Systems of four LS of order 10 with partial orthogonality


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Nataly-Mak
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Message 889 - Posted 22 Jun 2016 8:16:13 UTC

    Systems of four LS of order 10 with partial orthogonality

    Square A

    9 4 6 1 5 2 8 3 7 0
    8 3 7 4 1 6 9 0 5 2
    7 0 4 5 6 8 1 2 9 3
    6 5 9 7 3 1 2 8 0 4
    5 2 8 0 9 7 3 4 1 6
    4 9 0 6 7 3 5 1 2 8
    3 6 1 2 8 0 7 9 4 5
    2 7 3 8 0 9 4 5 6 1
    1 8 5 9 2 4 0 6 3 7
    0 1 2 3 4 5 6 7 8 9

    Square B

    0 1 2 3 4 5 6 7 8 9
    4 8 7 5 0 9 3 6 2 1
    1 0 4 6 5 8 7 2 9 3
    7 3 6 2 1 4 9 0 5 8
    5 7 3 1 8 0 2 9 6 4
    3 2 8 0 9 6 1 5 4 7
    9 6 1 8 2 3 5 4 7 0
    6 4 5 9 7 1 0 8 3 2
    8 5 9 7 3 2 4 1 0 6
    2 9 0 4 6 7 8 3 1 5

    Square C

    0 1 2 3 4 5 6 7 8 9
    7 3 6 2 1 4 9 0 5 8
    5 7 3 1 8 0 2 9 6 4
    3 2 8 0 9 6 1 5 4 7
    9 6 1 8 2 3 5 4 7 0
    6 4 5 9 7 1 0 8 3 2
    8 5 9 7 3 2 4 1 0 6
    2 9 0 4 6 7 8 3 1 5
    4 8 7 5 0 9 3 6 2 1
    1 0 4 6 5 8 7 2 9 3

    Square D

    0 1 2 3 4 5 6 7 8 9
    8 2 5 9 6 7 4 3 0 1
    2 8 0 5 3 9 7 4 1 6
    4 9 7 6 0 1 8 2 5 3
    1 0 3 7 5 4 9 6 2 8
    7 3 4 1 9 8 2 0 6 5
    5 6 8 2 1 0 3 9 4 7
    3 7 1 0 2 6 5 8 9 4
    9 4 6 8 7 2 1 5 3 0
    6 5 9 4 8 3 0 1 7 2

    Orthogonal pairs: A - B, A - C, A - D, partially orthogonal pairs: B - C, B - D, C - D.

    B – C

    00 11 22 33 44 55 66 77 88 99
    47 83 76 52 01 94 39 60 25 18
    15 07 43 61 58 80 72 29 96 34
    73 32 68 20 19 46 91 05 54 87
    59 76 31 18 82 03 25 94 67 40
    36 24 85 09 97 61 10 58 43 72
    98 65 19 87 23 32 54 41 70 06
    62 49 50 94 76 17 08 83 31 25
    84 58 97 75 30 29 43 16 02 61
    21 90 04 46 65 78 87 32 19 53

    00 01 02 03 04 05 06 07 08 09 10 11 15 16 17 18 19 20 21 22 23 24 25 29 30 31 32 33 34 36 39 40 41 43 44 46 47 49 50 52 53 54 55 58 59 60 61 62 65 66 67 68 70 72 73 75 76 77 78 80 82 83 84 85 87 88 90 91 94 96 97 98 99

    The number of unique ordered pairs: U = 73

    B – D

    00 11 22 33 44 55 66 77 88 99
    48 82 75 59 06 97 34 63 20 11
    12 08 40 65 53 89 77 24 91 36
    74 39 67 26 10 41 98 02 55 83
    51 70 33 17 85 04 29 96 62 48
    37 23 84 01 99 68 12 50 46 75
    95 66 18 82 21 30 53 49 74 07
    63 47 51 90 72 16 05 88 39 24
    89 54 96 78 37 22 41 15 03 60
    26 95 09 44 68 73 80 31 17 52

    00 01 02 03 04 05 06 07 08 09 10 11 12 15 16 17 18 20 21 22 23 24 26 29 30 31 33 34 36 37 39 40 41 44 46 47 48 49 50 51 52 53 54 55 59 60 62 63 65 66 67 68 70 72 73 74 75 77 78 80 82 83 84 85 88 89 90 91 95 96 97 98 99

    The number of unique ordered pairs: U = 73

    C – D

    00 11 22 33 44 55 66 77 88 99
    78 32 65 29 16 47 94 03 50 81
    52 78 30 15 83 09 27 94 61 46
    34 29 87 06 90 61 18 52 45 73
    91 60 13 87 25 34 59 46 72 08
    67 43 54 91 79 18 02 80 36 25
    85 56 98 72 31 20 43 19 04 67
    23 97 01 40 62 76 85 38 19 54
    49 84 76 58 07 92 31 65 23 10
    16 05 49 64 58 83 70 21 97 32

    00 01 02 03 04 05 06 07 08 09 10 11 13 15 16 18 19 20 21 22 23 25 27 29 30 31 32 33 34 36 38 40 43 44 45 46 47 49 50 52 54 55 56 58 59 60 61 62 64 65 66 67 70 72 73 76 77 78 79 80 81 83 84 85 87 88 90 91 92 94 97 98 99

    The number of unique ordered pairs: U = 73

    The number of unique ordered pairs simultaneously for B - C, B - D, C – D: R=55

    00 47 15 73 59 36 98 62 84 21 11 83 07 65 49 90 22 31 85 50 97 04 33 52 20 18 09 46 44 01 23 30 55 80 03 29 78 66 72 91 10 54 08 77 60 05 16 88 67 70 02 99 34 40 06

    We call R value - characteristic orthogonality of system with partial orthogonality.
    I was unable to find a system of four LS of order 10 with partial orthogonality, for which R> 55.



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    Message boards : Science : Systems of four LS of order 10 with partial orthogonality


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