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On the number of distinct k-decks: Enumeration and bounds

Parts of this paper were presented at the 19th International Symposium on Communications and Information Technologies (ISCIT), at Ho Chi Minh City, Vietnam [4].

The research of Alexander Vardy and Hanwen Yao was supported in part by the US National Science Foundation under grant CCF-1764104

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  • The $ k $-deck of a sequence is defined as the multiset of all its subsequences of length $ k $. Let $ D_k(n) $ denote the number of distinct $ k $-decks for binary sequences of length $ n $. For binary alphabet, we determine the exact value of $ D_k(n) $ for small values of $ k $ and $ n $, and provide asymptotic estimates of $ D_k(n) $ when $ k $ is fixed.

    Specifically, for fixed $ k $, we introduce a trellis-based method to compute $ D_k(n) $ in time polynomial in $ n $. We then compute $ D_k(n) $ for $ k \in \{3,4,5,6\} $ and $ k \leqslant n \leqslant 30 $. We also improve the asymptotic upper bound on $ D_k(n) $, and provide a lower bound thereupon. In particular, for binary alphabet, we show that $ D_k(n) = O\bigl(n^{(k-1)2^{k-1}+1}\bigr) $ and $ D_k(n) = \Omega(n^k) $. For $ k = 3 $, we moreover show that $ D_3(n) = \Omega(n^6) $ while the upper bound on $ D_3(n) $ is $ O(n^9) $.

    Mathematics Subject Classification: Primary: 68R05, 68R15.

    Citation:

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  • Figure 1.  Trellis section for $ k = 2 $ and levels $ i\in\{4,5\} $. Left vertices belong to $ \mathbb{D}_2(4) $, while right vertices belong to $ \mathbb{D}_2(5) $. Blue solid edges correspond to label '0', while red dashed edges correspond to label '1'

    Table 1.  Values of $ D_k(n) $ for $ 3 \leqslant k \leqslant 6 $ and $ k \leqslant n \leqslant 30 $. Values highlighted in bold correspond to $ D_k(S(k)) $

    $ n\backslash k $ 3 4 5 6
    3 8
    4 16 16
    5 32 32 32
    6 64 64 64 64
    7 126 128 128 128
    8 247 256 256 256
    9 480 512 512 512
    10 926 1024 1024 1024
    11 1764 2048 2048 2048
    12 3337 4092 4096 4096
    13 6208 8176 8192 8192
    14 11408 16328 16384 16384
    15 20608 32604 32768 32768
    16 36649 65075 65534 65536
    17 63976 129824 131064 131072
    18 109866 258906 262120 262144
    19 185012 516168 524212 524288
    20 306285 1028448 1048360 1048576
    21 497190 2048272 2096586 2097152
    22 792920 4077316 4192896 4194304
    23 1241936 8111400 8385216 8388608
    24 1913566 16124458 16769254 16777216
    25 2898574 32034016 33536094 33554432
    26 4323980 63579386 67067294 67108864
    27 6353060 126076522 134124596 134217728
    28 9206137 249736704 268228914 268435456
    29 13158574 494124382 536416730 536870912
    30 18576644 976302888 1072750464 1073741820
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