$ \mathbb{F}_2+u \mathbb{F}_2 $ | Symbol |
$ 0 $ | $ \texttt{0} $ |
$ 1 $ | $ \texttt{1} $ |
$ u $ | $ \texttt{2} $ |
$ 1+u $ | $ \texttt{3} $ |
In this work, we define a modification of a bordered construction for self-dual codes which utilises $ \lambda $-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.
Citation: |
Table 1.
Quaternary notation system for elements of
$ \mathbb{F}_2+u \mathbb{F}_2 $ | Symbol |
$ 0 $ | $ \texttt{0} $ |
$ 1 $ | $ \texttt{1} $ |
$ u $ | $ \texttt{2} $ |
$ 1+u $ | $ \texttt{3} $ |
Table 2.
Code of length 54 over
$ \mathcal{C}_{54,i}^* $ | $ {\bf{{a}}} $ | $ {\bf{{b}}} $ | $ {\bf{{c}}} $ | $ \boldsymbol{{\xi}} $ |
1 | $ \texttt{(0111000101101)} $ | $ \texttt{(1101110000100)} $ | $ \texttt{(0101111110011)} $ | $ \texttt{(001101)} $ |
Table 3.
New binary self-dual
$ \mathcal{C}_{54,i} $ | $ \mathcal{C}_{54,j}^* $ | $ {\bf{{x}}}_0 $ | $ W_{54,k} $ | $ \alpha $ | $ |\text{Aut}({\mathcal{C}_{54,i}})| $ |
1 | 1 | $ \texttt{(000001100101001000111101101)} $ | 1 | $ 23 $ | $ 3 $ |
Table 4.
New binary self-dual
$ \mathcal{C}_{68,i} $ | $ \lambda $ | $ \mu $ | $ {\bf{{a}}} $ | $ {\bf{{b}}} $ | $ {\bf{{c}}} $ | $ \boldsymbol{{\xi}} $ | $ W_{68,j} $ | $ \alpha $ | $ \beta $ | $ |\text{Aut}({\mathcal{C}_{68,i}})| $ |
1 | $ \texttt{1} $ | $ \texttt{1} $ | $ \texttt{(22120031)} $ | $ \texttt{(02331100)} $ | $ \texttt{(33331213)} $ | $ \texttt{(101132)} $ | 1 | $ 110 $ | $ - $ | $ 2 $ |
2 | $ \texttt{1} $ | $ \texttt{1} $ | $ \texttt{(10021300)} $ | $ \texttt{(31232012)} $ | $ \texttt{(30313131)} $ | $ \texttt{(120023)} $ | 1 | $ 124 $ | $ - $ | $ 2 $ |
3 | $ \texttt{1} $ | $ \texttt{1} $ | $ \texttt{(01323103)} $ | $ \texttt{(20022123)} $ | $ \texttt{(00300222)} $ | $ \texttt{(013332)} $ | 2 | $ 20 $ | $ 1 $ | $ 2 $ |
4 | $ \texttt{1} $ | $ \texttt{3} $ | $ \texttt{(01230200)} $ | $ \texttt{(13010312)} $ | $ \texttt{(22003002)} $ | $ \texttt{(102232)} $ | 2 | $ 28 $ | $ 1 $ | $ 2 $ |
5 | $ \texttt{1} $ | $ \texttt{1} $ | $ \texttt{(31221023)} $ | $ \texttt{(30003111)} $ | $ \texttt{(13012103)} $ | $ \texttt{(233310)} $ | 2 | $ 32 $ | $ 1 $ | $ 2 $ |
6 | $ \texttt{1} $ | $ \texttt{1} $ | $ \texttt{(03210210)} $ | $ \texttt{(32221121)} $ | $ \texttt{(13331101)} $ | $ \texttt{(122201)} $ | 2 | $ 34 $ | $ 1 $ | $ 2 $ |
7 | $ \texttt{1} $ | $ \texttt{1} $ | $ \texttt{(00030320)} $ | $ \texttt{(21031233)} $ | $ \texttt{(32100012)} $ | $ \texttt{(122201)} $ | 2 | $ 36 $ | $ 1 $ | $ 2 $ |
Table 5.
Code of length 34 over
$ \mathcal{C}_{34,i}^* $ | $ \lambda $ | $ \mu $ | $ {\bf{{a}}} $ | $ {\bf{{b}}} $ | $ {\bf{{c}}} $ | $ \boldsymbol{{\xi}} $ |
1 | $ \texttt{1} $ | $ \texttt{1} $ | $ \texttt{(01323103)} $ | $ \texttt{(20022123)} $ | $ \texttt{(00300222)} $ | $ \texttt{(013332)} $ |
Table 6.
New binary self-dual
$ \mathcal{C}_{68,i} $ | $ \mathcal{C}_{34,j}^* $ | $ {\bf{{x}}}_0 $ | $ W_{68,k} $ | $ \alpha $ | $ \beta $ | $ |\text{Aut}({\mathcal{C}_{68,i}})| $ |
8 | 1 | $ \texttt{(0101010011111010001101100011011100)} $ | 1 | $ 113 $ | $ - $ | $ 1 $ |
9 | 1 | $ \texttt{(1110010011100001110010110111100100)} $ | 1 | $ 114 $ | $ - $ | $ 1 $ |
10 | 1 | $ \texttt{(1010100100010111000000100111010111)} $ | 1 | $ 116 $ | $ - $ | $ 1 $ |
11 | 1 | $ \texttt{(0011000011011101010101010100010000)} $ | 1 | $ 118 $ | $ - $ | $ 1 $ |
12 | 1 | $ \texttt{(0101010001111010000101100011011111)} $ | 1 | $ 121 $ | $ - $ | $ 1 $ |
13 | 1 | $ \texttt{(0011001001011000000110010111110101)} $ | 1 | $ 123 $ | $ - $ | $ 1 $ |
14 | 1 | $ \texttt{(0101110101111010001101100011011101)} $ | 2 | $ 37 $ | $ 1 $ | $ 1 $ |
Table 7.
New binary self-dual
$ \mathcal{C}_{82,i} $ | $ {\bf{{a}}} $ | $ {\bf{{b}}} $ | $ {\bf{{c}}} $ | $ \boldsymbol{{\xi}} $ |
1 | $ \texttt{(00110011100000000110)} $ | $ \texttt{(00100110011101010011)} $ | $ \texttt{(00010010010001000001)} $ | $ \texttt{(101010)} $ |
2 | $ \texttt{(11001011011010110101)} $ | $ \texttt{(10011010011011010000)} $ | $ \texttt{(01010011100101001010)} $ | $ \texttt{(101010)} $ |
3 | $ \texttt{(00011110011001011110)} $ | $ \texttt{(01010101010011110100)} $ | $ \texttt{(10101110111000111011)} $ | $ \texttt{(101010)} $ |
4 | $ \texttt{(00000110100111111111)} $ | $ \texttt{(00110110000111101000)} $ | $ \texttt{(11111011010111011000)} $ | $ \texttt{(101001)} $ |
5 | $ \texttt{(11100011011110101011)} $ | $ \texttt{(11110001101100110011)} $ | $ \texttt{(00100010100000001010)} $ | $ \texttt{(101010)} $ |
6 | $ \texttt{(11111110010110010010)} $ | $ \texttt{(10001001101001001110)} $ | $ \texttt{(01111010111110011001)} $ | $ \texttt{(101001)} $ |
7 | $ \texttt{(00111010001011010100)} $ | $ \texttt{(11001010111101110001)} $ | $ \texttt{(10001100011010110001)} $ | $ \texttt{(101010)} $ |
8 | $ \texttt{(00110011011011110001)} $ | $ \texttt{(00101110100101000100)} $ | $ \texttt{(10110001110000000001)} $ | $ \texttt{(101110)} $ |
9 | $ \texttt{(10000011001000100011)} $ | $ \texttt{(00110001010001110100)} $ | $ \texttt{(00010001110001000101)} $ | $ \texttt{(101101)} $ |
10 | $ \texttt{(11101110100101100010)} $ | $ \texttt{(01110011001100110001)} $ | $ \texttt{(00010100000110011010)} $ | $ \texttt{(101101)} $ |
11 | $ \texttt{(00011011111101000011)} $ | $ \texttt{(11000000001100111001)} $ | $ \texttt{(10100000101010010010)} $ | $ \texttt{(101110)} $ |
12 | $ \texttt{(00011110101110000110)} $ | $ \texttt{(11000011010011000101)} $ | $ \texttt{(01001010001111101110)} $ | $ \texttt{(101110)} $ |
13 | $ \texttt{(00100000101100010000)} $ | $ \texttt{(11010101010010100011)} $ | $ \texttt{(01011101110000111001)} $ | $ \texttt{(101101)} $ |
14 | $ \texttt{(10001111010001011100)} $ | $ \texttt{(00000001010010011000)} $ | $ \texttt{(01101011111010000110)} $ | $ \texttt{(101101)} $ |
15 | $ \texttt{(10011111001010110001)} $ | $ \texttt{(11000010101110010110)} $ | $ \texttt{(01000011001011110111)} $ | $ \texttt{(101110)} $ |
16 | $ \texttt{(11100100001011100001)} $ | $ \texttt{(00101100110000110100)} $ | $ \texttt{(00011111001001111100)} $ | $ \texttt{(101101)} $ |
17 | $ \texttt{(10001110110000101100)} $ | $ \texttt{(00111010000111110010)} $ | $ \texttt{(01110111101001100001)} $ | $ \texttt{(101110)} $ |
18 | $ \texttt{(00001101111100100101)} $ | $ \texttt{(00011001110100011111)} $ | $ \texttt{(01001100001011101111)} $ | $ \texttt{(101110)} $ |
$\mathcal{C}_{82,i}$ | $W_{82,j}$ | $\alpha$ | $\beta$ | $|\text{Aut}({\mathcal{C}_{82,i}})|$ |
1 | 2 | $-738$ | $18$ | $1$ |
2 | 2 | $-736$ | $18$ | $1$ |
3 | 2 | $-734$ | $18$ | $1$ |
4 | 2 | $-714$ | $18$ | $1$ |
5 | 2 | $-706$ | $18$ | $1$ |
6 | 2 | $-688$ | $18$ | $1$ |
7 | 2 | $-662$ | $18$ | $1$ |
8 | 3 | $-828$ | $0$ | $1$ |
9 | 3 | $-816$ | $0$ | $1$ |
10 | 3 | $-812$ | $0$ | $1$ |
11 | 3 | $-798$ | $0$ | $1$ |
12 | 3 | $-786$ | $0$ | $1$ |
13 | 3 | $-778$ | $0$ | $1$ |
14 | 3 | $-776$ | $0$ | $1$ |
15 | 3 | $-818$ | $1$ | $1$ |
16 | 3 | $-838$ | $2$ | $1$ |
17 | 3 | $-818$ | $2$ | $1$ |
18 | 3 | $-854$ | $5$ | $1$ |
Table 8.
New binary self-dual
$ \mathcal{C}_{94,i} $ | $ {\bf{{a}}} $ | $ {\bf{{b}}} $ | $ {\bf{{c}}} $ | $ \boldsymbol{{\xi}} $ |
1 | $ \texttt{(01111111111001110101110)} $ | $ \texttt{(01101101000111011010001)} $ | $ \texttt{(00001000000000000000000)} $ | $ \texttt{(001110)} $ |
2 | $ \texttt{(10010111111101010000010)} $ | $ \texttt{(11100100111001001111001)} $ | $ \texttt{(00001000000000000000000)} $ | $ \texttt{(001110)} $ |
3 | $ \texttt{(01100111001001011111010)} $ | $ \texttt{(10110101001111101000010)} $ | $ \texttt{(11010111010100010110011)} $ | $ \texttt{(001110)} $ |
4 | $ \texttt{(10010101100111000001101)} $ | $ \texttt{(11010000110110110000001)} $ | $ \texttt{(01010001111011001010111)} $ | $ \texttt{(110010)} $ |
5 | $ \texttt{(00000111101001000010100)} $ | $ \texttt{(11110100110110100111000)} $ | $ \texttt{(01001111001111101100100)} $ | $ \texttt{(001101)} $ |
6 | $ \texttt{(11011110010100111000000)} $ | $ \texttt{(01110100011001101101111)} $ | $ \texttt{(01110000001111000111111)} $ | $ \texttt{(001101)} $ |
7 | $ \texttt{(01011011110110010001110)} $ | $ \texttt{(10010110110110001100101)} $ | $ \texttt{(00000100000000000000000)} $ | $ \texttt{(110010)} $ |
8 | $ \texttt{(01100001100001100101010)} $ | $ \texttt{(11111101000110000010101)} $ | $ \texttt{(00100000000000000000000)} $ | $ \texttt{(001101)} $ |
9 | $ \texttt{(00000111001111011011110)} $ | $ \texttt{(11100000000100010011010)} $ | $ \texttt{(01101111110111000010001)} $ | $ \texttt{(110010)} $ |
10 | $ \texttt{(01101101011111000010001)} $ | $ \texttt{(10100110011101001101101)} $ | $ \texttt{(01011000110000010010101)} $ | $ \texttt{(110010)} $ |
11 | $ \texttt{(11010010011100001111011)} $ | $ \texttt{(10001110000000010001110)} $ | $ \texttt{(11101110011100011101000)} $ | $ \texttt{(110010)} $ |
12 | $ \texttt{(10101100011011001010111)} $ | $ \texttt{(00010010000011111000010)} $ | $ \texttt{(00111100000011101111110)} $ | $ \texttt{(001101)} $ |
$ \mathcal{C}_{94,i} $ | $ W_{94,j} $ | $ \alpha $ | $ \beta $ | $ |\text{Aut}({\mathcal{C}_{94,i}})| $ |
1 | 1 | $ 4646 $ | $ -92 $ | $ 2\cdot 23 $ |
2 | 1 | $ 3450 $ | $ -46 $ | $ 2\cdot 23 $ |
3 | 1 | $ 3680 $ | $ -46 $ | $ 23 $ |
4 | 1 | $ 3772 $ | $ -46 $ | $ 23 $ |
5 | 1 | $ 4186 $ | $ -46 $ | $ 23 $ |
6 | 1 | $ 2944 $ | $ -23 $ | $ 23 $ |
7 | 1 | $ 3680 $ | $ -23 $ | $ 23 $ |
8 | 1 | $ 2346 $ | $ 0 $ | $ 2\cdot 23 $ |
9 | 1 | $ 2530 $ | $ 0 $ | $ 23 $ |
10 | 1 | $ 2576 $ | $ 0 $ | $ 23 $ |
11 | 1 | $ 3496 $ | $ 0 $ | $ 23 $ |
12 | 1 | $ 3588 $ | $ 0 $ | $ 23 $ |
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