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# Constructing self-dual codes from group rings and reverse circulant matrices

• * Corresponding author: Adrian Korban
• In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92.

Mathematics Subject Classification: 94B05, 16S34.

 Citation:

• Table 1.  Self-dual codes over $\mathbb{F}_{4}+u\mathbb{F}_{4}$ of length $64$ from $C_{2, 2}$

 $\mathcal{C}_{i}$ $r_{\sigma(v_1)}$ $r_{\sigma(v_2)}$ $r_C$ $|Aut(\mathcal{C}_i)|$ $\beta$ $1$ $(0, 9, 2, 1)$ $(0, 0, A, 4)$ $(3, C, 3, 3)$ $2^{4}$ $0$ $2$ $(0, 9, 4, F)$ $(0, 0, 0, 6)$ $(2, 5, 2, 2)$ $2^{5}$ $0$

Table 2.  Extremal Self-dual codes of length $68$ from Theorem 2.1

 $\mathcal{M}_{68, i}$ $\mathcal{C}_i$ $c$ $X$ $\gamma$ $\beta$ $|Aut(\mathcal{M}_{68, i})|$ $1$ $1$ $1$ $(3u0u01u103103030u01u0u0301u10013)$ $\bf{0}$ $\bf{40}$ $2$ $2$ $2$ $u+1$ $(33313311u3uu13110u1030u1u31u31u3)$ $\bf{3}$ $\bf{77}$ $2$

Table 3.  Self-dual codes over $\mathbb{F}_{2}+u\mathbb{F}_{2}$ of length $64$ from $C_{4,2}$

 $\mathcal{E}_{i}$ $r_{\sigma(v_1)}$ $r_{\sigma(v_2)}$ $r_C$ $|Aut(\mathcal{E}_i)|$ $\beta$ $1$ $(u,0,1,1,u,u,u,u)$ $(u,u,1,1,0,u,u,3)$ $(0,1,0,1,0,1,0,1)$ $2^5$ $0$ $2$ $(u,u,1,3,u,u,u,u)$ $(u,u,1,1,u,u,0,3)$ $(u,3,u,3,u,3,u,3)$ $2^6$ $0$ $3$ $(u,0,u,u,0,0,1,3)$ $(u,u,0,1,u,0,3,1)$ $(3,3,3,3,3,3,3,3)$ $2^7$ $80$

Table 4.  New codes of length 64 as neighbors

 $\mathcal{L}_{64,i}$ $\mathcal{E}_{i}$ $(x_{33},...,x_{64})$ $W_{64,i}$ $\beta$ $|Aut(\mathcal{L}_{64,i})|$ $1$ $3$ $(01110001001001101000011001011111)$ $1$ $\bf{58}$ $2^2$ $2$ $3$ $(11011001001101110110010110011010)$ $2$ $\bf{54}$ $2^3$ $3$ $3$ $(11111100101011111001111001010010)$ $2$ $\bf{62}$ $2$

Table 5.  Extremal Self-dual codes of length $68$ from Theorem 2.1

 $\mathcal{N}_{68,i}$ $\mathcal{E}_i$ $c$ $X$ $\gamma$ $\beta$ $|Aut(\mathcal{N}_{68,i})|$ $1$ $1$ $3$ $(01330u3131uuu3330uuuu000333u1u1u)$ $\bf{0}$ $\bf{39}$ $2$ $2$ $2$ $1$ $(0013u1111uu1u0uuuu101u1333330130)$ $\bf{3}$ $\bf{79}$ $2$ $3$ $2$ $1$ $(u30u1u03u10uu113uuu01131u111u030)$ $\bf{3}$ $\bf{85}$ $2$

Table 6.  $[80,40,14]$ Self-dual codes over $\mathbb{F}_{4}+u \mathbb{F}_{4}$ from $C_{5}$

 $\mathcal{D}_{i}$ $r_{\sigma(v_1)}$ $r_{\sigma(v_2)}$ $r_C$ $|Aut(\mathcal{D}_i)|$ $(\beta,\alpha)$ $1$ $(A,A,A,1,3)$ $(0,2,1,3,E)$ $(7,7,7,7,7)$ $2^3 \cdot 5$ $(0,-120)$ $2$ $(0,A,2,6,F)$ $(2,1,E,2,1)$ $(6,6,6,6,6)$ $2^2 \cdot 5$ $(0,-125)$ $3$ $(A,A,0,4,F)$ $(2,A,6,2,F)$ $(1,1,1,1,1)$ $2^2 \cdot 5$ $(0,-150)$ $4$ $(0,A,A,4,5)$ $(0,3,6,A,B)$ $(E,E,E,E,E)$ $2^2 \cdot 5$ $(0,-155)$ $5$ $(2,0,A,4,5)$ $(2,A,4,0,5)$ $(B,B,B,B,B)$ $2^2 \cdot 5$ $(0,-180)$ $6$ $(0,A,B,B,E)$ $(0,2,1,3,1)$ $(6,6,6,6,6)$ $2^2 \cdot 5$ $(0,-190)$ $7$ $(0,A,2,6,F)$ $(2,2,6,2,7)$ $(B,B,B,B,B)$ $2^2 \cdot 5$ $(0,-200)$ $8$ $(0,0,A,6,F)$ $(2,1,E,0,3)$ $(6,6,6,6,6)$ $2^2 \cdot 5$ $(0,-215)$ $9$ $(A,0,1,4,7)$ $(0,3,E,2,7)$ $(B,B,B,B,B)$ $2^2 \cdot 5$ $(0,-230)$ $10$ $(A,2,A,1,4)$ $(0,0,7,1,F)$ $(7,7,7,7,7)$ $2^2 \cdot 5$ $(0,-250)$ $11$ $(A,A,3,B,4)$ $(0,A,4,0,7)$ $(4,4,4,4,4)$ $2^2 \cdot 5$ $(0,-275)$ $12$ $(0,2,B,1,E)$ $(0,0,1,1,3)$ $(4,4,4,4,4)$ $2^2 \cdot 5$ $(10,-370)$

Table 7.  Self-dual codes over $\mathbb{F}_{2}$ of length $68$ from $C_{17}$

 $\mathcal{C}_{i}$ $r_{\sigma(v_1)}$ $r_{\sigma(v_2)}$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ $1$ $(0,0,0,0,0,0,0,1,1,0,1,1,0,1,1,1,1)$ $(0,0,0,1,0,0,0,1,1,1,0,0,1,0,1,1,1)$ $0$ $255$ $2 \cdot 17$ $2$ $(0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,1,1)$ $(0,0,0,1,0,1,1,1,1,0,1,1,0,1,1,1,1)$ $0$ $272$ $2^2 \cdot 17$

Table 8.  New codes of length 68 as neighbors

 $\mathcal{N}_{68,i}$ $\mathcal{C}_{i}$ $(x_{35},x_{36},...,x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{68,i}$ $\mathcal{C}_{i}$ $(x_{35},x_{36},...,x_{68})$ $\gamma$ $\beta$ $\mathcal{N}_{68,1}$ $\mathcal{C}_{1}$ $(1001110010101010011000001000111011)$ $\boldsymbol{0}$ $\boldsymbol{183}$ $\mathcal{N}_{68,2}$ $\mathcal{C}_{1}$ $(1101000000010001110101011010100001)$ $\boldsymbol{0}$ $\boldsymbol{185}$ $\mathcal{N}_{68,3}$ $\mathcal{C}_{1}$ $(0001000010011000110101100010101000)$ $\boldsymbol{0}$ $\boldsymbol{189}$ $\mathcal{N}_{68,4}$ $\mathcal{C}_{1}$ $(0100000001110110100011110011101111)$ $\boldsymbol{0}$ $\boldsymbol{191}$ $\mathcal{N}_{68,5}$ $\mathcal{C}_{1}$ $(0110110001001000110010110111100001)$ $\boldsymbol{0}$ $\boldsymbol{193}$ $\mathcal{N}_{68,6}$ $\mathcal{C}_{2}$ $(0000001110101000111001011000001101)$ $\boldsymbol{0}$ $\boldsymbol{195}$ $\mathcal{N}_{68,7}$ $\mathcal{C}_{2}$ $(1001000111000100110010000111111111)$ $\boldsymbol{0}$ $\boldsymbol{197}$ $\mathcal{N}_{68,8}$ $\mathcal{C}_{2}$ $(0110100100000001010000101001011100)$ $\boldsymbol{0}$ $\boldsymbol{199}$ $\mathcal{N}_{68,9}$ $\mathcal{C}_{2}$ $(1010111001110010001010100100011010)$ $\boldsymbol{0}$ $\boldsymbol{200}$ $\mathcal{N}_{68,10}$ $\mathcal{C}_{2}$ $(0000000100000111100111110000110110)$ $\boldsymbol{0}$ $\boldsymbol{203}$ $\mathcal{N}_{68,11}$ $\mathcal{C}_{1}$ $(1001010000011000011101100011101101)$ $\boldsymbol{1}$ $\boldsymbol{189}$ $\mathcal{N}_{68,12}$ $\mathcal{C}_{1}$ $(0110100111000110000001001001100011)$ $\boldsymbol{1}$ $\boldsymbol{201}$ $\mathcal{N}_{68,13}$ $\mathcal{C}_{1}$ $(1010011111110001111001110111001110)$ $\boldsymbol{1}$ $\boldsymbol{203}$ $\mathcal{N}_{68,14}$ $\mathcal{C}_{1}$ $(1111011111101101100101100000010101)$ $\boldsymbol{1}$ $\boldsymbol{205}$ $\mathcal{N}_{68,15}$ $\mathcal{C}_{1}$ $(1011110111111110101101111111101111)$ $\boldsymbol{1}$ $\boldsymbol{213}$ $\mathcal{N}_{68,16}$ $\mathcal{C}_{2}$ $(1010001111110100000010100011101001)$ $\boldsymbol{1}$ $\boldsymbol{216}$ $\mathcal{N}_{68,17}$ $\mathcal{C}_{1}$ $(1011110011111011001101111100111101)$ $\boldsymbol{1}$ $\boldsymbol{217}$ $\mathcal{N}_{68,18}$ $\mathcal{C}_{2}$ $(0000010011001100100101011101110101)$ $\boldsymbol{1}$ $\boldsymbol{233}$

Table 9.  Self-dual codes over $\mathbb{F}_{2}$ of length $92$ from $C_{23}$

 $\mathcal{C}_{i}$ $r_{\sigma(v_1)}$ $r_{\sigma(v_2)}$ $\gamma$ $\beta$ $|Aut(\mathcal{C}_i)|$ Type $1$ $(0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,0,1,1,1)$ $(0,0,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,0,1,1,1)$ $0$ $\textbf{759}$ $2 \cdot 23$ $W_{92,1}$ $3$ $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1)$ $(0,0,0,0,1,1,0,0,0,1,1,0,1,0,0,1,1,0,1,0,1,1,1)$ $0$ $\textbf{1012}$ $2 \cdot 23$ $W_{92,1}$ $13$ $(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,1)$ $(0,0,0,0,1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,1)$ $-46$ $\textbf{1564}$ $2^2 \cdot 23$ $W_{92,1}$ $16$ $(0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,1,1)$ $(0,0,0,0,0,0,1,0,0,0,0,1,1,0,1,0,1,1,1,0,0,0,1)$ $-46$ $\textbf{1978}$ $2 \cdot 23$ $W_{92,1}$
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