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Article Contents

# New type i binary [72, 36, 12] self-dual codes from composite matrices and R1 lifts

• * Corresponding author: Adrian Korban
• In this work, we define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form $[I_n \ | \ \Omega(v)],$ where $I_n$ is the identity matrix and $\Omega(v)$ is a composite matrix and search for binary self-dual codes with parameters $[36,18, 6 \ \text{or} \ 8].$ We next lift these codes over the ring $R_1 = \mathbb{F}_2+u\mathbb{F}_2$ to obtain codes whose binary images are self-dual codes with parameters $[72,36,12].$ Many of these codes turn out to have weight enumerators with parameters that were not known in the literature before. In particular, we find $30$ new Type I binary self-dual codes with parameters $[72,36,12].$

Mathematics Subject Classification: Primary: 94B05; Secondary: 16S34.

 Citation:

• Table 1.  Type I $[36,18,6-8]$ Codes from Theorem 4.1

 Type $r_B$ $r_C$ $|Aut(C_i)|$ $C_1$ $[36,18,6]$ $(0,0,0,0,0,1,0,1,1)$ $(1,0,1,1,1,0,1,0,1)$ $2^2 \cdot 3^2$ $C_2$ $[36,18,6]$ $(0,0,0,0,1,1,0,1,1)$ $(1,0,0,1,0,1,1,1,0)$ $2^2 \cdot 3^2$ $C_3$ $[36,18,8]$ $(0,0,1,0,0,1,0,0,1)$ $(1,0,0,1,1,0,1,1,1)$ $2^2 \cdot 3^2$ $C_4$ $[36,18,8]$ $(0,1,0,0,0,1,0,1,1)$ $(1,0,0,1,0,0,1,1,1)$ $2^2 \cdot 3^2$

Table 2.  New Type I $[72,36,12]$ Codes from $R_1$-lift of $C_1$

 Type $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_1$ $W_{72,1}$ $(u,0,u,u,u,1,u,u + 1,1)$ $(1,0,1,1,u + 1,0,1,u,1)$ $0$ $192$ $36$ $\mathcal{C}_2$ $W_{72,1}$ $(u,0,0,u,0,1,u,1,1)$ $(1,0,u + 1,1,u + 1,0,1,u,u + 1)$ $0$ $198$ $36$ $\mathcal{C}_3$ $W_{72,1}$ $(u,u,0,u,u,1,u,1,u + 1)$ $(1,u,u + 1,1,u + 1,0,1,0,u + 1)$ $0$ $336$ $36$ $\mathcal{C}_4$ $W_{72,1}$ $(0,u,0,0,0,1,0,1,u + 1)$ $(1,u,u + 1,1,u + 1,0,1,0,u + 1)$ $18$ $234$ $36$ $\mathcal{C}_5$ $W_{72,1}$ $(u,u,0,u,0,1,u,u + 1,1)$ $(1,u,u + 1,1,1,u,1,0,u + 1)$ $18$ $345$ $36$ $\mathcal{C}_6$ $W_{72,1}$ $(0,u,0,0,0,1,0,u + 1,1)$ $(1,u,1,1,u + 1,u,1,u,1)$ $18$ $378$ $36$ $\mathcal{C}_7$ $W_{72,1}$ $(u,u,u,u,0,1,u,u + 1,u + 1)$ $(1,u,1,1,u + 1,0,1,0,1)$ $18$ $396$ $36$ $\mathcal{C}_8$ $W_{72,1}$ $(0,0,u,0,u,1,0,1,u + 1)$ $(1,u,1,1,1,0,1,u,1)$ $18$ $441$ $36$ $\mathcal{C}_9$ $W_{72,1}$ $(u,u,0,u,0,1,u,1,u + 1)$ $(1,u,1,1,1,0,1,u,1)$ $18$ $453$ $36$

Table 3.  New Type I $[72,36,12]$ Codes from $R_1$-lift of $C_2$

 Type $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{10}$ $W_{72,1}$ $(0,u,0,0,1,1,0,1,u + 1)$ $(1,u,u,1,u,u + 1,1,1,0)$ $0$ $219$ $36$ $\mathcal{C}_{11}$ $W_{72,1}$ $(u,0,u,u,1,1,u,u + 1,u + 1)$ $(1,u,0,1,0,1,1,u + 1,u)$ $0$ $345$ $36$ $\mathcal{C}_{12}$ $W_{72,1}$ $(0,0,u,0,1,1,0,1,u + 1)$ $(1,u,0,1,0,1,1,1,0)$ $0$ $408$ $36$ $\mathcal{C}_{13}$ $W_{72,1}$ $(u,0,0,u,1,u + 1,u,u + 1,u + 1)$ $(1,0,u,1,0,u + 1,1,u + 1,0)$ $18$ $261$ $36$ $\mathcal{C}_{14}$ $W_{72,1}$ $(u,u,u,u,1,1,u,1,u + 1)$ $(1,0,0,1,u,1,1,u + 1,0)$ $18$ $270$ $36$ $\mathcal{C}_{15}$ $W_{72,1}$ $(u,0,u,u,1,u + 1,u,1,u + 1)$ $(1,0,u,1,u,1,1,u + 1,0)$ $18$ $357$ $36$

Table 4.  New Type I $[72,36,12]$ Codes from $R_1$-lift of $C_3$

 Type $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{16}$ $W_{72,1}$ $(u,u,1,u,0,1,u,0,1)$ $(1,u,0,1,u + 1,u,1,1,u + 1)$ $0$ $120$ $36$ $\mathcal{C}_{17}$ $W_{72,1}$ $(u,u,1,u,0,1,u,u,1)$ $(1,0,u,1,1,0,1,1,u + 1)$ $0$ $282$ $36$ $\mathcal{C}_{18}$ $W_{72,1}$ $(u,u,1,u,0,u + 1,u,0,1)$ $(1,u,0,1,u + 1,u,1,1,u + 1)$ $0$ $300$ $36$ $\mathcal{C}_{19}$ $W_{72,1}$ $(u,u,1,u,0,u + 1,u,0,1)$ $(1,u,u,1,1,0,1,1,1)$ $18$ $336$ $36$ $\mathcal{C}_{20}$ $W_{72,1}$ $(u,0,1,u,0,1,u,u,1)$ $(1,0,0,1,1,0,1,u + 1,u + 1)$ $36$ $435$ $36$

Table 5.  New Type I $[72,36,12]$ Codes from $R_1$-lift of $C_4$

 Type $r_B$ $r_C$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{21}$ $W_{72,1}$ $(0,1,u,0,u,1,0,u + 1,u + 1)$ $(1,u,u,1,u,u,1,u + 1,u + 1)$ $0$ $366$ $36$ $\mathcal{C}_{22}$ $W_{72,1}$ $(u,1,u,u,u,1,u,1,u + 1)$ $(1,u,0,1,0,0,1,u + 1,1)$ $0$ $372$ $36$ $\mathcal{C}_{23}$ $W_{72,1}$ $(0,1,u,0,u,1,0,u + 1,u + 1)$ $(1,0,0,1,0,0,1,u + 1,u + 1)$ $0$ $384$ $36$ $\mathcal{C}_{24}$ $W_{72,1}$ $(u,1,u,u,u,1,u,1,u + 1)$ $(1,u,u,1,u,0,1,u + 1,1)$ $0$ $390$ $36$ $\mathcal{C}_{25}$ $W_{72,1}$ $(u,1,0,u,0,1,u,u + 1,1)$ $(1,u,0,1,0,0,1,1,u + 1)$ $0$ $399$ $36$ $\mathcal{C}_{26}$ $W_{72,1}$ $(u,1,u,u,u,1,u,u + 1,1)$ $(1,0,u,1,0,0,1,u + 1,u + 1)$ $18$ $264$ $36$ $\mathcal{C}_{27}$ $W_{72,1}$ $(u,1,0,u,u,u + 1,u,u + 1,1)$ $(1,0,u,1,0,u,1,1,u + 1)$ $18$ $285$ $36$ $\mathcal{C}_{28}$ $W_{72,1}$ $(0,1,u,0,0,u + 1,0,1,1)$ $(1,u,u,1,u,0,1,1,1)$ $18$ $300$ $36$

Table 6.  Type I $[36,18,6-8]$ Codes from Theorem 4.2

 Type $r_B$ $r_C$ $r_D$ $|Aut(C_i)|$ $C_5$ $[36,18,6]$ $(0,0,0,0,0,1)$ $(1,1,1,0,0,1)$ $(1,1,1,0,1,0)$ $2^5 \cdot 3^4 \cdot 5$ $C_6$ $[36,18,6]$ $(0,0,0,0,1,1)$ $(0,0,0,0,1,1)$ $(1,1,1,1,0,1)$ $2^5 \cdot 3^4 \cdot 5$ $C_7$ $[36,18,6]$ $(0,0,0,0,1,1)$ $(0,1,1,0,1,1)$ $(0,1,1,1,0,0)$ $2^5 \cdot 3^2$ $C_8$ $[36,18,6]$ $(0,0,1,0,0,1)$ $(0,0,1,1,1,0)$ $(1,1,1,0,0,1)$ $2^5 \cdot 3^2$

Table 7.  New Type I $[72,36,12]$ Codes from $R_1$-lift of $C_7$

 Type $r_B$ $r_C$ $r_D$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{29}$ $W_{72,1}$ $(0,0,0,u,1,1)$ $(u,1,u + 1,u,1,1)$ $(u,u + 1,1,u + 1,0,0)$ $0$ $471$ $144$

Table 8.  Type I $[36,18,6-8]$ Codes from Theorem 4.3

 Type $r_B$ $r_C$ $r_D$ $|Aut(C_i)|$ $C_9$ $[36,18,6]$ $(0,0,0,0,0,1)$ $(0,1,1,0,1,1)$ $(1,0,1,1,0,1)$ $2^5 \cdot 3^2$ $C_{10}$ $[36,18,6]$ $(0,0,0,0,0,1)$ $(1,1,1,0,0,1)$ $(1,1,1,0,1,0)$ $2^5 \cdot 3^4 \cdot 5$ $C_{11}$ $[36,18,6]$ $(0,0,0,0,1,1)$ $(0,0,0,0,1,1)$ $(1,1,1,1,0,1)$ $2^5 \cdot 3^4 \cdot 5$ $C_{12}$ $[36,18,6]$ $(0,0,0,0,1,1)$ $(0,1,1,0,0,1)$ $(1,1,0,1,0,1)$ $2^5 \cdot 3^2$

Table 9.  New Type I $[72,36,12]$ Codes from $R_1$-lift of $C_9$

 Type $r_B$ $r_C$ $r_D$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $\mathcal{C}_{30}$ $W_{72,1}$ $(0,u,u,u,u,1)$ $(u,1,1,u,u + 1,1)$ $(u + 1,u,u + 1,1,u,u + 1)$ $0$ $621$ $432$
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