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2024, Volume 18, Issue 3: 784-813. Doi: 10.3934/amc.2022025
This issue Previous Article Construction of three classes of strictly optimal frequency-hopping sequence sets Next Article Quantum-safe identity-based broadcast encryption with provable security from multivariate cryptography

Constructions of optimal multiply constant-weight codes MCWC$ (3,n_1;1,n_2;1,n_3;8)s $

  • 1.

    School of Sciences, Nantong University, Nantong 226007, China

  • 2.

    School of Mathematics and Statistics, University of New South Wales, N.S.W. 2052, Australia

  • * Corresponding author: Jinhua Wang

    * Corresponding author: Jinhua Wang
Received:  November 2021
Revised:  March 2022
Early access:  April 28, 2022
Published online:  April 28, 2022

The third author's research is supported by the National Natural Science Foundation of China under Grant No.11671402.

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    Abstract

  • A binary code $ {\mathcal{C}} $ of length $ n = \sum_{i = 1}^{m}n_i $ and minimum distance $ d $ is said to be of multiply constant-weight and denoted by MCWC$ (w_1, n_1 $; $ w_2, n_2 $; $ \ldots $; $ w_m, n_m $; $ d) $, if each codeword has weight $ w_1 $ in the first $ n_1 $ coordinates, weight $ w_2 $ in the next $ n_2 $ coordinates, and so on and so forth. Multiply constant-weight codes (MCWCs) can be utilized to improve the reliability of certain physically unclonable function response and has been widely studied. Research showed that multiply constant-weight codes are equivalent to generalized packing designs and generalized Howell designs (GHDs) can be regarded as generalized packing designs with a special block type. In this paper, we give combinatorial constructions for optimal MCWC$ (3, n_1;1, n_2;1, n_3;8) $s by a class of generalized packing designs, which come from generalized Howell designs. Furthermore, for $ e = 3, 4, 5 $, we prove that there exists a GHD $ (n+e, 3n) $ if and only if $ n\ge 2e+1 $ leaving several possibly exceptions.

    Mathematics Subject Classification: Primary: 05B30, 05B40; Secondary: 94B25, 94C30.

    Citation:
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  • Table 1.  Values $ n\geq 7 $ for which no GHD $ (n+\frac{n-1}{2}, 3n) $ is known

    7 47 51 55 59 63 77 83 87 95 119
     | Show Table
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    Table 2.  Values $ n\geq 6 $ for which no GHD $ (n+\frac{n-2}{2}, 3n) $ is known

    48 50 52 56 58 60 62 64 68 72 74 76 78 80 82
    84 88 90 92 98 100 104 106 108 110 112 114 116 120 124
    126 128 130 132 136 138 142 144 146 152 154 156 158 160 168
    170 172 176 180 186 188 190 192 194 196 202 204 208 210 212
    218 220 222 224
     | Show Table
    DownLoad: CSV

    Table 3.  Values $ n\equiv 0\ (\bmod\ 4) $ for which no GHD $ (n+\frac{n-4}{2}, 3n) $ is known

    56 64 72 80 96 112 120 136 144 152 176 184 200 220 224
     | Show Table
    DownLoad: CSV

    Table 4.  Values $ n\geq 6 $ for which no GHD $ (n+\frac{n-6}{2}, 3n) $ is known

    42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
    74 76 78 80 84 88 90 94 96 104 106 108 118 120
     | Show Table
    DownLoad: CSV

    Table 5.  Values $ n\geq 7 $ for which no GHD $ (n+\frac{n-3}{2}, 3n) $ is known

    29 31 33 35 37 41 43 45 47 49 51 53 55 61 63
    69 77 95
     | Show Table
    DownLoad: CSV

    Table 6.  Values $ n\geq 5 $ for which no GHD $ (n+\frac{n-5}{2}, 3n) $ is known

    29 31 33 35 37 41 43 47 49 51 53 57 59 61
    63 69 73 77 79 81 83 89 91 93 99 101 107 109
    111 113 117 121 127 129 131 133 137 139 143 147 153 157
    159 161 169 171 173 177 181 187 189 191 193 197 203 209
    211 213 219 221 223 233 237 239 241 243 249 257 259 261
    267 269 271 275 277 281 283 287 289 297 299 301 303 307
    309 313 315 321 323 329 333 337 341 349 351
     | Show Table
    DownLoad: CSV

    Table 7.  Starters and adders for GHD $ (n+3, 3n) $s in Lemma 4.9

    R C
    $ 7_0 8_1 3_2 $ $ 2_0 6_1 3_2 $ A GHD $ (12, 27) $
    $ 4_0 6_1 7_2 $ $ 7_0 4_1 0_2 $ with an empty
    $ 8_0 4_1 6_2 $ $ 5_0 3_1 2_2 $ $ 3 \times 3 $ subarray
    S A S A S A S A S A
    $ 0_0 1_0 3_0 $ 0 $ 0_1 1_1 3_1 $ 8 $ 1_2 2_2 8_2 $ 5 $ 2_0 6_0 5_1 $ 2 $ 7_1 2_1 5_2 $ 3
    $ 5_0 0_2 4_2 $ 1
    R C
    $ 8_0 8_1 4_2 $ $ 9_0 0_1 3_2 $ A GHD $ (13, 30) $
    $ 6_0 2_1 3_2 $ $ 5_0 2_1 0_2 $ with an empty
    $ 7_0 6_1 8_2 $ $ 8_0 3_1 7_2 $ $ 3 \times 3 $ subarray
    S A S A S A S A S A
    $ 0_0 2_0 3_0 $ 0 $ 1_1 4_1 0_1 $ 7 $ 2_2 5_2 1_2 $ 4 $ 5_0 1_0 9_1 $ 6 $ 3_1 5_1 0_2 $ 1
    $ 7_2 9_2 9_0 $ 5 $ 4_0 7_1 6_2 $ 2
    R C
    $ 3_0 2_1 6_2 $ $ 10_0 4_1 6_2 $ A GHD $ (14, 33) $
    $ 1_0 9_1 3_2 $ $ 9_0 1_1 2_2 $ with an empty
    $ 8_0 3_1 9_2 $ $ 1_0 10_1 9_2 $ $ 3 \times 3 $ subarray
    S A S A S A S A S A
    $ 10_0 5_0 6_0 $ 9 $ 1_1 4_1 10_1 $ 4 $ 7_2 10_2 5_2 $ 5 $ 4_0 2_0 6_1 $ 3 $ 7_1 8_1 4_2 $ 10
    $ 1_2 2_2 7_0 $ 6 $ 0_0 0_1 0_2 $ 0 $ 9_0 5_1 8_2 $ 8
     | Show Table
    DownLoad: CSV

    Table 8.  Starters and adders for GHD $ (n+4, 3n) $s in Lemma 4.10

    R C
    $ 5_0 5_1 5_2 $ $ 9_0 10_1 2_2 $
    $ 4_0 8_1 9_2 $ $ 3_0 5_1 4_2 $ A GHD $ (15, 33) $
    $ 8_0 6_1 10_2 $ $ 8_0 0_1 6_2 $ with an empty
    $ 10_0 9_1 6_2 $ $ 2_0 8_1 10_2 $ $ 4 \times 4 $ subarray
    S A S A S A S A S A
    $ 2_0 7_0 9_0 $ 3 $ 1_1 2_1 4_1 $ 2 $ 1_2 2_2 4_2 $ 7 $ 0_0 7_1 3_2 $ 0 $ 3_0 6_0 0_1 $ 1
    $ 3_1 10_1 8_2 $ 10 $ 1_0 0_2 7_2 $ 5
    R C
    $ 8_0 0_1 1_2 $ $ 11_0 0_1 8_2 $
    $ 10_0 6_1 9_2 $ $ 8_0 10_1 4_2 $ A GHD $ (16, 36) $
    $ 5_0 2_1 6_2 $ $ 1_0 7_1 5_2 $ with an empty
    $ 9_0 7_1 0_2 $ $ 3_0 2_1 1_2 $ $ 4 \times 4 $ subarray
    S A S A S A S A S A
    $ 1_0 4_0 6_0 $ 1 $ 4_1 8_1 9_1 $ 7 $ 2_2 3_2 5_2 $ 9 $ 0_0 5_1 7_2 $ 0 $ 3_0 7_0 10_1 $ 3
    $ 1_1 3_1 10_2 $ 5 $ 4_2 8_2 2_0 $ 2 $ 11_0 11_1 11_2 $ 10
    R C
    $ 6_0 12_1 2_2 $ $ 6_0 6_1 11_2 $
    $ 2_0 9_1 3_2 $ $ 3_0 4_1 2_2 $ A GHD $ (17, 39) $
    $ 10_0 6_1 12_2 $ $ 12_0 3_1 5_2 $ with an empty
    $ 9_0 7_1 7_2 $ $ 10_0 5_1 1_2 $ $ 4 \times 4 $ subarray
    S A S A S A S A S A
    $ 4_0 7_0 11_0 $ 7 $ 0_1 1_1 8_1 $ 1 $ 1_2 9_2 10_2 $ 12 $ 3_1 5_1 4_2 $ 8 $ 0_0 8_0 10_1 $ 0
    $ 2_1 11_1 6_2 $ 10 $ 5_2 8_2 5_0 $ 2 $ 0_2 11_2 3_0 $ 6 $ 1_0 12_0 4_1 $ 3
    R C
    $ 2_0 3_1 10_2 $ $ 13_0 13_1 0_2 $
    $ 1_0 4_1 7_2 $ $ 12_0 5_1 3_2 $ An GHD $ (18, 42) $
    $ 12_0 2_1 8_2 $ $ 7_0 4_1 9_2 $ with an empty
    $ 3_0 1_1 12_2 $ $ 3_0 2_1 1_2 $ $ 4 \times 4 $ subarray
    S A S A S A S A S A
    $ 4_0 5_0 9_0 $ 6 $ 5_1 0_1 8_1 $ 12 $ 2_2 3_2 11_2 $ 10 $ 11_0 13_0 7_1 $ 7 $ 7_0 10_0 12_1 $ 9
    $ 11_1 13_1 13_2 $ 11 $ 1_2 5_2 8_0 $ 1 $ 6_1 10_1 0_2 $ 5 $ 6_2 9_2 6_0 $ 2 $ 0_0 9_1 4_2 $ 0
     | Show Table
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    Table 9.  Starter and adder for the GHD $ (22, 51) $s in Lemma 4.11

    R C
    $ 0_0 0_1 16_2 $ $ 5_0 7_1 3_2 $
    $ 13_0 14_1 3_2 $ $ 13_0 0_1 1_2 $ A GHD $ (22, 51) $
    $ 2_0 10_1 13_2 $ $ 16_0 5_1 0_2 $ with an empty
    $ 11_0 5_1 2_2 $ $ 10_0 2_1 6_2 $ $ 5 \times 5 $ subarray
    $ 7_0 3_1 10_2 $ $ 7_0 4_1 13_2 $
    S A S A S A S A S A
    $ 3_0 8_0 13_1 $ 0 $ 11_1 16_1 4_2 $ 12 $ 9_2 15_2 5_0 $ 6 $ 9_0 16_0 1_2 $ 10 $ 9_1 1_1 6_0 $ 9
    $ 14_2 6_2 12_1 $ 13 $ 4_0 10_0 12_0 $ 2 $ 4_1 6_1 7_1 $ 8 $ 8_2 11_2 12_2 $ 14 $ 1_0 14_0 15_0 $ 3
    $ 2_1 8_1 15_1 $ 1 $ 0_2 5_2 7_2 $ 7
     | Show Table
    DownLoad: CSV

    Table 10.  GHD $ (n+3, 3n) $s from 3 HMOLS of type $ h^m s^1 $ (with $ n = hm+s $) in Lemma 4.19

    $ n = hm+s $ $ h $ $ m $ $ s $ $ n = hm+s $ $ h $ $ m $ $ s $
    45, 50 9, 10 5 0 53-57 9 5 8-12
    58-62 10 5 8-12 63-67 11 5 8-12
    68-76 12 5 8-16 77-81 9 7 14-18
    82-93 9 8 10-21 94-106 12 7 10-22
     | Show Table
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    Table 11.  GHD $ (n+4, 3n) $s from 3 HMOLS of type $ h^m s^1 $ (with $ n = hm+s $) in Lemma 4.22

    $ n = hm+s $ $ h $ $ m $ $ s $ $ n = hm+s $ $ h $ $ m $ $ s $
    55 11 5 0 64-67 11 5 9-12
    69-76 12 5 9-16 77-81 13 5 12-16
    82-86 14 5 12-16 87-95 11 7 10-18
    96-108 12 7 12-24 109-116 12 8 13-20
     | Show Table
    DownLoad: CSV
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