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doi: 10.3934/amc.2022054
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New Classes of Asymptotically Optimal Spectrally-Constrained Sequences Derived from Cyclotomy

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

2. 

Department of Informatics, University of Bergen, Bergen, Norway

*Corresponding author: Zhengchun Zhou

Received  March 2022 Revised  June 2022 Early access July 2022

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.

Citation: Zhifan Ye, Zhengchun Zhou, Yang Yang, Tor Helleseth. New Classes of Asymptotically Optimal Spectrally-Constrained Sequences Derived from Cyclotomy. Advances in Mathematics of Communications, doi: 10.3934/amc.2022054
References:
[1]

K. T. ArasuC. DingT. HellesethP. V. Kumar and H. M. Martinsen, Almost difference sets and their sequences with optimal autocorrelation, IEEE Trans. Inform. Theory, 47 (2001), 2934-2943.  doi: 10.1109/18.959271.

[2]

A. AubryA. De MaioM. Piezzo and A. Farina, Radar waveform design in a spectrally crowded environment via nonconvex quadratic optimization, IEEE Trans. Aerosp. Electron. Syst., 50 (2014), 1138-1152.  doi: 10.1109/TAES.2014.120731.

[3]

B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, New York, NY, USA: Wiley, 1998.

[4]

C. Carlet and C. Ding, Highly nonlinear mappings, J. Complexity, 20 (2004), 205-244.  doi: 10.1016/j.jco.2003.08.008.

[5]

C. Ding, Autocorrelation values of the generalized cyclotomic sequences of order 2, IEEE Trans. Inform. Theory, 44 (1998), 1699-1702.  doi: 10.1109/18.681354.

[6]

C. Ding, Codes from Difference Sets, World Scientific, 2015.

[7]

C. Ding, Linear complexity of generalized cyclotomic binary sequences of order 2, Finite Fields Appl., 3 (1997), 159-174.  doi: 10.1006/ffta.1997.0181.

[8]

C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Cryptogr., 46 (2008), 113-126.  doi: 10.1007/s10623-007-9140-z.

[9]

C. DingT Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inform. Theory, 47 (2001), 428-433.  doi: 10.1109/18.904555.

[10]

P. Fan and M. Darnell, Sequence Design for Communications Applications, New York, NY, USA: Wiley, 1996.

[11]

C. Fan and G. Ge, A unified approach to Whiteman's and Ding-Helleseth's generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60 (2014), 1326-1336.  doi: 10.1109/TIT.2013.2290694.

[12]

R. Gerchberg and W. Saxton, A practical algorithm for the determination of the phase from image and diffraction plane pictures, Optik, 35 (1972), 237-250. 

[13] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.
[14]

B. GordonW. H. Mills and L. R. Welch, Some new difference sets, Canad. J. Math., 14 (1962), 614-625.  doi: 10.4153/CJM-1962-052-2.

[15]

S. Haykin, Cognitive radar: A way of the future, IEEE Signal Process. Mag., 23 (2006), 30-40.  doi: 10.1109/MSP.2006.1593335.

[16]

S. Haykin, Cognitive radio: Brain-empowered wireless communications, IEEE J. Sel. Areas Commun., 23 (2005), 201-220.  doi: 10.1109/JSAC.2004.839380.

[17]

H. He, P. Stoica and J. Li, Waveform design with stopband and correlation constraints for cognitive radar, In Proc. 2nd Int. Workshop Cognit. Inf. Process., Italy, (2010), 344–349. doi: 10.1109/CIP.2010.5604089.

[18]

S. HuZ. LiuY. L. GuanW. XiongG. Bi and S. Li, Sequence design for cognitive CDMA communications under arbitrary spectrum hole constraint, IEEE J. Sel. Areas Commun., 32 (2014), 1974-1986.  doi: 10.1109/JSAC.2014.141103.

[19]

L. Hu and Q. Yue, Gauss periods and codebooks from generalized cyclotomic sets of order four, Des. Codes Cryptogr., 69 (2013), 233-246.  doi: 10.1007/s10623-012-9648-8.

[20]

E. Lehmer, On residue difference sets, Canad. J. Math, 5 (1953), 425-432.  doi: 10.4153/cjm-1953-047-3.

[21]

Z. LiuY. L. GuanU. Parampalli and S. Hu, Spectrally-constrained sequences: Bounds and constructions, IEEE Trans. Inf. Theory, 64 (2018), 2571-2582.  doi: 10.1109/TIT.2018.2800012.

[22]

R. Paley, On orthogonal matrices, J. Math. Phys. MIT, 12 (1933), 311-320. 

[23]

R. A. PitavalB. M. PopovicP. Wang and B. Fredrik, Overcoming 5G PRACH capacity shortfall: Supersets of Zadoff-Chu sequences with low-correlation zone, IEEE Trans. Commun., 68 (2020), 5673-5688.  doi: 10.1109/TCOMM.2020.3003664.

[24]

B. M. Popovic, P. Wang, F. Berggren and R. A. Pitaval, Zero correlation zone sequences with flexible block-Repetitive spectral constraints, arXiv: 2007.08341, 2020.

[25]

W. RoweP. Stoica and J. Li, Spectrally constrained waveform design, IEEE Signal Process. Mag., 31 (2014), 157-162.  doi: 10.1109/MSP.2014.2301792.

[26]

D. Sarwate, Bounds on crosscorrelation and autocorrelation of sequences, IEEE Trans. Inf. Theory, 25 (1979), 720-724.  doi: 10.1109/TIT.1979.1056116.

[27]

M. Skolnik, Introduction to Radar Systems, 3$^{rd}$ edition, McGraw-Hill, New York, 2001.

[28]

J. SongP. Babu and D. P. Palomar, Sequence set design with good correlation properties via majorization-minimization, IEEE Trans. Signal Process., 64 (2016), 2866-2879.  doi: 10.1109/TSP.2016.2535312.

[29]

P. StoicaH. He and J. Li, New algorithms for designing unimodular sequences with good correlation properties, IEEE Trans. Signal Process., 57 (2009), 1415-1425.  doi: 10.1109/TSP.2009.2012562.

[30]

X. H. TangP. Z. Fan and S. Matsufuji, Lower bounds on the maximum correlation of sequence set with low or zero correlation zone, Electron. Lett., 36 (2000), 551-552. 

[31]

L. TianC. Xu and Y. Li, A family of single-channel spectrally-null-constrained sequences with low correlation, IEEE Signal Process. Lett., 27 (2020), 1645-1649.  doi: 10.1109/LSP.2020.3023858.

[32]

L. S. TsaiW. H. Chung and D. S. Shiu, Lower bounds on the correlation property for OFDM sequences with spectral-null constraints, IEEETrans. Wireless Commun., 10 (2011), 2652-2659.  doi: 10.1109/TWC.2011.060711.100626.

[33]

L. S. TsaiW. H. Chung and D. S. Shiu, Syntehsizing low autocorrelation and low PAPR OFDM sequences under spectral constraints through convex optimization and GS algorithm, IEEE Trans. Signal Process., 59 (2011), 2234-2243.  doi: 10.1109/TSP.2011.2108652.

[34]

L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inf. Theory, 20 (1974), 397-399.  doi: 10.1109/TIT.1974.1055219.

[35]

A. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), 107-121. 

[36]

T. Yucek and H. Arslan, A survey of spectrum sensing algorithms for cognitive radio applications, IEEE Commun. Surveys Tuts., 11 (2009), 116-130.  doi: 10.1109/SURV.2009.090109.

[37]

Q. Zhao and B. M. Sadler, A survey of dynamic spectrum access, IEEE Signal Process. Mag., 24 (2007), 79-89.  doi: 10.1109/MSP.2007.361604.

[38]

Z. ZhouT. Helleseth and U. Parampalli, A family of polyphase sequences with asymptotically optimal correlation, IEEE Trans. Inf. Theory, 64 (2018), 2896-2900.  doi: 10.1109/TIT.2018.2796597.

[39]

Y. ZhouZ. ZhouZ. LiuP. Fan and Y. L. Guan, Low-PMEPR preamble sequence design for dynamic spectrum allocation in OFDMA systems, IEEE Trans. Commun., 68 (2020), 2922-2933.  doi: 10.1109/TCOMM.2020.2973149.

show all references

References:
[1]

K. T. ArasuC. DingT. HellesethP. V. Kumar and H. M. Martinsen, Almost difference sets and their sequences with optimal autocorrelation, IEEE Trans. Inform. Theory, 47 (2001), 2934-2943.  doi: 10.1109/18.959271.

[2]

A. AubryA. De MaioM. Piezzo and A. Farina, Radar waveform design in a spectrally crowded environment via nonconvex quadratic optimization, IEEE Trans. Aerosp. Electron. Syst., 50 (2014), 1138-1152.  doi: 10.1109/TAES.2014.120731.

[3]

B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, New York, NY, USA: Wiley, 1998.

[4]

C. Carlet and C. Ding, Highly nonlinear mappings, J. Complexity, 20 (2004), 205-244.  doi: 10.1016/j.jco.2003.08.008.

[5]

C. Ding, Autocorrelation values of the generalized cyclotomic sequences of order 2, IEEE Trans. Inform. Theory, 44 (1998), 1699-1702.  doi: 10.1109/18.681354.

[6]

C. Ding, Codes from Difference Sets, World Scientific, 2015.

[7]

C. Ding, Linear complexity of generalized cyclotomic binary sequences of order 2, Finite Fields Appl., 3 (1997), 159-174.  doi: 10.1006/ffta.1997.0181.

[8]

C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Cryptogr., 46 (2008), 113-126.  doi: 10.1007/s10623-007-9140-z.

[9]

C. DingT Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inform. Theory, 47 (2001), 428-433.  doi: 10.1109/18.904555.

[10]

P. Fan and M. Darnell, Sequence Design for Communications Applications, New York, NY, USA: Wiley, 1996.

[11]

C. Fan and G. Ge, A unified approach to Whiteman's and Ding-Helleseth's generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60 (2014), 1326-1336.  doi: 10.1109/TIT.2013.2290694.

[12]

R. Gerchberg and W. Saxton, A practical algorithm for the determination of the phase from image and diffraction plane pictures, Optik, 35 (1972), 237-250. 

[13] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.
[14]

B. GordonW. H. Mills and L. R. Welch, Some new difference sets, Canad. J. Math., 14 (1962), 614-625.  doi: 10.4153/CJM-1962-052-2.

[15]

S. Haykin, Cognitive radar: A way of the future, IEEE Signal Process. Mag., 23 (2006), 30-40.  doi: 10.1109/MSP.2006.1593335.

[16]

S. Haykin, Cognitive radio: Brain-empowered wireless communications, IEEE J. Sel. Areas Commun., 23 (2005), 201-220.  doi: 10.1109/JSAC.2004.839380.

[17]

H. He, P. Stoica and J. Li, Waveform design with stopband and correlation constraints for cognitive radar, In Proc. 2nd Int. Workshop Cognit. Inf. Process., Italy, (2010), 344–349. doi: 10.1109/CIP.2010.5604089.

[18]

S. HuZ. LiuY. L. GuanW. XiongG. Bi and S. Li, Sequence design for cognitive CDMA communications under arbitrary spectrum hole constraint, IEEE J. Sel. Areas Commun., 32 (2014), 1974-1986.  doi: 10.1109/JSAC.2014.141103.

[19]

L. Hu and Q. Yue, Gauss periods and codebooks from generalized cyclotomic sets of order four, Des. Codes Cryptogr., 69 (2013), 233-246.  doi: 10.1007/s10623-012-9648-8.

[20]

E. Lehmer, On residue difference sets, Canad. J. Math, 5 (1953), 425-432.  doi: 10.4153/cjm-1953-047-3.

[21]

Z. LiuY. L. GuanU. Parampalli and S. Hu, Spectrally-constrained sequences: Bounds and constructions, IEEE Trans. Inf. Theory, 64 (2018), 2571-2582.  doi: 10.1109/TIT.2018.2800012.

[22]

R. Paley, On orthogonal matrices, J. Math. Phys. MIT, 12 (1933), 311-320. 

[23]

R. A. PitavalB. M. PopovicP. Wang and B. Fredrik, Overcoming 5G PRACH capacity shortfall: Supersets of Zadoff-Chu sequences with low-correlation zone, IEEE Trans. Commun., 68 (2020), 5673-5688.  doi: 10.1109/TCOMM.2020.3003664.

[24]

B. M. Popovic, P. Wang, F. Berggren and R. A. Pitaval, Zero correlation zone sequences with flexible block-Repetitive spectral constraints, arXiv: 2007.08341, 2020.

[25]

W. RoweP. Stoica and J. Li, Spectrally constrained waveform design, IEEE Signal Process. Mag., 31 (2014), 157-162.  doi: 10.1109/MSP.2014.2301792.

[26]

D. Sarwate, Bounds on crosscorrelation and autocorrelation of sequences, IEEE Trans. Inf. Theory, 25 (1979), 720-724.  doi: 10.1109/TIT.1979.1056116.

[27]

M. Skolnik, Introduction to Radar Systems, 3$^{rd}$ edition, McGraw-Hill, New York, 2001.

[28]

J. SongP. Babu and D. P. Palomar, Sequence set design with good correlation properties via majorization-minimization, IEEE Trans. Signal Process., 64 (2016), 2866-2879.  doi: 10.1109/TSP.2016.2535312.

[29]

P. StoicaH. He and J. Li, New algorithms for designing unimodular sequences with good correlation properties, IEEE Trans. Signal Process., 57 (2009), 1415-1425.  doi: 10.1109/TSP.2009.2012562.

[30]

X. H. TangP. Z. Fan and S. Matsufuji, Lower bounds on the maximum correlation of sequence set with low or zero correlation zone, Electron. Lett., 36 (2000), 551-552. 

[31]

L. TianC. Xu and Y. Li, A family of single-channel spectrally-null-constrained sequences with low correlation, IEEE Signal Process. Lett., 27 (2020), 1645-1649.  doi: 10.1109/LSP.2020.3023858.

[32]

L. S. TsaiW. H. Chung and D. S. Shiu, Lower bounds on the correlation property for OFDM sequences with spectral-null constraints, IEEETrans. Wireless Commun., 10 (2011), 2652-2659.  doi: 10.1109/TWC.2011.060711.100626.

[33]

L. S. TsaiW. H. Chung and D. S. Shiu, Syntehsizing low autocorrelation and low PAPR OFDM sequences under spectral constraints through convex optimization and GS algorithm, IEEE Trans. Signal Process., 59 (2011), 2234-2243.  doi: 10.1109/TSP.2011.2108652.

[34]

L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inf. Theory, 20 (1974), 397-399.  doi: 10.1109/TIT.1974.1055219.

[35]

A. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), 107-121. 

[36]

T. Yucek and H. Arslan, A survey of spectrum sensing algorithms for cognitive radio applications, IEEE Commun. Surveys Tuts., 11 (2009), 116-130.  doi: 10.1109/SURV.2009.090109.

[37]

Q. Zhao and B. M. Sadler, A survey of dynamic spectrum access, IEEE Signal Process. Mag., 24 (2007), 79-89.  doi: 10.1109/MSP.2007.361604.

[38]

Z. ZhouT. Helleseth and U. Parampalli, A family of polyphase sequences with asymptotically optimal correlation, IEEE Trans. Inf. Theory, 64 (2018), 2896-2900.  doi: 10.1109/TIT.2018.2796597.

[39]

Y. ZhouZ. ZhouZ. LiuP. Fan and Y. L. Guan, Low-PMEPR preamble sequence design for dynamic spectrum allocation in OFDMA systems, IEEE Trans. Commun., 68 (2020), 2922-2933.  doi: 10.1109/TCOMM.2020.2973149.

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 $D$.
$ (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 $D$.
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
$ 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
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
$ 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
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
$ 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
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
$ 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
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
$ 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
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