1,0,1,0,0,0,1) $	$ (w,1,0,0,1,1,1,1) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,1,1,0,0,1,0,0) $	$ (w,0,1,0,1,1,0,1) $
$ (w,0,0,1,1,1,0,0) $	$ (w,1,0,1,0,1,0,0) $
$ (w,0,0,1,0,1,1,0) $	$ (w,1,0,1,1,1,0,0) $
$ (w,0,0,1,0,1,1,0) $	$ (w,0,1,1,1,1,0,1) $
$ (w,0,0,1,0,1,1,0) $	$ (w,1,1,1,1,0,1,1) $
$ (w,1,0,0,0,1,0,1) $	$ (w,1,1,1,0,0,0,1) $
$ (w,1,0,0,0,1,0,1) $	$ (w,1,1,1,0,1,0,0) $
$ (w,1,0,0,0,0,1,1) $	$ (w,1,1,0,0,1,1,0) $
$ (w,1,0,0,0,0,1,1) $	$ (w,1,1,0,1,1,0,0) $
$ (w,1,0,0,0,0,1,1) $	$ (w,1,1,0,1,0,0,1) $
$ (w,1,0,1,0,1,0,0) $	$ (w,0,1,1,0,1,0,1) $
$ (w,1,0,0,1,1,0,0) $	$ (w,0,1,0,1,1,1,0) $
$ (w,1,0,0,1,1,0,0) $	$ (w,1,1,1,0,1,1,0) $
$ (w,1,0,0,1,1,0,0) $	$ (w,1,1,0,1,1,1,0) $
$ (w,0,0,0,1,0,1,1) $	$ (w,1,1,1,0,1,1,0) $
$ (w,0,0,0,1,0,1,1) $	$ (w,1,1,0,1,1,1,1) $
$ (w,0,0,0,1,1,1,0) $	$ (w,0,1,1,1,0,1,0) $
$ (w,0,0,0,1,1,1,0) $	$ (w,0,1,1,1,0,1,1) $
$ (w,1,1,0,0,0,0,1) $	$ (w,1,0,1,1,1,0,0) $
$ (w,1,1,0,0,0,0,1) $	$ (w,1,0,1,1,1,0,1) $
$ (w,0,1,0,1,1,0,0) $	$ (w,0,0,1,1,0,1,1) $
$ (w,0,1,0,0,1,1,0) $	$ (w,1,1,1,1,0,0,1) $
$ (w,0,1,0,1,1,0,1) $	$ (w,0,0,1,1,0,1,1) $
$ (w,0,0,1,0,1,1,1) $	$ (w,1,0,1,1,1,0,0) $
$ (w,0,0,1,0,1,1,1) $	$ (w,1,0,1,0,1,1,0) $
$ (w,0,0,1,1,1,1,0) $	$ (w,0,1,1,0,1,0,1) $
$ (w,0,0,1,1,1,1,0) $	$ (w,0,1,1,0,1,1,1) $
$ (w,0,0,1,1,1,1,0) $	$ (w,0,1,1,1,1,0,1) $
$ (w,0,1,1,0,1,0,1) $	$ (w,1,1,1,1,0,1,0) $
$ (w,0,0,1,1,0,1,1) $	$ (w,0,1,1,1,0,1,1) $
$ (w,0,0,1,1,0,1,1) $	$ (w,1,0,1,1,1,1,1) $
$ (w,0,1,0,1,1,1,0) $	$ (w,0,1,1,0,1,1,1) $
$ (w,1,0,1,0,0,1,1) $	$ (w,0,1,1,1,1,1,0) $
$ (w,1,0,1,1,1,0,0) $	$ (w,1,1,0,0,1,1,0) $
$ (w,1,1,0,0,1,1,0) $	$ (w,1,1,0,1,0,1,1) $
$ (w,1,0,1,1,0,1,0) $	$ (w,1,1,1,0,1,1,0) $
$ (w,1,0,1,1,0,1,0) $	$ (w,1,1,1,0,0,1,1) $
$ (w,1,1,0,0,0,1,1) $	$ (w,1,1,0,1,0,1,1) $
$ (w,1,1,0,0,0,1,1) $	$ (w,1,1,0,1,1,1,0) $
$ (w,1,1,0,0,1,0,1) $	$ (w,1,1,0,0,1,0,1) $
$ (w,1,1,0,0,1,0,1) $	$ (w,1,1,1,0,0,1,1) $
$ (w,1,0,0,1,1,0,1) $	$ (w,1,0,0,1,1,0,1) $
$ (w,1,0,0,1,1,0,1) $	$ (w,1,1,1,1,0,1,1) $
$ (w,1,0,1,1,1,0,1) $	$ (w,0,1,0,1,1,1,1) $
$ (w,0,1,1,1,0,1,1) $	$ (w,1,0,1,0,1,1,1) $
$ (w,0,1,1,0,1,1,1) $	$ (w,1,1,1,0,1,0,1) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,1,1,0,1,0,0,0,0) $	$ (w,0,0,0,0,1,0,1,1) $
$ (w,0,1,1,1,0,0,0,0) $	$ (w,0,0,0,0,1,1,1,0) $
$ (w,1,1,1,0,1,0,0,0) $	$ (w,0,0,0,1,0,1,1,1) $
$ (w,1,0,0,1,1,0,0,0) $	$ (w,0,0,0,1,1,0,0,1) $
$ (w,0,1,0,1,1,0,0,0) $	$ (w,0,0,0,1,1,0,1,0) $
$ (w,0,0,1,1,1,0,0,0) $	$ (w,0,0,0,1,1,1,0,0) $
$ (w,1,1,1,0,0,1,0,0) $	$ (w,0,0,1,0,0,1,1,1) $
$ (w,1,0,0,1,1,1,0,0) $	$ (w,0,0,1,1,1,0,0,1) $
$ (w,0,0,1,1,1,1,0,0) $	$ (w,0,0,1,1,1,1,0,0) $
$ (w,0,1,1,1,1,1,0,0) $	$ (w,0,0,1,1,1,1,1,0) $
$ (w,1,1,0,1,0,0,1,0) $	$ (w,0,1,0,0,1,0,1,1) $
$ (w,1,1,1,1,0,0,1,0) $	$ (w,0,1,0,0,1,1,1,1) $
$ (w,1,1,0,1,1,0,1,0) $	$ (w,0,1,0,1,1,0,1,1) $
$ (w,1,1,0,1,0,1,1,0) $	$ (w,0,1,1,0,1,0,1,1) $
$ (w,1,1,1,1,0,1,1,0) $	$ (w,0,1,1,0,1,1,1,1) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,1,0,1,1,0,1,0,0,0) $	$ (w,0,0,0,1,0,1,1,0,1) $
$ (w,1,0,1,0,0,1,1,0,0) $	$ (w,0,0,1,1,0,0,1,0,1) $
$ (w,0,1,1,0,1,1,1,0,0) $	$ (w,0,0,1,1,1,0,1,1,0) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,0,0,1,0,0,1,1,0,1,0) $	$ (w,0,1,0,1,1,0,0,1,0,0) $
$ (w,0,1,0,0,0,0,0,1,1,1) $	$ (w,1,1,1,0,0,0,0,0,1,0) $
$ (w,1,0,1,0,1,1,0,0,0,0) $	$ (w,0,0,0,0,1,1,0,1,0,1) $
$ (w,0,1,1,0,1,0,0,0,1,0) $	$ (w,0,1,0,0,0,1,0,1,1,0) $
$ (w,0,0,0,1,1,0,1,1,0,0) $	$ (w,0,0,1,1,0,1,1,0,0,0) $
$ (w,0,0,1,0,0,1,0,0,1,1) $	$ (w,1,1,0,0,1,0,0,1,0,0) $
$ (w,1,0,1,1,1,0,0,0,0,0) $	$ (w,0,0,0,0,0,1,1,1,0,1) $
$ (w,0,1,1,1,0,0,1,0,0,1) $	$ (w,1,0,0,1,0,0,1,1,1,0) $
$ (w,1,1,0,1,1,0,0,1,0,0) $	$ (w,0,0,1,0,0,1,1,0,1,1) $
$ (w,1,1,0,0,1,0,1,1,0,0) $	$ (w,0,0,1,1,0,1,0,0,1,1) $
$ (w,1,0,1,1,0,0,0,1,1,0) $	$ (w,0,1,1,0,0,0,1,1,0,1) $
$ (w,0,0,1,1,0,0,1,0,1,1) $	$ (w,1,1,0,1,0,0,1,1,0,0) $
$ (w,0,1,0,0,0,1,1,1,0,1) $	$ (w,1,0,1,1,1,0,0,0,1,0) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,0,1,0,0,0,0,0,1,1,1) $	$ (w,1,1,1,0,1,0,1,1,0,1) $
$ (w,0,1,1,1,0,0,1,0,0,0) $	$ (w,1,0,1,0,1,0,0,0,1,1) $
$ (w,1,1,0,0,0,1,0,1,0,0) $	$ (w,1,0,1,0,1,0,0,0,1,1) $
$ (w,1,1,0,1,0,0,0,0,1,0) $	$ (w,1,1,0,0,0,0,1,0,1,1) $
$ (w,0,1,1,0,1,0,0,0,1,0) $	$ (w,1,1,0,1,1,1,1,0,0,0) $
$ (w,1,0,0,0,0,1,0,0,1,1) $	$ (w,1,1,1,1,0,1,1,1,1,0) $
$ (w,0,0,0,0,1,0,1,1,0,1) $	$ (w,1,1,1,1,0,1,0,0,0,0) $
$ (w,0,0,0,1,1,0,0,1,1,0) $	$ (w,1,1,1,0,0,1,1,0,0,0) $
$ (w,0,0,1,0,1,0,0,1,0,1) $	$ (w,1,0,1,0,1,1,0,1,0,0) $
$ (w,0,1,0,0,0,1,1,0,1,0) $	$ (w,0,1,1,0,1,0,0,1,1,0) $
$ (w,0,0,1,0,0,0,1,1,1,0) $	$ (w,1,1,0,1,1,0,1,0,1,0) $
$ (w,1,0,1,0,0,1,0,1,0,0) $	$ (w,0,0,1,0,1,1,0,1,0,1) $
$ (w,1,1,0,0,0,0,1,0,0,1) $	$ (w,1,1,1,0,1,1,1,1,0,0) $
$ (w,1,1,0,1,0,1,0,0,0,0) $	$ (w,0,0,0,0,1,1,1,0,1,1) $
$ (w,1,1,0,0,0,0,0,1,1,0) $	$ (w,0,1,1,0,0,1,0,1,1,0) $
$ (w,0,0,1,1,0,0,0,1,1,0) $	$ (w,1,0,1,1,1,1,1,0,1,1) $
$ (w,0,1,0,0,1,1,0,1,0,0) $	$ (w,0,1,0,0,1,1,1,1,1,1) $
$ (w,0,1,0,0,1,1,1,0,0,0) $	$ (w,0,0,0,1,1,1,0,0,1,1) $
$ (w,0,0,0,1,0,1,1,0,0,1) $	$ (w,1,0,0,1,1,0,1,1,0,0) $
$ (w,0,0,0,0,1,0,1,0,1,1) $	$ (w,1,1,0,1,1,1,0,0,0,0) $
$ (w,1,0,0,0,1,1,0,0,0,1) $	$ (w,1,1,0,0,1,1,0,0,0,0) $
$ (w,1,0,0,0,1,1,0,0,0,1) $	$ (w,1,1,0,0,1,1,0,0,0,1) $
$ (w,1,1,0,0,1,1,0,0,0,0) $	$ (w,1,0,0,0,1,1,0,0,1,1) $
$ (w,0,0,1,0,1,1,0,1,0,0) $	$ (w,1,0,1,0,1,1,0,1,0,0) $
$ (w,1,0,1,0,0,1,0,0,1,0) $	$ (w,1,0,1,0,0,1,0,0,1,0) $
$ (w,1,0,1,1,0,0,0,1,0,0) $	$ (w,1,1,1,1,0,1,0,0,1,1) $
$ (w,1,0,1,1,0,0,0,1,1,0) $	$ (w,1,0,1,1,1,1,0,1,1,0) $
$ (w,0,1,1,0,0,1,0,1,0,1) $	$ (w,1,1,1,1,1,0,0,1,1,0) $
$ (w,0,0,1,1,1,0,1,1,0,0) $	$ (w,1,0,1,1,0,1,1,1,0,0) $
$ (w,1,0,0,1,1,0,0,0,1,1) $	$ (w,0,1,1,0,1,0,1,0,1,0) $
$ (w,0,0,0,1,1,0,1,1,1,0) $	$ (w,1,0,0,1,1,0,1,1,0,0) $
$ (w,0,1,0,0,1,0,1,1,1,0) $	$ (w,0,1,0,1,1,1,1,0,0,1) $
$ (w,1,0,0,1,1,0,0,1,0,1) $	$ (w,0,1,0,1,1,1,1,1,0,1) $
$ (w,0,0,0,1,1,1,0,1,0,1) $	$ (w,1,0,1,0,1,0,1,1,0,0) $
$ (w,0,0,0,1,0,1,1,0,1,1) $	$ (w,1,1,0,1,1,0,1,0,0,1) $
$ (w,0,1,1,1,1,0,0,0,1,0) $	$ (w,1,1,0,0,0,1,1,1,1,0) $
$ (w,0,1,1,0,0,0,1,1,0,1) $	$ (w,0,0,1,1,0,1,1,1,0,1) $
$ (w,1,1,1,0,1,1,0,0,0,1) $	$ (w,0,1,0,1,1,1,1,1,0,0) $
$ (w,0,1,0,1,1,1,1,0,0,1) $	$ (w,1,0,1,1,1,0,1,1,1,1) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,0,0,1,0,1,1,1,0,1,0,0) $	$ (w,0,0,1,0,1,1,1,0,1,0,0) $
$ (w,0,1,1,0,1,1,1,1,0,1,0) $	$ (w,0,1,0,1,1,1,1,0,1,1,0) $

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

Upper Generator Vectors	Lower Generator Vectors
$ (w,1,0,1,0,0,1,1,1,0,0,0,0) $	$ (w,0,0,0,0,1,1,1,0,0,1,0,1) $
$ (w,1,1,1,0,1,1,1,1,1,0,1,0) $	$ (w,0,1,0,1,1,1,1,1,0,1,1,1) $

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References

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

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

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