$ 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 $ |
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.
Citation: |
Table 1.
Self-dual code over
$ 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 $ |
Table 2.
Self-dual code over
$ 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 $ |
Table 3.
Self-dual code over
$ 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 $ |
Table 4.
Self-dual code of length
$ 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 $ |
Table 5.
Self-dual codes over
$ 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 $ |
Table 6.
New codes of length 68 from neighbours of
$ 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} $ |
Table 7.
New codes of length 68 from neighbours of
$ 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} $ |
Table 8.
New codes of length 68 from neighbours of
$ 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} $ |
Table 9.
Code of length 68 from the neighbours of
$ 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} $ |
Table 10.
Code of length 68 from the neighbours of
$ 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} $ |
Table 11.
New codes of length 68 from the neighbours of
$ 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} $ |
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 |
$1$ | $2$ | $(0110111111010100011101010011010101)$ | $1$ | $\textbf{7}$ | $\textbf{155}$ | $W_{68, 2}$ |
$2$ | $2$ | $(0101010101010001001010011101110010)$ | $1$ | $\textbf{7}$ | $\textbf{156}$ | $W_{68, 2}$ |
$3$ | $2$ | $(0010011101010101010111011110110110)$ | $1$ | $\textbf{7}$ | $\textbf{157}$ | $W_{68, 2}$ |
$4$ | $2$ | $(1101111110110111001111110101101100)$ | $1$ | $\textbf{7}$ | $\textbf{159}$ | $W_{68, 2}$ |
$5$ | $2$ | $(1001011111000110001111101100101110)$ | $1$ | $\textbf{7}$ | $\textbf{160}$ | $W_{68, 2}$ |
$6$ | $2$ | $(1100000100100000010100101100011010)$ | $1$ | $\textbf{7}$ | $\textbf{162}$ | $W_{68, 2}$ |
$7$ | $2$ | $(1000010000010110000111110010011111)$ | $1$ | $\textbf{7}$ | $\textbf{164}$ | $W_{68, 2}$ |
$8$ | $2$ | $(0100001001101111111010000101010001)$ | $1$ | $\textbf{7}$ | $\textbf{165}$ | $W_{68, 2}$ |
$9$ | $2$ | $(0011101000100011011101001111101111)$ | $1$ | $\textbf{7}$ | $\textbf{167}$ | $W_{68, 2}$ |
Table 13.
New singly-even binary self-dual
$ 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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