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

• * Corresponding author: Hayrullah Özimamoğlu
• 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.

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

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

 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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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