$ A_0 $ | $ A_1 $ | $ A_2 $ | $ A_3 $ | $ A_4 $ | |
$ A_0 \cdot A_0 $ | 1 | 0 | 0 | 0 | 0 |
$ A_1 \cdot A_1 $ | 9 | 0 | 1 | 0 | 0 |
$ A_2 \cdot A_2 $ | 72 | 16 | 21 | 15 | 24 |
$ A_3 \cdot A_3 $ | 144 | 48 | 84 | 68 | 80 |
$ A_4 \cdot A_4 $ | 54 | 6 | 12 | 9 | 12 |
We give two methods for a construction of self-orthogonal linear codes from equitable partitions of distance-regular graphs. By applying these methods, we construct self-orthogonal codes from equitable partitions of the graph of unitals in $ PG(2,4) $ and the only known strongly regular graph with parameters $ (216,40,4,8) $. Some of the codes obtained are optimal.
Citation: |
Table 1.
The coefficients
$ A_0 $ | $ A_1 $ | $ A_2 $ | $ A_3 $ | $ A_4 $ | |
$ A_0 \cdot A_0 $ | 1 | 0 | 0 | 0 | 0 |
$ A_1 \cdot A_1 $ | 9 | 0 | 1 | 0 | 0 |
$ A_2 \cdot A_2 $ | 72 | 16 | 21 | 15 | 24 |
$ A_3 \cdot A_3 $ | 144 | 48 | 84 | 68 | 80 |
$ A_4 \cdot A_4 $ | 54 | 6 | 12 | 9 | 12 |
Table 2.
Self-orthogonal codes from the matrices
i | $ H\leq P \Gamma L(3,4).Z_2 $ | orbit lengths | code | construction |
$ 3 $ | $ I $ | $ 280\times 1 $ | $ [280,18,112]_2 $ | Cor. 1 |
$ 3 $ | $ Z_3 $ | $ 10\times 1,90\times 3 $ | $ [100,10,40]_2 $ | Cor. 1 |
$ 3 $ | $ Z_3 $ | $ 7\times 1,91\times 3 $ | $ [98,6,48]_2* $ | Cor. 1 |
$ 3 $ | $ Z_3 $ | $ 1\times 1,93\times 3 $ | $ [94,6,38]_2 $ | Cor. 1 |
$ 3 $ | $ Z_5 $ | $ 56\times 5 $ | $ [56,2,24]_2 $ | Cor. 1 |
$ 3 $ | $ E_9 $ | $ 1\times 1,9\times 3,28\times 9 $ | $ [38,6,16]_2 $ | Cor. 1 |
$ 3 $ | $ E_9 $ | $ 1\times 1,6\times 3,29\times 9 $ | $ [36,2,18]_2 $ | Cor. 1 |
$ 3 $ | $ E_9 $ | $ 1\times 1,31\times9 $ | $ [32,2,16]_2 $ | Cor. 1 |
$ 3 $ | $ Z_{15} $ | $ 2\times 5,18\times 15 $ | $ [20,2,8]_2 $ | Cor. 1 |
$ 3 $ | $ E_9:Z_3 $ | $ 1\times 1,7\times 9,8\times 27 $ | $ [16,2,8]_2 $ | Cor. 1 |
$ 3 $ | $ Z_2 $ | $ 28\times 1,126\times 2 $ | $ [28,6,12]_2* $, $ [126,6,60]_2 $ | Thm. 2.5 |
$ 3 $ | $ Z_2 $ | $ 10\times 1,135\times 2 $ | $ [135,9,56]_2 $ | Thm. 2.5 |
$ 3 $ | $ Z_2 $ | $ 10\times 1,135\times 2 $ | $ [10,6,2]_2 $, $ [135,12,5]_2 $ | Thm. 2.5 |
$ 3 $ | $ Z_2 $ | $ 8\times 1,136\times 2 $ | $ [136,8,56]_2 $ | Thm. 2.5 |
$ 3 $ | $ Z_4 $ | $ 4\times 1,2\times 2,68\times 4 $, $ 4\times 2,68\times 4 $ | $ [68,4,28]_2 $ | Thm. 2.5 |
$ 3 $ | $ Z_4 $ | $ 6\times 1,1\times 2,68\times 4 $ | $ [68,4,32]_2 $ | Thm. 2.5 |
$ 3 $ | $ E_4 $ | $ 4\times 1,18\times 2,60\times 4 $ | $ [18,3,6]_2 $ $ [60,3,30]_2 $ | Thm. 2.5 |
$ 3 $ | $ E_4 $ | $ 4\times 1,26\times 2,56\times 4 $ | $ [26,4,8]_2 $, $ [56,2,32]_2 $ | Thm. 2.5 |
$ 3 $ | $ E_4 $ | $ 12\times 2,64\times 4 $ | $ [64,2,28]_2 $ | Thm. 2.5 |
$ 3 $ | $ E_4 $ | $ 12\times 2,64\times 4 $ | $ [64,2,40]_2 $ | Thm. 2.5 |
$ 3 $ | $ E_4 $ | $ 2\times 1,11\times 2,64\times 4 $ | $ [64,4,30]_2 $ | Thm. 2.5 |
$ 4 $ | $ Z_7 $ | $ 40\times 7 $ | $ [40,12,12]_3 $ | Thm. 2.2 |
$ 4 $ | $ Z_3 $ | $ 1\times 1,93\times 3 $ | $ [93,27,15]_3 $ | Thm. 2.5 |
$ 4 $ | $ Z_3 $ | $ 7\times 1,91\times 3 $ | $ [91,28,18]_3 $ | Thm. 2.5 |
$ 4 $ | $ Z_3 $ | $ 10\times 1,90\times 3 $ | $ [90,25,18]_3 $ | Thm. 2.5 |
$ 4 $ | $ E_9 $ | $ 1\times 1,31\times 9 $ | $ [31,8,6]_3 $ | Thm. 2.5 |
$ 4 $ | $ E_9 $ | $ 1\times 1,6\times 3,29\times 9 $ | $ [29,9,6]_3 $ | Thm. 2.5 |
$ 4 $ | $ E_9 $ | $ 1\times 1,9\times 3,28\times 9 $ | $ [28,7,6]_3 $ | Thm. 2.5 |
Table 3.
The coefficients
$ A_0 $ | $ A_1 $ | $ A_2 $ | |
$ A_0 \cdot A_0 $ | 1 | 0 | 0 |
$ A_1 \cdot A_1 $ | 40 | 4 | 8 |
$ A_2 \cdot A_2 $ | 175 | 140 | 142 |
Table 4.
Binary self-orthogonal codes from the matrices
$ H\leq Aut(\Gamma_{216}) $ | orbit lengths | code |
$ I $ | $ 216\times 1 $ | $ [ 216,60,32 ] $ |
$ Z_3 $ | $ 72\times 3 $ | $ [ 72, 18, 16 ] $ |
$ Z_3 $ | $ 72\times 3 $ | $ [ 72, 16, 16 ] $ |
$ Z_3 $ | $ 9\times 1, 69\times 3 $ | $ [ 78, 20, 16 ] $ |
$ Z_5 $ | $ 1\times 1, 43\times 5 $ | $ [ 44, 12, 8 ] $ |
$ E_9 $ | $ 6\times 3, 22\times 9 $ | $ [ 28, 4, 8 ] $ |
$ E_9 $ | $ 9\times 3, 21\times 9 $ | $ [ 30, 6, 8 ] $ |
$ E_9 $ | $ 3\times 3, 23\times 9 $ | $ [ 26, 4, 8 ] $ |
$ Z_9 $ | $ 24\times 9 $ | $ [ 24, 6, 8 ] $ |
Table 5.
Binary self-orthogonal codes from the matrices
$ H\leq Aut(\Gamma_{216}) $ | orbit lengths | code |
$ Z_2 $ | $ 6\times 1,105\times 2 $ | $ [ 105, 26, 20 ] $ |
$ Z_2 $ | $ 48 \times 1, 84\times 2 $ | $ [ 48, 12, 12 ], [ 84, 24, 16 ] $ |
$ Z_2 $ | $ 12\times 1,102\times 2 $ | $ [ 102, 28, 16 ] $ |
$ Z_2 $ | $ 10\times 1,103\times 2 $ | $ [ 103, 30, 16 ] $ |
$ Z_4 $ | $ 24\times 2, 42\times 4 $ | $ [ 24, 4, 8 ], [ 42, 12, 8 ] $ |
$ Z_4 $ | $ 6\times 2, 51\times 4 $ | $ [ 51, 14, 8 ] $ |
$ E_4 $ | $ 18\times 2, 45\times 4 $ | $ [ 45, 10, 8 ] $ |
$ E_4 $ | $ 12\times 1, 36\times 2, 33\times 4 $ | $ [ 36, 8, 8 ], [ 33, 10, 8 ] $ |
$ E_4 $ | $ 12\times 2, 48\times 4 $ | $ [ 48, 12, 12 ] $ |
$ E_4 $ | $ 36\times 2, 36\times 4 $ | $ [ 36, 10, 8 ], [ 36, 4, 8 ] $ |
$ E_4 $ | $ 4\times 1, 26\times 2, 40 \times 4 $ | $ [ 26, 6, 6 ], [ 40, 10, 12 ] $ |
$ E_4 $ | $ 2\times 1, 15\times 2, 46\times 4 $ | $ [ 46, 12, 8 ] $ |
$ Z_4 $ | $ 4\times 1, 4\times 2, 51\times 4 $ | $ [ 51, 14, 8 ] $ |
$ E_4 $ | $ 4\times 1, 10\times 2, 48\times 4 $ | $ [ 48, 12, 12 ] $ |
$ Z_4 $ | $ 2\times 1, 5\times 2, 51\times 4 $ | $ [ 51, 14, 8 ] $ |
$ E_4 $ | $ 2\times 1, 11\times 2, 48\times 4 $ | $ [ 48, 12, 12 ] $ |
$ E_4 $ | $ 2\times 1, 13\times 2, 47\times 4 $ | $ [ 47, 14, 8 ] $ |
$ E_4 $ | $ 4\times 1, 28\times 2, 39\times 4 $ | $ [ 39, 12, 8 ],[ 28, 6, 6 ] $ |
$ S_3 $ | $ 6\times 3, 33\times 6 $ | $ [ 33, 4, 12 ] $ |
$ Z_6 $ | $ 16\times 3, 28\times 6 $ | $ [ 28, 8, 8 ] $ |
$ Z_6 $ | $ 2\times 3, 35\times 6 $ | $ [ 35, 6, 8 ] $ |
$ S_3 $ | $ 10\times 3, 31\times 6 $ | $ [ 31, 8, 8 ] $ |
$ S_3 $ | $ 12\times 3, 30\times 6 $ | $ [ 30, 6, 8 ] $ |
$ Z_6 $ | $ 16\times 3, 28\times 6 $ | $ [ 28, 6, 8 ], [ 16, 4, 8 ] * $ |
$ S_3 $ | $ 10\times 3, 31\times 6 $ | $ [ 31, 8, 8 ] $ |
$ Z_6 $ | $ 4\times3, 34\times 6 $ | $ [ 34, 8, 8 ] $ |
$ S_3 $ | $ 10\times 3, 31\times 6 $ | $ [ 31, 9, 8 ] $ |
$ Q_8 $ | $ 12\times 4, 21\times 8 $ | $ [ 21, 4, 8 ] $ |
$ E_8 $ | $ 4\times 2, 24\times 4, 14\times 8 $ | $ [ 14, 4, 4 ] $ |
$ D_8 $ | $ 6\times 2, 21\times 4, 15\times 8 $ | $ [ 15, 4, 8 ]*, [ 21, 4, 8 ] $ |
$ E_8 $ | $ 18\times 2, 21\times 4, 12\times 8 $ | $ [ 12, 4, 4 ] $ |
$ Z_4\times Z_2 $ | $ 18\times 4, 18\times 8 $ | $ [ 18, 4, 4 ] $ |
$ E_8 $ | $ 4\times 2, 10\times 4, 21\times 8 $ | $ [ 21, 6, 8 ]* $ |
$ E_8 $ | $ 6\times 2, 9\times 4, 21\times 8 $ | $ [ 21, 4, 8 ] $ |
$ Z_4\times Z_2 $ | $ 2\times 2, 17\times 4, 18\times 8 $ | $ [ 18, 4, 8 ] * $ |
$ D_8 $ | $ 2\times 2, 11\times 4, 21\times 8 $ | $ [ 21, 4, 8 ] $ |
$ E_8 $ | $ 8\times 2, 14\times 4, 18\times 8 $ | $ [ 18, 4, 8 ] * $ |
$ Z_4\times Z_2 $ | $ 4\times 2, 16\times 4, 18\times 8 $ | $ [ 18, 4, 8 ]* $ |
$ D_8 $ | $ 4\times 1, 4\times 2, 21\times 4, 15\times 8 $ | $ [ 21, 4, 8 ] $ |
$ D_8 $ | $ 24\times 4, 15\times 8 $ | $ [ 15, 4, 4 ] $ |
$ E_8 $ | $ 10\times 2, 13\times 4, 18\times 8 $ | $ [ 18, 4, 8 ]* $ |
$ Z_4\times Z_2 $ | $ 2\times 2, 17\times 4, 18\times 8 $ | $ [ 18, 4, 4 ] $ |
$ D_8 $ | $ 2\times 2, 9\times 4, 22\times 8 $ | $ [ 22, 6, 8 ] $ |
$ D_8 $ | $ 4\times 2, 10\times 4, 21\times 8 $ | $ [ 21, 4, 8 ] $ |
$ D_8 $ | $ 2\times 2, 11\times 4, 21\times 8 $ | $ [ 21, 6, 8 ]* $ |
$ Z_8 $ | $ 12\times 4, 21\times 8 $ | $ [ 21, 6, 4 ] $ |
$ D_8 $ | $ 2\times 1, 5\times 2, 23\times 4, 14\times 8 $ | $ [ 23, 4, 8 ] $ |
$ E_8 $ | $ 2\times 1, 9\times 2, 19\times 4, 15\times 8 $ | $ [ 19, 4, 4 ] $ |
$ Z_4\times Z_2 $ | $ 2\times 2, 5\times 4, 24\times 8 $ | $ [ 24, 6, 8 ] $ |
$ D_8 $ | $ 2\times 2, 13\times 4, 20\times 8 $ | $ [ 20, 4, 4 ] $ |
$ D_8 $ | $ 4\times 2, 16\times 4, 18\times 8 $ | $ [ 18, 6, 4 ] $ |
$ Z_6\times Z_2 $ | $ 4\times 3, 12\times 6, 11\times 12 $ | $ [ 12, 4, 4 ] $ |
$ Z_{12} $ | $ 8\times 6, 14\times 12 $ | $ [ 14, 4, 4 ] $ |
$ D_{12} $ | $ 2\times 3, 9\times 6, 13\times 12 $ | $ [ 13, 4, 4 ] $ |
$ Z_{12} $ | $ 2\times 6, 17\times 12 $ | $ [ 17, 4, 4 ] $ |
$ Z_3:Z_4 $ | $ 2\times 3, 1\times 6, 17\times 12 $ | $ [ 17, 4, 4 ] $ |
$ D_{12} $ | $ 4\times 3, 12\times 6, 11\times 12 $ | $ [ 11, 4, 4 ] $ |
$ D_{12} $ | $ 4\times 3, 12\times 6, 11\times 12 $ | $ [ 11, 3, 4 ] $ |
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