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The $[46, 9, 20]_2$ code is unique

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  • The minimum distance of all binary linear codes with dimension at most eight is known. The smallest open case for dimension nine is length $ n = 46 $ with known bounds $ 19\le d\le 20 $. Here we present a $ [46,9,20]_2 $ code and show its uniqueness. Interestingly enough, this unique optimal code is asymmetric, i.e., it has a trivial automorphism group. Additionally, we show the non-existence of $ [47,10,20]_2 $ and $ [85,9,40]_2 $ codes.

    Mathematics Subject Classification: Primary: 94B65; Secondary: 94B05.

    Citation:

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  • Table 1.  Number of $ [n,k,\{20,24,28,32\}]_2 $ codes

    k/n 20 24 28 30 32 34 35 36 37 38 39 40 41 42 43 44
    1 1 1 1 0 1 0 0 0 0 0
    2 1 1 2 0 3 0 3 0
    3 1 1 2 4 6 9
    4 1 4 13 26
    5 3 15 163
    6 24 3649
    7 5 337794
     | Show Table
    DownLoad: CSV

    Table 2.  Number of $ [n,k,\{40,48,56\}]_2 $ codes

    k/n 40 48 56 60 64 68 70 72 74 75 76 77 78 79 80 81 82 83
    1 1 1 1 0 0 0 0 0 0 0 0 0
    2 1 1 2 0 2 0 0 2 0 0
    3 1 1 2 0 3 0 5 0
    4 1 1 2 3 6 10
    5 1 3 11 16
    6 2 8 106
    7 7 5613
     | Show Table
    DownLoad: CSV

    Table 3.  Number of $ [84,8,\{40,48,56\}]_2 $ codes per $ A_{56} $

    $ A_{56} $ 3 4 5 6 7 8 9
    25773 48792 26091 5198 450 17 1
     | Show Table
    DownLoad: CSV
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