\`x^2+y_1+z_12^34\`
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On finite length nonbinary sequences with large nonlinear complexity over the residue ring $ \mathbb{Z}_{m} $

  • * Corresponding author: Xiangyong Zeng

    * Corresponding author: Xiangyong Zeng 
L. Yi and Z. Sun were supported by the Natural Science Foundation of Hubei province of China (2019CFB543). X. Zeng was supported by Major Technological Innovation Special Project of Hubei Province (No. 2019ACA144)
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  • In this paper, we characterize all nonbinary sequences of length $ n $ with nonlinear complexity $ n-4 $ for $ n\geq9 $ and establish a formula on the number of such sequences. More generally, we characterize other finite length nonbinary sequences with large nonlinear complexity over $ \mathbb{Z}_{m} $.

    Mathematics Subject Classification: Primary: 94A55; Secondary: 94A60.

    Citation:

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  • Table 1.  Binary sequences of length 13 with nonlinear complexity 9 and their distribution

    set Sequences $ \# $ Seq.
    $ E_3(A_{10}) $ 0000000001000, 0000000001100, 0000000001010, 0000000001110, 0000000001001
    0000000001101, 0000000001011, 0000000001111, 1111111110000, 1111111110100 16
    1111111110010, 1111111110110, 1111111110001, 1111111110101, 1111111110011
    1111111110111
    $ E_2(A_{11}) $ 0101010101100, 0101010101110, 0101010101101, 0101010101111, 0111111111000
    0111111111010, 0111111111001, 0111111111011, 1010101010000, 1010101010010 16
    1010101010001, 1010101010011, 1000000000100, 1000000000110, 1000000000101
    1000000000111
    $ E_1(A_{12}) $ 0010010010000, 0010010010001, 0010101010110, 0010101010111, 0011111111100
    0011111111101, 1101101101110, 1101101101111, 1101010101000, 1101010101001 24
    1100000000010, 1100000000011, 0100100100110, 0100100100111, 0100000000010
    0100000000011, 1011011011000, 1011011011001, 1011111111100, 1011111111101
    0110110110100, 0110110110101, 1001001001010, 1001001001011
    $ B_1 $ 0011001100111, 0011011011010, 1100110011000, 1100100100101, 0110011001101 8
    0110000000001, 1001100110010, 1001111111110
    $ B_2 $ 0001000100011, 0001001001000, 0001010101011, 0001111111110, 1110111011100
    1110110110111, 1110101010100, 1110000000001, 0010001000101, 0010000000001
    1101110111010, 1101111111110, 0100010001001, 1011101110110, 0101101101100 22
    0101111111110, 1010010010011, 1010000000001, 0110101010100, 1001010101011
    0111011101111, 1000100010000
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