New self-dual codes from $ 2 \times 2 $ block circulant matrices, group rings and neighbours of neighbours

Joe Gildea; Abidin Kaya; Adam Michael Roberts; Rhian Taylor; Alexander Tylyshchak

doi:10.3934/amc.2021039

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2021, Volume 0. Doi: 10.3934/amc.2021039

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New self-dual codes from $ 2 \times 2 $ block circulant matrices, group rings and neighbours of neighbours

1.
Department of Mathematics, Faculty of Science and Engineering, University of Chester, England
2.
Harmony Public Schools, Houston, TX, USA
3.
Department of Algebra, Uzhgorod National University, Uzhgorod, Ukraine

Received: February 28, 2021

Revised: May 31, 2021

Early access: September 2021

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Abstract

In this paper, we construct new self-dual codes from a construction that involves a unique combination; $ 2 \times 2 $ block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings $ \mathbb{F}_2 $, $ \mathbb{F}_2+u \mathbb{F}_2 $ and $ \mathbb{F}_4+u \mathbb{F}_4 $. Using extensions and neighbours of codes, we construct $ 32 $ new self-dual codes of length $ 68 $. We construct 48 new best known singly-even self-dual codes of length 96.

Keywords:

Mathematics Subject Classification: 94B05, 20C05, 16S34, 15B33.

Citation:

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Table 1. Self-dual code over $ \mathbb{F}_4+u \mathbb{F}_4 $ of length $ 64 $ from $ C_{4} $ and $ C_{4} $

$ A_{i} $	$ v_1 \in C_{4} $	$ v_2 \in C_{4} $	$ r_A $	$ \|Aut(A_i)\| $	$ \beta $
$ 1 $	$ (8966) $	$ (0000) $	$ (A617) $	$ 2^4 $	$ 0 $

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Table 2. Self-dual code over $ \mathbb{F}_2+u \mathbb{F}_2 $ of length $ 64 $ from $ C_{8} $ and $ C_{8} $

$ B_{i} $	$ v_1 \in C_{8} $	$ v_2 \in C_{8} $	$ r_A $	$ \|Aut(B_i)\| $	$ \beta $
$ 1 $	$ (uuu10311) $	$ (uu011uu0) $	$ (u0300013) $	$ 2^3 $	$ 0 $

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Table 3. Self-dual code over $ \mathbb{F}_2+u \mathbb{F}_2 $ of length $ 64 $ from $ C_{2\cdot 4} $ and $ C_{2\cdot 4} $

$ C_{i} $	$ v_1 \in C_{2\cdot 4} $	$ v_2 \in C_{2\cdot 4} $	$ r_A $	$ \|Aut(C_i)\| $	$ \beta $
$ 1 $	$ (uu01u0u1) $	$ (u0u11u31) $	$ (u3u3u3u0) $	$ 2^{4} $	$ 48 $

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Table 4. Self-dual code of length $ 68 $ from extensions of $ C_1 $, $ C_2 $ and $ C_3 $

$ D_i $	Code	$ c $	$ X $	$ \gamma $	$ \beta $	$ \|Aut(E_i)\| $
$ 1 $	$ A_1 $	$ 1 $	$ (0133010303011u1001333u01031uuu1u) $	$ 4 $	$ 113 $	$ 2 $
$ 2 $	$ B_1 $	$ u+1 $	$ (013011030003013301111030uuu13u10) $	$ \textbf{2} $	$ \textbf{61} $	$ 2 $
$ 3 $	$ C_1 $	$ u+1 $	$ (0u10303u110333001103u00130103303) $	$ \textbf{1} $	$ \textbf{179} $	$ 2 $

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Table 5. Self-dual codes over $ \mathbb{F}_2 $ of length $ 68 $ $ (W_{68, 2}) $ from $ C_{17} $ and $ C_{17} $

$ E_{i} $	$ v_1 \in C_{17} $	$ v_2 \in C_{17} $	$ r_A $	$ \|Aut(D_i)\| $	$ \gamma $	$ \beta $
$ 1 $	(00000000000011011)	$ (00000000000000000) $	$ (00100110010110111) $	$ 2^2 \cdot 17 $	$ 0 $	$ 238 $
$ 2 $	(00000000110001111)	$ (00000000000000000) $	$ (00100100101010101) $	$ 2^2 \cdot 17 $	$ 0 $	$ 272 $

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Table 6. New codes of length 68 from neighbours of $ E_1 $ and $ E_2 $

$ F_{i} $	$ E_{i} $	$ (x_{35}, x_{36}, ..., x_{68}) $	$ \|Aut(F_i) \| $	$ \gamma $	$ \beta $	Type
$ 1 $	$ 2 $	$ (0111011100100011000001001000100110) $	$ 2 $	$ \textbf{0} $	$ \textbf{208} $	$ W_{68, 2} $
$ 2 $	$ 2 $	$ (1110000011111000011000011110011000) $	$ 1 $	$ \textbf{0} $	$ \textbf{214} $	$ W_{68, 2} $
$ 3 $	$ 2 $	$ (0001000100001110111100001010011010) $	$ 2 $	$ \textbf{1} $	$ \textbf{191} $	$ W_{68, 2} $
$ 4 $	$ 2 $	$ (0010111111111110001111001010111001) $	$ 2 $	$ \textbf{1} $	$ \textbf{202} $	$ W_{68, 2} $
$ 5 $	$ 1 $	$ (1001101111101110011000101000010110) $	$ 1 $	$ \textbf{1} $	$ \textbf{210} $	$ W_{68, 2} $
$ 6 $	$ 2 $	$ (0101001000111001100011110011000101) $	$ 1 $	$ \textbf{1} $	$ \textbf{211} $	$ W_{68, 2} $
$ 7 $	$ 2 $	$ (0010101101010100111100000001010001) $	$ 1 $	$ \textbf{1} $	$ \textbf{229} $	$ W_{68, 2} $
$ 8 $	$ 2 $	$ (1111111111111111111011101111111111) $	$ 1 $	$ {} $	$ \textbf{317} $	$ W_{68, 1} $

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Table 7. New codes of length 68 from neighbours of $ F_7 $ and $ F_8 $

$ G_{i} $	$ F_{i} $	$ (x_{35}, x_{36}, ..., x_{68}) $	$ \|Aut(G_i) \| $	$ \gamma $	$ \beta $	Type
$ 1 $	$ 8 $	$ (0001001101110000000000101011001100) $	$ 1 $	$ \textbf{0} $	$ \textbf{218} $	$ W_{68, 2} $
$ 2 $	$ 7 $	$ (0110000010001000111000111000100010) $	$ 1 $	$ \textbf{1} $	$ \textbf{193} $	$ W_{68, 2} $
$ 3 $	$ 7 $	$ (1000100101011000011011110011000000) $	$ 1 $	$ \textbf{1} $	$ \textbf{195} $	$ W_{68, 2} $
$ 4 $	$ 7 $	$ (0101001010010010000100100101001001) $	$ 1 $	$ 1 $	$ 233 $	$ W_{68, 2} $
$ 5 $	$ 7 $	$ (0111010010001001001000000100101010) $	$ 1 $	$ \textbf{2} $	$ \textbf{193} $	$ W_{68, 2} $
$ 6 $	$ 7 $	$ (1100010011000010110111011101101111) $	$ 1 $	$ \textbf{2} $	$ \textbf{195} $	$ W_{68, 2} $

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Table 8. New codes of length 68 from neighbours of $ G_5 $

$ H_{i} $	$ G_{i} $	$ (x_{35}, x_{36}, ..., x_{68}) $	$ \|Aut(H_i) \| $	$ \gamma $	$ \beta $	Type
$ 1 $	$ 5 $	$ (0010010110011000000010111001111110) $	$ 1 $	$ \textbf{1} $	$ \textbf{197} $	$ W_{68, 2} $
$ 2 $	$ 5 $	$ (0100001011001011101010110111011111) $	$ 1 $	$ \textbf{1} $	$ \textbf{199} $	$ W_{68, 2} $
$ 3 $	$ 5 $	$ (1101001011101101011111110111100111) $	$ 1 $	$ \textbf{2} $	$ \textbf{199} $	$ W_{68, 2} $
$ 4 $	$ 5 $	$ (0011000011001110011000001100000001) $	$ 1 $	$ \textbf{2} $	$ \textbf{191} $	$ W_{68, 2} $
$ 5 $	$ 5 $	$ (0001100100110010010101000111100100) $	$ 1 $	$ \textbf{2} $	$ \textbf{204} $	$ W_{68, 2} $
$ 6 $	$ 5 $	$ (1011101001000001101001010111011101) $	$ 1 $	$ \textbf{2} $	$ \textbf{218} $	$ W_{68, 2} $

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Table 9. Code of length 68 from the neighbours of $ D_1 $

$ I_{i} $	$ D_{i} $	$ (x_{35}, x_{36}, ..., x_{68}) $	$ \|Aut(I_i) \| $	$ \gamma $	$ \beta $	Type
$ 1 $	$ 1 $	$ (1111000110110011110111001010111101) $	$ 1 $	$ 5 $	$ 133 $	$ W_{68, 2} $

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Table 10. Code of length 68 from the neighbours of $ I_1 $

$ J_{i} $	$ I_{i} $	$ (x_{35}, x_{36}, ..., x_{68}) $	$ \|Aut(J_i) \| $	$ \gamma $	$ \beta $	Type
$ 1 $	$ 1 $	$ (0000100001011000111001010100001100 $	$ 1 $	$ 6 $	$ 141 $	$ W_{68, 2} $

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Table 11. New codes of length 68 from the neighbours of $ J_1 $

$ K_{i} $	$ J_{i} $	$ (x_{35}, x_{36}, ..., x_{68}) $	$ \|Aut(K_i) \| $	$ \gamma $	$ \beta $	Type
$ 1 $	$ 1 $	$ (1111111101001100010100001000010100) $	$ 1 $	$ \textbf{6} $	$ \textbf{131} $	$ W_{68, 2} $
$ 2 $	$ 1 $	$ (0000001110010111101110011111001111) $	$ 1 $	$ \textbf{7} $	$ \textbf{158} $	$ W_{68, 2} $

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Table 12. New codes of length 68 from the neighbours of $K_2$

$L_{i}$	$K_{i}$	$(x_{35}, x_{36}, ..., x_{68})$	$\|Aut(L_i) \|$	$\gamma$	$\beta$	Type
$1$	$2$	$(0110111111010100011101010011010101)$	$1$	$\textbf{7}$	$\textbf{155}$	$W_{68, 2}$
$2$	$2$	$(0101010101010001001010011101110010)$	$1$	$\textbf{7}$	$\textbf{156}$	$W_{68, 2}$
$3$	$2$	$(0010011101010101010111011110110110)$	$1$	$\textbf{7}$	$\textbf{157}$	$W_{68, 2}$
$4$	$2$	$(1101111110110111001111110101101100)$	$1$	$\textbf{7}$	$\textbf{159}$	$W_{68, 2}$
$5$	$2$	$(1001011111000110001111101100101110)$	$1$	$\textbf{7}$	$\textbf{160}$	$W_{68, 2}$
$6$	$2$	$(1100000100100000010100101100011010)$	$1$	$\textbf{7}$	$\textbf{162}$	$W_{68, 2}$
$7$	$2$	$(1000010000010110000111110010011111)$	$1$	$\textbf{7}$	$\textbf{164}$	$W_{68, 2}$
$8$	$2$	$(0100001001101111111010000101010001)$	$1$	$\textbf{7}$	$\textbf{165}$	$W_{68, 2}$
$9$	$2$	$(0011101000100011011101001111101111)$	$1$	$\textbf{7}$	$\textbf{167}$	$W_{68, 2}$

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Table 13. New singly-even binary self-dual $ [96, 48, 16] $ codes from $ C_6 $ and $ C_6 $ over $ \mathbb{F}_4+u \mathbb{F}_4 $

$ C_{96, i} $	$ v_1 \in C_{6} $	$ v_2 \in C_{6} $	$ r_A $	$ \|Aut(C_{96, i})\| $	$ \alpha $	$ \beta $	$ \gamma $	Type
$ 1 $	$ (17DD00) $	$ (DC34EB) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11104} $	$ -\textbf{68} $	$ \textbf{0} $	$ W_{96, 2} $
$ 2 $	$ (C00E11) $	$ (C8BDA9) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{10208} $	$ -\textbf{52} $	$ \textbf{0} $	$ W_{96, 2} $
$ 3 $	$ (6482FF) $	$ (0D0D0D) $	$ (7C111C) $	$ 2^{4}\cdot 3 $	$ \textbf{11328} $	$ -\textbf{28} $	$ \textbf{0} $	$ W_{96, 2} $
$ 4 $	$ (1236FC) $	$ (914FD8) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{11312} $	$ -\textbf{108} $	$ \textbf{2} $	$ W_{96, 2} $
$ 5 $	$ (3E222F) $	$ (8EBA97) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{11728} $	$ -\textbf{100} $	$ \textbf{2} $	$ W_{96, 2} $
$ 6 $	$ (C6EB5F) $	$ (EA56C1) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11184} $	$ -\textbf{84} $	$ \textbf{2} $	$ W_{96, 2} $
$ 7 $	$ (B88D66) $	$ (99680F) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{10592} $	$ -\textbf{80} $	$ \textbf{2} $	$ W_{96, 2} $
$ 8 $	$ (1D271F) $	$ (A7870E) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{11184} $	$ -\textbf{76} $	$ \textbf{2} $	$ W_{96, 2} $
$ 9 $	$ (0A7B3D) $	$ (126325) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{11488} $	$ -\textbf{72} $	$ \textbf{2} $	$ W_{96, 2} $
$ 10 $	$ (535DD1) $	$ (F1CECB) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{10624} $	$ -\textbf{64} $	$ \textbf{2} $	$ W_{96, 2} $
$ 11 $	$ (C2F3D9) $	$ (1EDF0A) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{10944} $	$ -\textbf{60} $	$ \textbf{2} $	$ W_{96, 2} $
$ 12 $	$ (D4787D) $	$ (9FCD5D) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{11224} $	$ -\textbf{56} $	$ \textbf{2} $	$ W_{96, 2} $
$ 13 $	$ (344A57) $	$ (47F231) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{10728} $	$ -\textbf{48} $	$ \textbf{2} $	$ W_{96, 2} $
$ 14 $	$ (D399AB) $	$ (6DB3F0) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{12320} $	$ -\textbf{156} $	$ \textbf{4} $	$ W_{96, 2} $
$ 15 $	$ (F7A016) $	$ (AE0EBF) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{11104} $	$ -\textbf{140} $	$ \textbf{4} $	$ W_{96, 2} $
$ 16 $	$ (EF2862) $	$ (8867A5) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{11528} $	$ -\textbf{136} $	$ \textbf{4} $	$ W_{96, 2} $
$ 17 $	$ (A56B03) $	$ (317717) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11472} $	$ -\textbf{132} $	$ \textbf{4} $	$ W_{96, 2} $
$ 18 $	$ (4250B6) $	$ (979C73) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{11728} $	$ -\textbf{120} $	$ \textbf{4} $	$ W_{96, 2} $
$ 19 $	$ (01A176) $	$ (CA0455) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{11360} $	$ -\textbf{116} $	$ \textbf{4} $	$ W_{96, 2} $
$ 20 $	$ (FE26F3) $	$ (23B01B) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{11160} $	$ -\textbf{112} $	$ \textbf{4} $	$ W_{96, 2} $
$ 21 $	$ (6C02AE) $	$ (6F098F) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{11328} $	$ -\textbf{112} $	$ \textbf{4} $	$ W_{96, 2} $
$ 22 $	$ (F79924) $	$ (AA77C9) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{11568} $	$ -\textbf{112} $	$ \textbf{4} $	$ W_{96, 2} $
$ 23 $	$ (5FFB7B) $	$ (4A6DD5) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11088} $	$ -\textbf{108} $	$ \textbf{4} $	$ W_{96, 2} $
$ 24 $	$ (3522FB) $	$ (C05E9F) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{11488} $	$ -\textbf{108} $	$ \textbf{4} $	$ W_{96, 2} $
$ 25 $	$ (9E88C6) $	$ (07DE86) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11072} $	$ -\textbf{104} $	$ \textbf{4} $	$ W_{96, 2} $
$ 26 $	$ (088C5F) $	$ (77601A) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{10672} $	$ -\textbf{100} $	$ \textbf{4} $	$ W_{96, 2} $
$ 27 $	$ (313674) $	$ (343BD9) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{10944} $	$ -\textbf{100} $	$ \textbf{4} $	$ W_{96, 2} $
$ 28 $	$ (35EA9C) $	$ (930785) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11048} $	$ -\textbf{96} $	$ \textbf{4} $	$ W_{96, 2} $
$ 29 $	$ (505084) $	$ (57696E) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{11064} $	$ -\textbf{88} $	$ \textbf{4} $	$ W_{96, 2} $
$ 30 $	$ (6D4401) $	$ (92206E) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{11504} $	$ -\textbf{84} $	$ \textbf{4} $	$ W_{96, 2} $
$ 31 $	$ (58263B) $	$ (D98510) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{10888} $	$ -\textbf{80} $	$ \textbf{4} $	$ W_{96, 2} $
$ 32 $	$ (9AE7CA) $	$ (74D032) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{12504} $	$ -\textbf{160} $	$ \textbf{6} $	$ W_{96, 2} $
$ 33 $	$ (73A8CF) $	$ (D46308) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{11552} $	$ -\textbf{156} $	$ \textbf{6} $	$ W_{96, 2} $
$ 34 $	$ (F97D3B) $	$ (6B7D82) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{11872} $	$ -\textbf{156} $	$ \textbf{6} $	$ W_{96, 2} $
$ 35 $	$ (B4196E) $	$ (97B0E5) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{11376} $	$ -\textbf{148} $	$ \textbf{6} $	$ W_{96, 2} $
$ 36 $	$ (47E5CD) $	$ (CECECE) $	$ (6B6DBD) $	$ 2^{4}\cdot 3 $	$ \textbf{11736} $	$ -\textbf{148} $	$ \textbf{6} $	$ W_{96, 2} $
$ 37 $	$ (6B78E6) $	$ (113CD9) $	$ (F656F5) $	$ 2^{4} $	$ \textbf{11576} $	$ -\textbf{140} $	$ \textbf{6} $	$ W_{96, 2} $
$ 38 $	$ (B1C856) $	$ (F7452D) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{12448} $	$ -\textbf{140} $	$ \textbf{6} $	$ W_{96, 2} $
$ 39 $	$ (FC0863) $	$ (18BD3B) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{11008} $	$ -\textbf{132} $	$ \textbf{6} $	$ W_{96, 2} $
$ 40 $	$ (DC4A91) $	$ (A58C34) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{11304} $	$ -\textbf{132} $	$ \textbf{6} $	$ W_{96, 2} $
$ 41 $	$ (8798CD) $	$ (FD6017) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11312} $	$ -\textbf{120} $	$ \textbf{6} $	$ W_{96, 2} $
$ 42 $	$ (9217CF) $	$ (DCD676) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{12928} $	$ -\textbf{192} $	$ \textbf{8} $	$ W_{96, 2} $
$ 43 $	$ (C620D5) $	$ (EAE546) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11768} $	$ -\textbf{172} $	$ \textbf{8} $	$ W_{96, 2} $
$ 44 $	$ (3617E2) $	$ (19B065) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11272} $	$ -\textbf{168} $	$ \textbf{8} $	$ W_{96, 2} $
$ 45 $	$ (3BAE33) $	$ (5F852E) $	$ (7C111C) $	$ 2^{4} $	$ \textbf{11968} $	$ -\textbf{168} $	$ \textbf{8} $	$ W_{96, 2} $
$ 46 $	$ (E90589) $	$ (D62FE2) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{12896} $	$ -\textbf{260} $	$ \textbf{12} $	$ W_{96, 2} $
$ 47 $	$ (B89454) $	$ (F5F331) $	$ (D4DE6E) $	$ 2^{4} $	$ \textbf{12288} $	$ -\textbf{244} $	$ \textbf{12} $	$ W_{96, 2} $
$ 48 $	$ (E9DA51) $	$ (6D030D) $	$ (6B6DBD) $	$ 2^{4} $	$ \textbf{12320} $	$ -\textbf{244} $	$ \textbf{12} $	$ W_{96, 2} $

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References

[1]	D. Anev, M. Harada and N. Yankov, New extremal singly even self-dual codes of lengths 64 and 66, J. Algebra Comb. Discrete Struct. Appl., 5 (2018), 143-151. doi: 10.13069/jacodesmath.458601.
[2]	E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's theorem on self-dual codes, IEEE Trans. Inform. Theory, IT-18 (1972), 409-414. doi: 10.1109/tit.1972.1054817.
[3]	F. Bernhardt, P. Landrock and O. Manz, The extended golay codes considered as ideals, J. Combin. Theory Ser. A, 55 (1990), 235-246. doi: 10.1016/0097-3165(90)90069-9.
[4]	W. Bosma, J. Cannon and C. Playoust, The magma algebra system. I. the user language,, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[5]	C. L. Chen, W. W Peterson and E. J. Weldon. Jr, Some results on quasi-cyclic codes, Information and Control, 15 (1969), 407-423. doi: 10.1016/S0019-9958(69)90497-5.
[6]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[7]	P. J. Davis, Circulant Matrices, A Wiley-Interscience Publication; Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 1979.
[8]	S. T. Dougherty, P. Gaborit, M. Harada and P. Solé, Type ii codes over ${\mathbf f}_2+u{\mathbf f}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45. doi: 10.1109/18.746770.
[9]	S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, A. Tylyshchak and B. Yildiz, Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes, Finite Fields Appl., 57 (2019), 108-127. doi: 10.1016/j.ffa.2019.02.004.
[10]	S. T. Dougherty, J. Gildea, R. Taylor and A. Tylyshchak, Group rings, $g$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138. doi: 10.1007/s10623-017-0440-7.
[11]	S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047. doi: 10.1109/18.641574.
[12]	S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $r_k$, gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219. doi: 10.1016/j.ffa.2010.11.002.
[13]	S. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $r_k$ and binary self-dual codes, Eur. J. Pure Appl. Math., 6 (2013), 89-106.
[14]	P. Gaborit, Quadratic double circulant codes over fields, J. Combin. Theory Ser. A, 97 (2002), 85-107. doi: 10.1006/jcta.2001.3198.
[15]	S. D. Georgiou and E. Lappas, Self-dual codes from circulant matrices, Des. Codes Cryptogr., 64 (2012), 129-141. doi: 10.1007/s10623-011-9510-4.
[16]	J. Gildea, H. Hamilton, A. Kaya and B. Yildiz, Modified quadratic residue constructions and new extremal binary self-dual codes of lengths 64, 66 and 68, Information Processing Letters, 157. doi: 10.1016/j.ipl.2020.105927.
[17]	J. Gildea, A. Kaya, A. Korban and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. Math. Commun., 15 (2021), 471-485. doi: 10.3934/amc.2020077.
[18]	J. Gildea, A. Korban and A. M. Roberts, New binary self-dual codes of lengths 80, 84 and 96 from composite matrices, 2021, https://arXiv.org/abs/2106.12355.
[19]	A. M. Gleason, Weight polynomials of self-dual codes and the macwilliams identities, 1971,211–215.
[20]	T. A. Gulliver and M. Harada, On extremal double circulant self-dual codes of lengths 90–96, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 403-415. doi: 10.1007/s00200-019-00381-3.
[21]	M. Gürel and N. Yankov, Self-dual codes with an automorphism of order 17, Math. Commun., 21 (2016), 97-107.
[22]	T. Hurley, Group rings and rings of matrices, Int. J. Pure Appl. Math., 31 (2006), 319-335.
[23]	S. Karadeniz and B. Yildiz, New extremal binary self-dual codes of length 66 as extensions of self-dual codes over $R_k$, J. Franklin Inst., 350, (2013), 1963–1973 doi: 10.1016/j.jfranklin.2013.05.015.
[24]	M. Karlin, New binary coding results by circulants, IEEE Trans. Inform. Theory, IT-15 (1969), 81-92. doi: 10.1109/tit.1969.1054261.
[25]	A. Kaya, B. Yildiz and I. Siap, Quadratic residue codes over $\mathbb{F}_p+v\mathbb{F}_p$ and their Gray images, J. Pure Appl. Algebra, 218 (2014), 1999-2011. doi: 10.1016/j.jpaa.2014.03.002.
[26]	J.-L. Kim, New extremal self-dual codes of lengths 36, 38, and 58, IEEE Trans. Inform. Theory, 47 (2001), 386-393. doi: 10.1109/18.904540.
[27]	S. Ling and P. Solé, Type ii codes over $\mathbf { F_4+u} f_4$, European J. Combin., 22 (2001), 983-997. doi: 10.1006/eujc.2001.0509.
[28]	F. J. MacWilliams, C. L. Mallows and N. J. A Sloane, Generalizations of gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, IT-18 (1972), 794-805. doi: 10.1109/tit.1972.1054898.
[29]	F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist, Discrete Math., 3 (1972), 153-162. doi: 10.1016/0012-365X(72)90030-1.
[30]	E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.
[31]	A. M. Roberts, Weight Enumerator Parameter Database for Binary Self-Dual Codes, 2021, https://amr-wepd-bsdc.netlify.app
[32]	N. Yankov and D. Anev, On the self-dual codes with an automorphism of order 5, Appl. Algebra Engrg. Comm. Comput., 32 (2021), 97-111. doi: 10.1007/s00200-019-00403-0.
[33]	N. Yankov, M. Ivanova and M. H. Lee, Self-dual codes with an automorphism of order 7 and $s$-extremal codes of length 68, Finite Fields Appl., 51 (2018), 17-30. doi: 10.1016/j.ffa.2017.12.001.
[34]	R. Yorgova and A. Wassermann, Binary self-dual codes with automorphisms of order 23, Des. Codes Cryptogr., 48 (2008), 155-164. doi: 10.1007/s10623-007-9152-8.