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Binary subspace codes in small ambient spaces

The authors were supported by the DFG project "Ganzzahlige Optimierungsmodelle für Subspace Codes und endliche Geometrie" (DFG grants KU 2430/3-1, WA 1666/9-1)

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  • Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for projective dimensions of at most $ 7 $, i.e., affine dimensions of at most $ 8 $. We obtain several improvements of the bounds and perform two classifications of optimal subspace codes, which are unknown so far in the literature.

    Mathematics Subject Classification: Primary: 94B05, 05B25, 51E20; Secondary: 51E14, 51E22, 51E23.

    Citation:

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  • Table 1.  $\text{A}_2(v, d)\label{tbl:smax2}$ and isomorphism types of optimal codes for $v\leq 8$

    $ v\backslash d $12345678
    12(1)
    25(1)3(1)
    316(1)8(2)2(2)
    467(1)37(1)5(4)5(1)
    5374(1)187(2)18(48217)9(14)2(3)
    62825(1)1521(1)108-11777(5)9(5)9(1)
    729212(1)14606(2)614-776334-40734(39)17(1856)2(4)
    8417199(1)222379(2)5687-92684803-6479263-326257(8)17(572)17(8)
     | Show Table
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    Table 2.  Details for the ILP computations

    IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$Orbits of phase 2 $\max z_8^{\operatorname{LP}}(\text{"31"})$ $\max z_8^{\operatorname{BLP}}(\text{"31"})$
    116960272271.1856 $ 16^{2}, 240^{6}, 480^{47}, 960^{242} $263.0287799257
    216384266.26086957267.4646 $ 96^{6}, 192^{91}, 384^{711} $206.04279728
    3164270.83786676265.3281 $ 1^{13}, 2^{29}, 4^{2638} $257.20717665254
    41648271.43451032262.082 $ 4^{3}, 12^{11}, 24^{59}, 48^{1104} $200.5850228
    5162263.8132689259.8044 $ 1^{5}, 2^{59966} $206.39304042
    61620267.53272206259.394 $ 5, 10^{9}, 20^{1843} $199.98690666
    71764282.96047431259.1063 $ 16^{10}, 32^{145}, 64^{6293} $259.45364626257
    81732268.0388109257.2408 $\le 255$ by a separate argumentation
    IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$IndexTypeAut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$
    9161263.82742528 $\le$ 25510161263.36961743 $\le$ 255
    11161264.25957151 $\le$ 25412161263.85869815 $\le$ 254
    13162263.07052878 $\le$ 254141612261.91860556 $\le$ 254
    15164261.62648174 $\le$ 254161612261.31512837 $\le$ 254
    17174261.11518721 $\le$ 25418161260.96388752 $\le$ 254
    19161260.82432878 $\le$ 25420162260.65762276 $\le$ 254
    21164260.43036283 $\le$ 25422162260.19475349 $\le$ 254
    23161260.08583792 $\le$ 25424161260.04857193 $\le$ 254
    25161259.75041996 $\le$ 25426162259.55230081 $\le$ 254
    27162259.46335297 $\le$ 254281612259.11945025 $\le$ 254
    29161258.89395938 $\le$ 254301724258.75142045 $\le$ 254
    31168258.35689437 $\le$ 25432161257.81420526 $\le$ 254
    33162257.75126819 $\le$ 25434164257.63965018 $\le$ 254
    35161257.57663803 $\le$ 25436161257.2820438 $\le$ 254
    37164257.01931801 $\le$ 2543817128257 $\le$ 254
    391612256.83887168 $\le$ 254401612256.31380897 $\le$ 254
    41166256.22093781 $\le$ 254421612256.10154389 $\le$ 254
    43161255.87957119
     | Show Table
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