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Lengths of divisible codes with restricted column multiplicities

  • *Corresponding author: Sascha Kurz

    *Corresponding author: Sascha Kurz
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  • We determine the minimum possible column multiplicity of even, doubly-, and triply-even codes given their length. This refines a classification result for the possible lengths of $ q^r $-divisible codes over $ \mathbb{F}_q $. We also give a few computational results for field sizes $ q>2 $. Non-existence results of divisible codes with restricted column multiplicities for a given length have applications e.g. in Galois geometry and can be used for upper bounds on the maximum cardinality of subspace codes.

    Mathematics Subject Classification: Primary: 51E23; Secondary: 05B40.

    Citation:

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  • Table 1.  Combinatorial data of 4-divisible multisets of points of cardinality 17

    $ n $ $ k $ $ \Delta $ $ \gamma_1 $ $ \lambda_1 $ $ \lambda_2 $ $ \lambda_3 $ spectrum
    17 3 4 7 4 0 2 $ (a_{5},a_{9},a_{13})=(4,2,1) $
    17 3 4 5 3 0 3 $ (a_{5},a_{9},a_{13})=(3,4,0) $
    17 4 4 7 6 2 0 $ (a_{5},a_{9},a_{13})=(8,3,4) $
    17 4 4 6 6 1 1 $ (a_{5},a_{9},a_{13})=(7,5,3) $
    17 4 4 5 5 2 1 $ (a_{5},a_{9},a_{13})=(6,7,2) $
    17 4 4 4 6 2 1 $ (a_{5},a_{9},a_{13})=(5,9,1) $
    17 4 4 4 4 0 3 $ (a_{5},a_{9},a_{13})=(6,7,2) $
    17 4 4 3 4 2 3 $ (a_{5},a_{9},a_{13})=(5,9,1) $
    17 4 4 3 6 4 1 $ (a_{5},a_{9},a_{13})=(4,11,0) $
    17 5 4 7 10 0 0 $ (a_{5},a_{9},a_{13})=(16,5,10) $
    17 5 4 6 7 2 0 $ (a_{5},a_{9},a_{13})=(14,9,8) $
    17 5 4 5 9 0 1 $ (a_{5},a_{9},a_{13})=(12,13,6) $
    17 5 4 5 6 3 0 $ (a_{5},a_{9},a_{13})=(12,13,6) $
    17 5 4 4 7 3 0 $ (a_{5},a_{9},a_{13})=(10,17,4) $
    17 5 4 4 10 0 1 $ (a_{5},a_{9},a_{13})=(10,17,4) $
    17 5 4 4 6 2 1 $ (a_{5},a_{9},a_{13})=(11,15,5) $
    17 5 4 3 5 3 2 $ (a_{5},a_{9},a_{13})=(10,17,4) $
    17 5 4 3 9 1 2 $ (a_{5},a_{9},a_{13})=(9,19,3) $
    17 5 4 3 10 2 1 $ (a_{5},a_{9},a_{13})=(8,21,2) $
    17 5 4 3 6 4 1 $ (a_{5},a_{9},a_{13})=(9,19,3) $
    17 5 4 2 7 5 0 $ (a_{5},a_{9},a_{13})=(8,21,2) $
    17 5 4 2 11 3 0 $ (a_{5},a_{9},a_{13})=(7,23,1) $
    17 5 4 2 15 1 0 $ (a_{5},a_{9},a_{13})=(6,25,0) $
    17 6 4 4 10 0 1 $ (a_{5},a_{9},a_{13})=(21,31,11) $
    17 6 4 4 7 3 0 $ (a_{5},a_{9},a_{13})=(21,31,11) $
    17 6 4 3 11 0 2 $ (a_{5},a_{9},a_{13})=(18,37,8) $
    17 6 4 3 10 2 1 $ (a_{5},a_{9},a_{13})=(17,39,7) $
    17 6 4 3 6 4 1 $ (a_{5},a_{9},a_{13})=(19,35,9) $
    17 6 4 3 12 1 1 $ (a_{5},a_{9},a_{13})=(16,41,6) $
    17 6 4 2 7 5 0 $ (a_{5},a_{9},a_{13})=(17,39,7) $
    17 6 4 2 11 3 0 $ (a_{5},a_{9},a_{13})=(15,43,5) $
    17 6 4 2 13 2 0 $ (a_{5},a_{9},a_{13})=(14,45,4) $
    17 6 4 2 15 1 0 $ (a_{5},a_{9},a_{13})=(13,47,3) $
    17 6 4 1 17 0 0 $ (a_{5},a_{9},a_{13})=(12,49,2) $
    17 7 4 2 11 3 0 $ (a_{5},a_{9},a_{13})=(31,83,13) $
    17 7 4 2 7 5 0 $ (a_{5},a_{9},a_{13})=(35,75,17) $
    17 7 4 2 13 2 0 $ (a_{5},a_{9},a_{13})=(29,87,11) $
    17 7 4 2 15 1 0 $ (a_{5},a_{9},a_{13})=(27,91,9) $
    17 7 4 1 17 0 0 $ (a_{5},a_{9},a_{13})=(25,95,7) $
    17 8 4 1 17 0 0 $ (a_{5},a_{9},a_{13})=(51,187,17) $
     | Show Table
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    Table 2.  Number of even codes per dimension $ k $ and effective length $ n $

    n / k 1 2 3 4 5 6 7 8 9
    2 1
    3 1
    4 1 1 1
    5 1 1 1
    6 1 2 3 2 1
    7 2 4 4 2 1
    8 1 3 8 10 7 3 1
    9 3 9 18 16 9 3 1
    10 1 4 17 37 46 30 13 4 1
     | Show Table
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    Table 3.  Number of doubly-even codes per dimension $ k $ and effective length $ n $

    n / k 1 2 3 4 5 6 7 8 9
    4 1
    6 1
    7 1
    8 1 1 1 1
    10 1 1 1
    11 1 1
    12 1 2 3 4 2
    13 1 1 2
    14 2 4 6 5 4
    15 3 6 6 4 2
    16 1 3 8 18 21 15 7 2
    17 2 7 14 11 5 1
    18 3 9 27 44 45 21 6
    19 6 22 52 62 40 10
    20 1 4 17 64 149 212 156 65 10
     | Show Table
    DownLoad: CSV

    Table 4.  Number of triply-even codes per dimension $ k $ and effective length $ n $

    n / k 1 2 3 4 5 6 7 8 9 10 11
    8 1
    12 1
    14 1
    15 1
    16 1 1 1 1 1
    20 1 1 1
    22 1 1
    23 1 1
    24 1 2 3 4 4 1
    26 1 1 2
    27 1 1 1
    28 2 4 6 7 6 1
    29 1 1 2 1
    30 3 6 8 7 6 2
    31 4 8 8 6 4 1
    32 1 3 8 18 32 34 24 13 5 1
    34 2 7 14 11 5 1
    35 3 7 7 3 1
    36 3 9 27 54 65 36 11 1
    37 2 5 8 5 1
    38 6 22 57 79 61 21 2
    39 10 36 57 49 30 10 1
    40 1 4 17 64 194 347 323 187 59 11 1
    41 2 12 29 26 12 3
     | Show Table
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    Table 5.  Number of 16-divisible codes per dimension $ k $ and effective length $ n $

    n / k 1 2 3 4 5 6 7 8 9 10 11 12
    16 1
    24 1
    28 1
    30 1
    31 1
    32 1 1 1 1 1 1
    40 1 1 1
    44 1 1
    46 1 1
    47 1 1
    48 1 2 3 4 4 3 1
    52 1 1 2
    54 1 1 1
    55 1 1 1
    56 2 4 6 7 8 3 1
    58 1 1 2 1
    59 1 1 1 1
    60 3 6 8 9 8 4 1
    61 1 1 2 1 1
    62 4 8 10 9 8 4 2
    63 5 10 10 8 6 3 1
    64 1 3 8 18 32 48 48 35 21 11 4 1
    68 2 7 14 11 5 1
    70 3 7 7 3 1
    71 3 7 7 3 1
    72 3 9 27 54 75 56 26 6 1
    74 2 5 8 5 1
    75 2 5 5 4 1
    76 6 22 59 86 75 34 9 1
    77 2 5 8 6 4 1
    78 10 36 64 66 52 28 11 2
    79 14 47 71 63 44 23 8 1
     | Show Table
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    Table 6.  Number of 3-divisible ternary linear codes per dimension $ k $ and effective length $ n $

    n / k 1 2 3
    3 1
    4 1
    6 1 1
    7 1 1
     | Show Table
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    Table 7.  Number of 4-divisible quaternary linear codes per dimension $ k $ and effective length $ n $

    n / k 1 2 3 4 5 6 7
    4 1
    5 1
    8 1 1
    9 1 1
    10 1 1 1
    12 1 2 2
    13 2 3 1
    14 1 5 3 1
    15 1 3 6 2 1
    16 1 4 9 7 2
    17 3 12 9 2
    18 2 18 25 8 1
    19 1 14 42 25 6 1
     | Show Table
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  • [1] K. Betsumiya and A. Munemasa, On triply even binary codes, Journal of the London Mathematical Society, 86 (2012), 1-16.  doi: 10.1112/jlms/jdr054.
    [2] I. BouyuklievS. Bouyuklieva and S. Kurz, Computer classification of linear codes, IEEE Transactions on Information Theory, 67 (2021), 7807-7814.  doi: 10.1109/TIT.2021.3114280.
    [3] R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bulletin of the London Mathematical Society, 18 (1986), 97-122. 
    [4] J. De BeuleJ. DemeyerS. Mattheus and P. Sziklai, On the cylinder conjecture, Designs, Codes and Cryptography, 87 (2019), 879-893.  doi: 10.1007/s10623-018-0571-5.
    [5] S. Dodunekov and J. Simonis, Codes and projective multisets, The Electronic Journal of Combinatorics, 5 (1998), 23 pp. doi: 10.37236/1375.
    [6] C. F. DoranM. G. FauxS. JamesG. D. Landweber and R. L. Miller, Codes and supersymmetry in one dimension, Advances in Theoretical and Mathematical Physics, 15 (2011), 1909-1970. 
    [7] T. Etzion, Covering of subspaces by subspaces, Designs, Codes and Cryptography, 72 (2014), 405-421.  doi: 10.1007/s10623-012-9766-3.
    [8] T. EtzionS. KurzK. Otal and F. Özbudak, Subspace packings: Constructions and bounds, Designs, Codes and Cryptography, 88 (2020), 1781-1810.  doi: 10.1007/s10623-020-00732-z.
    [9] D. Heinlein, T. Honold, M. Kiermaier, S. Kurz and A. Wassermann, Projective divisible binary codes, in The Tenth International Workshop on Coding and Cryptography 2017 : WCC Proceedings, IEEE Information Theory Society, Saint-Petersburg, 2017, https://eref.uni-bayreuth.de/40887/.
    [10] T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, (2018), 131-170.
    [11] T. HonoldM. Kiermaier and S. Kurz, Classification of large partial plane spreads in PG(6, 2) and related combinatorial objects, Journal of Geometry, 110 (2019), 1-31.  doi: 10.1007/s00022-018-0459-6.
    [12] T. Honold, M. Kiermaier and S. Kurz, Johnson type bounds for mixed dimension subspace codes, The Electronic Journal of Combinatorics, 26 (2019), 21 pp. doi: 10.37236/8188.
    [13] T. HonoldM. KiermaierS. Kurz and A. Wassermann, The lengths of projective triply-even binary codes, IEEE Transactions on Information Theory, 66 (2020), 2713-2716.  doi: 10.1109/tit.2019.2940967.
    [14] M. Kiermaier and S. Kurz, On the lengths of divisible codes, IEEE Transactions on Information Theory, 66 (2020), 4051-4060.  doi: 10.1109/TIT.2020.2968832.
    [15] M. Kiermaier and S. Kurz, Classification of δ-divisible linear codes spanned by codewords of weight δ, IEEE Transactions on Information Theory, 69 (2023), 3544-3551. 
    [16] S. Kurz, No projective 16-divisible binary linear code of length 131 exists, IEEE Communications Letters, 25 (2020), 38-40. 
    [17] S. Kurz and S. Mattheus, A generalization of the cylinder conjecture for divisible codes, IEEE Transactions on Information Theory, 68 (2022), 2281-2289. 
    [18] I. Landjev and A. Rousseva, Divisible arcs, divisible codes, and the extension problem for arcs and codes, Problems of Information Transmission, 55 (2019), 226-240.  doi: 10.1134/s0555292319030033.
    [19] T. Miezaki and H. Nakasora, The support designs of the triply even binary codes of length 48, Journal of Combinatorial Designs, 27 (2019), 673-681.  doi: 10.1002/jcd.21670.
    [20] B. G. Rodrigues, On some projective triply-even binary codes invariant under the conway group Co1, International Journal of Group Theory, 11 (2022), 23-35. 
    [21] H. N. Ward, Divisible codes, Archiv der Mathematik, 36 (1981), 485-494.  doi: 10.1007/BF01223730.
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