Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

Joe Gildea; Adrian Korban; Adam M. Roberts; Alexander Tylyshchak

doi:10.3934/amc.2022021

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2022. Doi: 10.3934/amc.2022021

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Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

1.
Department of Computing Science and Mathematics, School of Informatics and Creative Arts, Dundalk Institute of Technology, Dundalk, County Louth, A91 K584, Ireland
2.
Department of Physical, Mathematical and Engineering Sciences, University of Chester, Exton Park, Chester, CH1 4AR, United Kingdom
3.
Ferenc Rákóczi II Transcarpathian Hungarian College of Higher Education, Berehove, Zakarpattia Oblast, 90201, Ukraine

^*Corresponding author: Adam M. Roberts
^*Corresponding author: Adam M. Roberts

Received: October 31, 2021

Revised: January 31, 2022

Early access: April 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 94B05, 15B10; Secondary: 15B33.

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Table 1. Quaternary notation system for elements of $ \mathbb{F}_2+u \mathbb{F}_2 $

$ \mathbb{F}_2+u \mathbb{F}_2 $	Symbol
$ 0 $	$ \texttt{0} $
$ 1 $	$ \texttt{1} $
$ u $	$ \texttt{2} $
$ 1+u $	$ \texttt{3} $

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Table 2. Code of length 54 over $ \mathbb{F}_2 $ from Theorem 3.1 to which we apply Remark 4.1 to obtain the code in Table 3, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $

$ \mathcal{C}_{54,i}^* $	$ {\bf{{a}}} $	$ {\bf{{b}}} $	$ {\bf{{c}}} $	$ \boldsymbol{{\xi}} $
1	$ \texttt{(0111000101101)} $	$ \texttt{(1101110000100)} $	$ \texttt{(0101111110011)} $	$ \texttt{(001101)} $

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Table 3. New binary self-dual $ [54,27,10] $ code from searching for neighbours of $ \mathcal{C}_{54,j}^* $ as given in Table 2 using Remark 4.1 with $ {\bf{{x}}} = ({\bf{{0}}},{\bf{{x}}}_0) $

$ \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 $

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Table 4. New binary self-dual $ [68,34,12] $ codes from Theorem 3.1 over $ \mathbb{F}_2+u \mathbb{F}_2 $, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $

$ \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 $

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Table 5. Code of length 34 over $ \mathbb{F}_2+u \mathbb{F}_2 $ from Theorem 3.1 to the image of which under $ \varphi_{ \mathbb{F}_2+u \mathbb{F}_2} $ we then apply Remark 4.1 to obtain the codes in Table 6, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $

$ \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)} $

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Table 6. New binary self-dual $ [68,34,12] $ codes from searching for neighbours of $ \varphi_{ \mathbb{F}_2+u \mathbb{F}_2}(\mathcal{C}_{34,j}^*) $ using Remark 4.1 with $ {\bf{{x}}} = ({\bf{{0}}},{\bf{{x}}}_0) $, where $ \mathcal{C}_{34,j}^* $ are as given in Table 5

$ \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 $

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Table 7. New binary self-dual $ [82,41,14] $ codes from Theorem 3.1 over $ \mathbb{F}_2 $, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $

$ \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	$-854$	$5$	$1$

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Table 8. New binary self-dual $ [94,47,16] $ codes from Theorem 3.1 over $ \mathbb{F}_2 $, where $ \boldsymbol{{\xi}} = (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) $

$ \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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