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New Classes of Asymptotically Optimal Spectrally-Constrained Sequences Derived from Cyclotomy

  • *Corresponding author: Zhengchun Zhou

    *Corresponding author: Zhengchun Zhou 
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  • As numerous applications in wireless communications and radar sensing all rely on the finite and precious spectral resource, contiguous spectrum allocation schemes have become very difficult to continue nowadays. Spectrally constrained sequences (SCSs) are specially designed sequences which display low correlation sidelobes to effectively utilize the increasingly congested and fragmented spectrum. Recently, Liu et. al (IEEE Trans. Inf. Theory 64(4):2571-2582, 2018) proposed a lower bound of the maximal correlation for SCSs and constructed two classes of optimal SCSs by using Singer difference sets and perfect ternary sequences. For the application of SCSs, it is desirable that the spectrally constrained position should be as flexible as possible, and the correlation tolerance should be as small as possible. In this paper, we present some systematic constructions based on cyclotomy which generate some asymptotically optimal SCSs with respect to Liu's bound. The proposed constructions result in SCSs with new parameters and are more flexible in terms of the "position of the null constraint". In addition, we propose a framework which control the power of new SCSs while maintaining the correlation magnitudes by using cyclic algorithm-new (CAN) algorithm.

    Mathematics Subject Classification: Primary: 94A05; Secondary: 60G35.

    Citation:

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  • Figure 1.  Asymptotically optimal (13, 3) SCS in Example 6

    Table 1.  Some Known Asymptotically optimal SCSs

    $ (N,E) $ $ \overline{\Omega} $ $ \delta_{\max}/N $ Restriction Reference
    $ \frac{r^{k}-1}{r-1}, \frac{r^{k-1}-1}{r-1} $ $ \mathcal{D} $ $ 1 $ (optimal) (i) [21]
    $ p, \frac{p-1}{2} $ $ C_{i}^{(2,p)} $ $ \frac{1}{\sqrt{p}-1} $ (ii) Const.1-A
    $ p, \frac{p+1}{2} $ $ C_{i}^{(2,p)}\cup\{0\} $ $ \frac{\sqrt{p}+1}{p+1} $ (ii) Const.1-B
    $ p, \frac{p-1}{2} $ $ C_{i}^{(4,p)}\cup C_{i+1}^{(4,p)} $ $ \frac{\sqrt{ p+1+2y \sqrt{p}}}{p-1} $ (iii) Const.2-B
    $ p,(p-1) / 4 $ $ C_{i}^{(4,p)} $ $ \frac{\sqrt{3 p+1+2|1+x| \sqrt{p}}}{p-1} $ (iii) Const.2-A
    $ p,(p+3) / 4 $ $ C_{i}^{(4,p)} \cup\{0\} $ $ \frac{\sqrt{3 p+9+2|3-x| \sqrt{p}}}{p+3} $ (iii) Const.2-C
    $ p q, \frac{(p-1)(q-1)}{4} $ $ H_{i}^{(4,pq)} $ $ \max \left\{\frac{1}{p-1}, \frac{1}{q-1}, \frac{\sqrt{3 p q+1+2 \sqrt{p q} |a-1|}}{\left(p-1\right)\left(q-1\right)}\right\} $ (iv) Const.3-A
    $ p q, \frac{(p-1)(q-1)}{2} $ $ H_{i}^{(4,pq)}\cup H_{i+1}^{(4,pq)} $ $ \max \left\{\frac{1}{p-1}, \frac{1}{q-1}, \frac{\sqrt{p q+1+2|b| \sqrt{p q} }}{\left(p-1\right)\left(q-1\right)}\right\} $ (iv) Const.3-B
    $ p q, \frac{(p-1)(q-1)}{2} $ $ H_{i}^{(2,pq)} $ $ \max \left\{\frac{1}{p-1}, \frac{1}{q-1}, \frac{\sqrt{pq}+1}{(p-1)(q-1)}\right\} $ (iv) Const.3-B, Const.4
    $ p q, \frac{3p+1+pq-q}{4} $ $ H_{i}^{(4,pq)}\cup Q \cup\{0\} $ $ \max \big\{\frac{3p+1}{3p+1+pq-q}, \frac{q-1}{3p+1+pq-q},\\ \frac{\sqrt{3 p q+1+2 \sqrt{p q} |a-1|}}{3p+1+pq-q}\big\} $ (iv) Const.3-C
    $ 2p, p-1 $ $ \left[\{0\} \times\left(C_{i}^{(4, p)} \cup C_{j}^{(4, p)}\right)\right] \\ \cup\left[\{1\} \times\left(C_{\ell}^{(4, p)} \cup C_{j}^{(4, p)}\right)\right] $ $ \frac{\sqrt{4+2p+2|x|\sqrt{p}}}{p-1}, \text{ if } i-l\equiv 2 \pmod 4 \\ \max\left\{\frac{p\sqrt{4+2p+|4- 2y|\sqrt{p}}}{p-1},\frac{p\sqrt{2p+2|y|\sqrt{p}}}{p-1}\right\} , \\ \text{ if } i-l\equiv \pm 1 \pmod 4 $ (v) Const.5
    $ 4 \ell, 2 \ell+1 $ $ \left[\{0\} \times D\right] \cup\left[\{1,2,3\} \times \overline{D} \right] $ $ \frac{\sqrt{\ell+1}}{2 \ell+1} $ (vi) Const.6
    (i) $r$ is prime and $\mathcal{D}$ is a $\left(\frac{r^{k}-1}{r-1}, \frac{r^{k-1}-1}{r-1}, \frac{r^{k-2}-1}{r-1}\right)$ Singer difference set.
    (ii) $p$ is prime with $p \equiv1\pmod 4$.
    (iii) $p$ is prime with $p \equiv5\pmod 8$ and $p=x^{2}+4 y^{2}$ with $x \equiv -1\pmod 4$.
    (iv) $p$ and $q$ are prime with $p \equiv 1\pmod 8$ and $q \equiv5\pmod 8$ with $|p-q|$ small.
    (v) $p$ is prime with $p \equiv 5\pmod 8$ and $p=x^{2}+4 y^{2}$ with $x \equiv \pm1\pmod 4$.
    (vi) There exists a cyclic $(\ell,(\ell-1)/2,(\ell-3)/4)$ difference set $D$.
     | Show Table
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    Table 2.  Parameters of SCS From Construction 1-A

    $ N $ $ E $ $ \delta_{\max } $ $ \delta_{\text{opt }} $ $ \rho $
    13 6 4.9893 4.0535 1.2309
    41 20 7.5882 6.6428 1.1664
    73 36 9.6766 8.7218 1.1347
    109 54 11.5462 10.5852 1.1149
    149 74 13.2958 12.3302 1.1010
    193 96 14.9700 14.0009 1.0906
     | Show Table
    DownLoad: CSV

    Table 3.  Parameters of SCS From Construction 2-A

    $ N $ $ E $ $ \delta_{\max } $ $ \delta_{\text{opt }} $ $ \rho $
    13 3 8.9887 6.8516 1.3119
    109 27 20.4738 18.2785 1.1437
    1453 363 68.3474 66.0758 1.0432
    3373 843 102.9121 100.628 1.0292
    3853 963 109.831 107.5454 1.0257
    4909 1227 123.6719 121.3837 1.0233
     | Show Table
    DownLoad: CSV

    Table 4.  Parameters of SCS From Construction 3-A

    $ p $ $ q $ $ N $ $ E $ $ \delta_{\max } $ $ \delta_{\text{opt }} $ $ \rho $
    5 17 85 16 26.6789 19.2594 1.3852
    13 17 221 48 32.2071 28.2868 1.1386
    17 29 493 112 44.8036 40.9938 1.0929
    29 73 2117 504 83.6923 82.3313 1.0165
    41 61 2501 600 90.2711 89.0347 1.0139
     | Show Table
    DownLoad: CSV

    Table 5.  Parameters of SCS From Construction 5

    $ p $ $ N $ $ E $ $ \delta_{\max } $ $ \delta_{\text{opt }} $ $ \rho $
    5 10 4 5.3724 4.0825 1.3160
    37 74 36 9.7593 8.8984 1.0968
    101 202 100 15.187 14.3898 1.0554
    197 394 196 20.7468 19.9758 1.0386
    677 1354 676 37.606 36.8647 1.0201
     | Show Table
    DownLoad: CSV

    Table 6.  Parameters of SCS From Construction 6

    $ p $ $ N $ $ E $ $ \delta_{\max } $ $ \delta_{\text{opt }} $ $ \rho $
    7 28 15 5.2797 5.0165 1.0525
    19 76 39 8.7149 8.5477 1.0196
    31 124 63 11.1341 11.0018 1.0120
    47 188 95 13.7105 13.6024 1.0079
    59 236 119 15.3617 15.2650 1.0063
    71 284 143 16.8519 16.7636 1.0053
     | Show Table
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