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Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme

  • *Corresponding author: Serap Șahinkaya

    *Corresponding author: Serap Șahinkaya 

The third author is supported by TUBITAK grant no: 1059B192000947

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  • We present a generator matrix of the form $ [ \sigma(v_1) \ | \ \sigma(v_2)] $, where $ v_1 \in RG $ and $ v_2\in RH $, for finite groups $ G $ and $ H $ of order $ n $ for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring $ R $. In general, many of the constructions to produce self-dual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve $ 10 $ of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain $ 82 $ new binary linear complementary dual codes, $ 50 $ of which are either optimal or near optimal of lengths $ 41 \leq n \leq 61 $ which are new to the literature.

    Mathematics Subject Classification: Primary: 94B05; Secondary: 94B15.

    Citation:

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  • Table 1.  New Binary Type Ⅰ Codes of Length 72

    Generator Matrix Type $ \gamma $ $ \beta $ $ |Aut(C_i)| $
    $ C_{1} $ $ \mathcal{G}_{1} $ $ W_{72, 1} $ $ 2 $ $ 226 $ $ 1 $
    $ C_{2} $ $ \mathcal{G}_{1} $ $ W_{72, 1} $ $ 4 $ $ 230 $ $ 1 $
    $ C_{3} $ $ \mathcal{G}_{2} $ $ W_{72, 1} $ $ 14 $ $ 248 $ $ 1 $
    $ C_{4} $ $ \mathcal{G}_{2} $ $ W_{72, 1} $ $ 1 $ $ 184 $ $ 1 $
    $ C_{5} $ $ \mathcal{G}_{3} $ $ W_{72, 1} $ $ 10 $ $ 318 $ $ 4 $
    $ C_{6} $ $ \mathcal{G}_{3} $ $ W_{72, 1} $ $ 4 $ $ 217 $ $ 4 $
    $ C_{7} $ $ \mathcal{G}_{4} $ $ W_{72, 1} $ $ 12 $ $ 315 $ $ 6 $
    $ C_{8} $ $ \mathcal{G}_{4} $ $ W_{72, 1} $ $ 6 $ $ 295 $ $ 12 $
    $ C_{9} $ $ \mathcal{G}_{5} $ $ W_{72, 1} $ $ 2 $ $ 229 $ $ 2 $
    $ C_{10} $ $ \mathcal{G}_{5} $ $ W_{72, 1} $ $ 14 $ $ 267 $ $ 2 $
    $ C_{11} $ $ \mathcal{G}_{6} $ $ W_{72, 1} $ $ 2 $ $ 225 $ $ 2 $
    $ C_{12} $ $ \mathcal{G}_{6} $ $ W_{72, 1} $ $ 4 $ $ 281 $ $ 4 $
    $ C_{13} $ $ \mathcal{G}_{6} $ $ W_{72, 1} $ $ 4 $ $ 236 $ $ 4 $
    $ C_{14} $ $ \mathcal{G}_{7} $ $ W_{72, 1} $ $ 18 $ $ 297 $ $ 18 $
     | Show Table
    DownLoad: CSV

    Table 2.  LCD Codes, where $ 41 \leq n \leq 61 $ and $ 15 \leq k \leq 30 $

    Generator Matrix $ [n, k, d] $ Optimal/Near Optimal Generator Matrix $ [n, k, d] $ Optimal/Near Optimal
    $ \mathcal{G}_8 $ $ [30, 15, 7] $ Near Optimal $ \mathcal{G}_8 $ $ [32, 16, 8] $ Optimal
    Extended $ \mathcal{G}_8 $ $ [33, 16, 8] $ Optimal Punctered $ \mathcal{G}_8 $ $ [31, 16, 7] $ Near Optimal
    Shortened $ \mathcal{G}_8 $ $ [31, 15, 8] $ Optimal $ \mathcal{G}_8 $ $ [34, 17, 8] $ Optimal
    Punctered $ \mathcal{G}_8 $ $ [33, 17, 7] $ Near Optimal Shortened $ \mathcal{G}_8 $ $ [33, 16, 8] $ Optimal
    $ \mathcal{G}_8 $ $ [36, 18, 8] $ Optimal Shortened $ \mathcal{G}_8 $ $ [35, 17, 8] $ Optimal
    Extended $ \mathcal{G}_8 $ $ [41, 16, 10] $ $ - $ $ \mathcal{G}_8 $ $ [42, 18, 10] $ -
    Extended $ \mathcal{G}_8 $ $ [43, 18, 10] $ $ - $ Punctered $ \mathcal{G}_8 $ $ [41, 18, 9] $ -
    $ \mathcal{G}_8 $ $ [42, 20, 10] $ Optimal Extended $ \mathcal{G}_8 $ $ [43, 20, 10] $ Near Optimal
    Punctered $ \mathcal{G}_8 $ $ [41, 20, 9] $ Near Optimal $ \mathcal{G}_8 $ $ [42, 19, 9] $ -
    Punctered $ \mathcal{G}_8 $ $ [41, 19, 8] $ - $ \mathcal{G}_8 $ $ [42, 21, 9] $ Near Optimal
    Punctered $ \mathcal{G}_8 $ $ [41, 21, 8] $ Near Optimal $ \mathcal{G}_8 $ $ [44, 20, 10] $ -
    Extended $ \mathcal{G}_8 $ $ [45, 20, 10] $ - Punctered $ \mathcal{G}_8 $ $ [43, 20, 9] $ -
    $ \mathcal{G}_8 $ $ [44, 22, 9] $ Near Optimal Extended $ \mathcal{G}_8 $ $ [45, 22, 10] $ Near Optimal
    Punctered $ \mathcal{G}_8 $ $ [43, 22, 8] $ Near Optimal Shortened $ \mathcal{G}_8 $ $ [43, 21, 9] $ Near Optimal
    $ \mathcal{G}_9 $ $ [46, 22, 10] $ - Extended $ \mathcal{G}_8 $ $ [47, 22, 10] $ -
    Punctered $ \mathcal{G}_8 $ $ [45, 22, 9] $ - $ \mathcal{G}_8 $ $ [46, 23, 9] $ -
    Punctered $ \mathcal{G}_8 $ $ [45, 23, 8] $ - $ \mathcal{G}_8 $ $ [48, 24, 9] $ -
    Extended $ \mathcal{G}_8 $ $ [49, 24, 10] $ - Punctered $ \mathcal{G}_8 $ $ [47, 24, 8] $ -
    Shortened $ \mathcal{G}_8 $ $ [47, 23, 9] $ - $ \mathcal{G}_8 $ $ [50, 21, 12] $ Optimal
    Punctered $ \mathcal{G}_8 $ $ [49, 21, 11] $ Optimal Shortened $ \mathcal{G}_8 $ $ [49, 20, 12] $ Optimal
    $ \mathcal{G}_8 $ $ [50, 24, 10] $ - Extended $ \mathcal{G}_8 $ $ [51, 24, 10] $ -
    Punctered $ \mathcal{G}_8 $ $ [49, 24, 9] $ - $ \mathcal{G}_8 $ $ [50, 25, 10] $ Optimal
    Punctered $ \mathcal{G}_8 $ $ [49, 25, 9] $ Near Optimal Shortened $ \mathcal{G}_8 $ $ [49, 24, 10] $ -
    $ \mathcal{G}_8 $ $ [52, 24, 12] $ Optimal Extended $ \mathcal{G}_8 $ $ [53, 24, 12] $ Optimal
    Punctered $ \mathcal{G}_8 $ $ [51, 24, 11] $ Optimal $ \mathcal{G}_8 $ $ [52, 26, 10] $ Optimal
    Extended $ \mathcal{G}_8 $ $ [53, 26, 10] $ Near Optimal Punctered $ \mathcal{G}_9 $ $ [51, 26, 9] $ Near Optimal
    Shortened $ \mathcal{G}_8 $ $ [51, 25, 10] $ Near Optimal $ \mathcal{G}_8 $ $ [54, 26, 12] $ Optimal
    Extended $ \mathcal{G}_8 $ $ [55, 26, 12] $ Optimal Punctered $ \mathcal{G}_8 $ $ [53, 26, 11] $ Optimal
    $ \mathcal{G}_8 $ $ [54, 24, 12] $ Optimal Extended $ \mathcal{G}_8 $ $ [55, 24, 12] $ Optimal
    Punctered $ \mathcal{G}_8 $ $ [53, 24, 11] $ Near Optimal $ \mathcal{G}_8 $ $ [54, 25, 11] $ Near Optimal
    Punctered $ \mathcal{G}_8 $ $ [53, 25, 10] $ - Shortened $ \mathcal{G}_8 $ $ [53, 24, 11] $ Near Optimal
    $ \mathcal{G}_8 $ $ [54, 27, 10] $ Near Optimal Punctered $ \mathcal{G}_8 $ $ [53, 27, 9] $ Near Optimal
    Shortened $ \mathcal{G}_8 $ $ [53, 26, 10] $ Near Optimal $ \mathcal{G}_8 $ $ [56, 24, 12] $ Optimal
    Extended $ \mathcal{G}_8 $ $ [57, 24, 12] $ Near Optimal Punctered $ \mathcal{G}_8 $ $ [55, 24, 11] $ Near Optimal
    $ \mathcal{G}_8 $ $ [56, 28, 11] $ Near Optimal Extended $ \mathcal{G}_8 $ $ [57, 28, 12] $ Optimal
    Punctered $ \mathcal{G}_8 $ $ [55, 28, 10] $ Near Optimal Shortened $ \mathcal{G}_8 $ $ [55, 27, 11] $ Near Optimal
    $ \mathcal{G}_8 $ $ [58, 28, 12] $ Optimal Extended $ \mathcal{G}_8 $ $ [59, 28, 12] $ Optimal
    Punctered $ \mathcal{G}_8 $ $ [57, 28, 11] $ Optimal $ \mathcal{G}_8 $ $ [58, 29, 11] $ Near Optimal
    Punctered $ \mathcal{G}_8 $ $ [57, 29, 10] $ - Shortened $ \mathcal{G}_8 $ $ [57, 28, 11] $ Near Optimal
    $ \mathcal{G}_8 $ $ [60, 28, 12] $ Optimal Extended $ \mathcal{G}_8 $ $ [61, 28, 12] $ Near Optimal
    Punctered $ \mathcal{G}_8 $ $ [59, 28, 11] $ Near Optimal $ \mathcal{G}_8 $ $ [60, 30, 11] $ Near Optimal
    Extended $ \mathcal{G}_8 $ $ [61, 30, 12] $ Optimal Punctured $ \mathcal{G}_8 $ $ [59, 30, 10] $ -
    Shortened $ \mathcal{G}_8 $ $ [59, 29, 11] $ Near Optimal $ \mathcal{G}_8 $ $ [60, 26, 11] $ -
    Extended $ \mathcal{G}_8 $ $ [61, 26, 12] $ - Punctered $ \mathcal{G}_8 $ $ [59, 26, 10] $ -
    Shortened $ \mathcal{G}_8 $ $ [59, 25, 11] $ - $ \mathcal{G}_9 $ $ [60, 24, 10] $ -
    Extended $ \mathcal{G}_9 $ $ [61, 24, 10] $ - Punctered $ \mathcal{G}_9 $ $ [59, 24, 10] $ -
     | Show Table
    DownLoad: CSV
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