May  2018, 12(2): 387-413. doi: 10.3934/amc.2018024

New families of strictly optimal frequency hopping sequence sets

Department of Mathematics, Ningbo University, Ningbo 315211, China

Received  June 2017 Revised  November 2017 Published  March 2018

Fund Project: The first author is supported by the NSFC under Grants 11701303, and K.C.Wong Magna Fund in Ningbo University

Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHS sets with optimal partial Hamming correlation. We present several direct constructions for balanced nested cyclic difference packings (BNCDPs) and balanced nested cyclic relative difference packings (BNCRDPs) by using trace functions and discrete logarithm. We also show three recursive constructions for FHS sets with partial Hamming correlation, which are based on cyclic difference matrices and discrete logarithm. Combing these BNCDPs, BNCRDPs and three recursive constructions, we obtain infinitely many new strictly optimal FHS sets with respect to the Peng-Fan bounds.

Citation: Jingjun Bao. New families of strictly optimal frequency hopping sequence sets. Advances in Mathematics of Communications, 2018, 12 (2) : 387-413. doi: 10.3934/amc.2018024
References:
[1]

J. Bao and L. Ji, Frequency hopping sequences with optimal partial hamming correlation, IEEE Trans. Inf. Theory, 62 (2016), 3768-3783. doi: 10.1109/TIT.2016.2551225. Google Scholar

[2]

J. Bao and L. Ji, New families of optimal frequency hopping sequence sets, IEEE Trans. Inf. Theory, 62 (2016), 5209-5224. doi: 10.1109/TIT.2016.2589258. Google Scholar

[3]

Bluetooth Special Interest Group (SIG), Washington, DC, USA. (2003, Nov.). Specification of the Bluetooth Systems-Core [Online]. Available: http://www.bluetooth.orgGoogle Scholar

[4]

H. CaiZ. ZhouY. Yang and X. Tang, A new construction of frequency hopping sequences with optimal partial Hamming correlation, IEEE Trans. Inf. Theory, 60 (2014), 5782-5790. doi: 10.1109/TIT.2014.2332996. Google Scholar

[5]

H. CaiY. YangZ. Zhou and X. Tang, Strictly optimal frequency-hopping sequence sets with optimal family sizes, IEEE Trans. Inf. Theory, 62 (2016), 1087-1093. doi: 10.1109/TIT.2015.2512859. Google Scholar

[6]

B. ChenL. LinS. Ling and H. Liu, Three new classes of optimal frequency-hopping sequence sets, Des. Codes Cryptogr., 83 (2017), 219-232. doi: 10.1007/s10623-016-0220-9. Google Scholar

[7]

W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141. doi: 10.1109/TIT.2004.842708. Google Scholar

[8]

J.-H. ChungG. Gong and K. Yang, New families of optimal frequency-hopping sequences by composite lengths, IEEE Trans. Inf. Theory, 60 (2014), 3688-3697. doi: 10.1109/TIT.2014.2315207. Google Scholar

[9]

J.-H. ChungY. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving techniques, IEEE Trans. Inf. Theory, 55 (2009), 5783-5791. doi: 10.1109/TIT.2009.2032742. Google Scholar

[10]

J.-H. Chung and K. Yang, $k$ -fold cyclotomy and its application to frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 2306-2317. doi: 10.1109/TIT.2011.2112235. Google Scholar

[11]

J.-H. Chung and K. Yang, Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inf. Theory, 56 (2010), 1685-1693. doi: 10.1109/TIT.2010.2040888. Google Scholar

[12]

C. J. Colbourn, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 2010. doi: 10. 1201/9781420049954. Google Scholar

[13]

C. DingR. Fuji-HaraY. FujiwaraM. Jimbo and M. Mishima, Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inf. Theory, 55 (2009), 3297-3304. doi: 10.1109/TIT.2009.2021366. Google Scholar

[14]

C. DingM. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 53 (2007), 2606-2610. doi: 10.1109/TIT.2007.899545. Google Scholar

[15]

C. DingY. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 3605-3612. doi: 10.1109/TIT.2010.2048504. Google Scholar

[16]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 54 (2008), 3741-3745. doi: 10.1109/TIT.2008.926410. Google Scholar

[17]

Y.-C. EunS.-Y. JinY.-P. Hong and H.-Y. Song, Frequency hopping sequences with optimal partial autocorrelation properties, IEEE Trans. Inf. Theory, 50 (2004), 2438-2442. doi: 10.1109/TIT.2004.834792. Google Scholar

[18]

C. FanH. Cai and X. Tang, A combinatorial construction for strictly optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 62 (2016), 4769-4774. doi: 10.1109/TIT.2016.2556710. Google Scholar

[19]

P. Fan and M. Darnell, Sequence Design for Communications Applications, London, U. K. : Wiley, 1996.Google Scholar

[20]

R. Fuji-HaraY. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inf. Theory, 50 (2004), 2408-2420. doi: 10.1109/TIT.2004.834783. Google Scholar

[21]

G. GeR. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency-hopping sequences, J. Combin. Theory Ser. A, 113 (2006), 1699-1718. doi: 10.1016/j.jcta.2006.03.019. Google Scholar

[22]

G. GeY. Miao and Z. Yao, Optimal frequency hopping sequences: Auto-and cross-correlation properties, IEEE Trans. Inf. Theory, 55 (2009), 867-879. doi: 10.1109/TIT.2008.2009856. Google Scholar

[23]

A. Lempel and H. Greenberger, Families of sequences with optimal Hamming-correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94. Google Scholar

[24]

D. Peng and P. Fan, Lower bounds on the Hamming auto-and cross correlations of frequency-hopping sequences, IEEE Trans. Inf. Theory, 50 (2004), 2149-2154. doi: 10.1109/TIT.2004.833362. Google Scholar

[25]

W. RenF. Fu and Z. Zhou, New sets of frequency-hopping sequences with optimal Hamming correlation, Des. Codes Cryptogr., 72 (2014), 423-434. doi: 10.1007/s10623-012-9774-3. Google Scholar

[26]

P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Trans. Inf. Theory, 44 (1998), 1492-1503. doi: 10.1109/18.681324. Google Scholar

[27]

L. Yang and G. B. Giannakis, Ultra-wideband communications: An idea whose time has come, IEEE Signal Process. Mag., 21 (2004), 26-54. Google Scholar

[28]

Y. YangX. TangP. Udaya and D. Peng, New bound on frequency hopping sequence sets and its optimal constructions, IEEE Trans. Inf. Theory, 57 (2011), 7605-7613. doi: 10.1109/TIT.2011.2162571. Google Scholar

[29]

X. ZengH. CaiX. Tang and Y. Yang, A class of optimal frequency hopping sequences with new parameters, IEEE Trans. Inf. Theory, 58 (2012), 4899-4907. doi: 10.1109/TIT.2012.2195771. Google Scholar

[30]

X. ZengH. CaiX. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248. doi: 10.1109/TIT.2013.2237754. Google Scholar

[31]

Z. ZhouX. TangD. Peng and U. Parampalli, New constructions for optimal sets of frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 3831-3840. doi: 10.1109/TIT.2011.2137290. Google Scholar

[32]

Z. ZhouX. TangX. Niu and U. Parampalli, New classes of frequency hopping sequences with optimal partial correlation, IEEE Trans. Inf. Theory, 58 (2012), 453-458. doi: 10.1109/TIT.2011.2167126. Google Scholar

show all references

References:
[1]

J. Bao and L. Ji, Frequency hopping sequences with optimal partial hamming correlation, IEEE Trans. Inf. Theory, 62 (2016), 3768-3783. doi: 10.1109/TIT.2016.2551225. Google Scholar

[2]

J. Bao and L. Ji, New families of optimal frequency hopping sequence sets, IEEE Trans. Inf. Theory, 62 (2016), 5209-5224. doi: 10.1109/TIT.2016.2589258. Google Scholar

[3]

Bluetooth Special Interest Group (SIG), Washington, DC, USA. (2003, Nov.). Specification of the Bluetooth Systems-Core [Online]. Available: http://www.bluetooth.orgGoogle Scholar

[4]

H. CaiZ. ZhouY. Yang and X. Tang, A new construction of frequency hopping sequences with optimal partial Hamming correlation, IEEE Trans. Inf. Theory, 60 (2014), 5782-5790. doi: 10.1109/TIT.2014.2332996. Google Scholar

[5]

H. CaiY. YangZ. Zhou and X. Tang, Strictly optimal frequency-hopping sequence sets with optimal family sizes, IEEE Trans. Inf. Theory, 62 (2016), 1087-1093. doi: 10.1109/TIT.2015.2512859. Google Scholar

[6]

B. ChenL. LinS. Ling and H. Liu, Three new classes of optimal frequency-hopping sequence sets, Des. Codes Cryptogr., 83 (2017), 219-232. doi: 10.1007/s10623-016-0220-9. Google Scholar

[7]

W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141. doi: 10.1109/TIT.2004.842708. Google Scholar

[8]

J.-H. ChungG. Gong and K. Yang, New families of optimal frequency-hopping sequences by composite lengths, IEEE Trans. Inf. Theory, 60 (2014), 3688-3697. doi: 10.1109/TIT.2014.2315207. Google Scholar

[9]

J.-H. ChungY. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving techniques, IEEE Trans. Inf. Theory, 55 (2009), 5783-5791. doi: 10.1109/TIT.2009.2032742. Google Scholar

[10]

J.-H. Chung and K. Yang, $k$ -fold cyclotomy and its application to frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 2306-2317. doi: 10.1109/TIT.2011.2112235. Google Scholar

[11]

J.-H. Chung and K. Yang, Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inf. Theory, 56 (2010), 1685-1693. doi: 10.1109/TIT.2010.2040888. Google Scholar

[12]

C. J. Colbourn, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 2010. doi: 10. 1201/9781420049954. Google Scholar

[13]

C. DingR. Fuji-HaraY. FujiwaraM. Jimbo and M. Mishima, Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inf. Theory, 55 (2009), 3297-3304. doi: 10.1109/TIT.2009.2021366. Google Scholar

[14]

C. DingM. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 53 (2007), 2606-2610. doi: 10.1109/TIT.2007.899545. Google Scholar

[15]

C. DingY. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 3605-3612. doi: 10.1109/TIT.2010.2048504. Google Scholar

[16]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 54 (2008), 3741-3745. doi: 10.1109/TIT.2008.926410. Google Scholar

[17]

Y.-C. EunS.-Y. JinY.-P. Hong and H.-Y. Song, Frequency hopping sequences with optimal partial autocorrelation properties, IEEE Trans. Inf. Theory, 50 (2004), 2438-2442. doi: 10.1109/TIT.2004.834792. Google Scholar

[18]

C. FanH. Cai and X. Tang, A combinatorial construction for strictly optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 62 (2016), 4769-4774. doi: 10.1109/TIT.2016.2556710. Google Scholar

[19]

P. Fan and M. Darnell, Sequence Design for Communications Applications, London, U. K. : Wiley, 1996.Google Scholar

[20]

R. Fuji-HaraY. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inf. Theory, 50 (2004), 2408-2420. doi: 10.1109/TIT.2004.834783. Google Scholar

[21]

G. GeR. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency-hopping sequences, J. Combin. Theory Ser. A, 113 (2006), 1699-1718. doi: 10.1016/j.jcta.2006.03.019. Google Scholar

[22]

G. GeY. Miao and Z. Yao, Optimal frequency hopping sequences: Auto-and cross-correlation properties, IEEE Trans. Inf. Theory, 55 (2009), 867-879. doi: 10.1109/TIT.2008.2009856. Google Scholar

[23]

A. Lempel and H. Greenberger, Families of sequences with optimal Hamming-correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94. Google Scholar

[24]

D. Peng and P. Fan, Lower bounds on the Hamming auto-and cross correlations of frequency-hopping sequences, IEEE Trans. Inf. Theory, 50 (2004), 2149-2154. doi: 10.1109/TIT.2004.833362. Google Scholar

[25]

W. RenF. Fu and Z. Zhou, New sets of frequency-hopping sequences with optimal Hamming correlation, Des. Codes Cryptogr., 72 (2014), 423-434. doi: 10.1007/s10623-012-9774-3. Google Scholar

[26]

P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Trans. Inf. Theory, 44 (1998), 1492-1503. doi: 10.1109/18.681324. Google Scholar

[27]

L. Yang and G. B. Giannakis, Ultra-wideband communications: An idea whose time has come, IEEE Signal Process. Mag., 21 (2004), 26-54. Google Scholar

[28]

Y. YangX. TangP. Udaya and D. Peng, New bound on frequency hopping sequence sets and its optimal constructions, IEEE Trans. Inf. Theory, 57 (2011), 7605-7613. doi: 10.1109/TIT.2011.2162571. Google Scholar

[29]

X. ZengH. CaiX. Tang and Y. Yang, A class of optimal frequency hopping sequences with new parameters, IEEE Trans. Inf. Theory, 58 (2012), 4899-4907. doi: 10.1109/TIT.2012.2195771. Google Scholar

[30]

X. ZengH. CaiX. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248. doi: 10.1109/TIT.2013.2237754. Google Scholar

[31]

Z. ZhouX. TangD. Peng and U. Parampalli, New constructions for optimal sets of frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 3831-3840. doi: 10.1109/TIT.2011.2137290. Google Scholar

[32]

Z. ZhouX. TangX. Niu and U. Parampalli, New classes of frequency hopping sequences with optimal partial correlation, IEEE Trans. Inf. Theory, 58 (2012), 453-458. doi: 10.1109/TIT.2011.2167126. Google Scholar

Table Ⅰ.  SOME KNOWN FHS SETS WITH OPTIMA HAMMING CORRELATION
LengthNumber of sequences $H_{max}$Alphabet sizeConstraintsReference
$p(q-1)$ $\left\lfloor \frac{c}{p}\right\rfloor$ $pd$ $c+1$ $q-1=cd$, $c\geq p(d+1)$, gcd$(p, q-1)=1$[25]
$q^m-1$ $q^u$ $q^{m-u}$ $q^u$ $m > u \geq 1$[31]
$\frac{q^m-1}{d}$ $d$ $\frac{q^{m-u}-1}{d}$ $q^u$ $d|q-1$, $ m > u \geq 1$, gcd$(d, m)=1$[31]
$\frac{q+1}{d}$ $d(q-1)$ $1$ $q$ $q+1\equiv d \pmod {2d}$[15]
$2^{2u}+1$ $2^{2u}-1$ $2^u+1$ $2^u$[13]
$up^2$ $\left\lfloor \frac{p}{u}\right\rfloor$ $up$ $p$ $ p>u\geq 2$[9]
$u\frac{q^m-1}{d}$ $\left\lfloor \frac{d}{u}\right\rfloor$ $u\frac{q^{m-1}-1}{d}$ $q$ $ d\geq u \geq 2$, $d|q-1$, gcd$(m, d)=1$[9]
$v$ $f$ $e$ $\frac{v-1}{e}+1$ $v$ is not a prime or $v$ is a prime with $f\geq e >1$[30]
$\frac{w(q^m-1)}{d}$ $d$ $\frac{q^{m-u}-1}{d}$ $wq^u$ $d|q-1$, gcd$(d, m)=1$, $q_1>q^{m-u}$, $m > u \geq 1$[2]
$vw$ $f$ $e$ $\frac{v-1}{e}w+\frac{w-1}{e^{'}}+1$ $e\geq e'\geq 2$, $v\geq e^2$, $q_1\geq p_1> 2e$[2]
$vwu(q-1)$ $\lfloor \frac{c}{u}\rfloor$ $ud$ $cvw+\frac{(v-1)w}{e}+\frac{w-1}{e^{'}}+1$ $q-1=cd$, $c\geq u(d+1)$, $d\geq e\geq e'\geq 2$, gcd$(u, q-1)=1$, $q_1 \geq p_1> q-1$[2]
$vu(q-1)$ $\lfloor\frac{f}{u}\rfloor$ $eu$ $\frac{vq-q}{e}+c+1$ $q-1=cd$, $c \geq u(d+1)$, gcd$(u, q-1)=1$, $q\geq \max\{p_1-1, 4ue\}$, $e\geq d$, $p_1> 2ue$, $v \geq 4e^2u$[2]
$m, u, c$ and $d$ are positive integers;
$p$ is a prime and $q$ is a prime power;
$v$ is an integer with prime factor decomposition $v=p_1^{m_1}p_2^{m_2}\cdots p_s^{m_s}$ with $p_1 <p_2<\ldots<p_s$;
$e$ is an integer such that $e|gcd(p_1-1, p_2-1, \ldots, p_s-1)$, and $f=\frac{p_1-1}{e}$;
$w$ is an integer with prime factor decomposition $w=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}$ with $q_1<q_2<\ldots<q_t$;
$e'$ is an integer such that $e'|gcd(q_1-1, q_2-1, \ldots, q_t-1)$
LengthNumber of sequences $H_{max}$Alphabet sizeConstraintsReference
$p(q-1)$ $\left\lfloor \frac{c}{p}\right\rfloor$ $pd$ $c+1$ $q-1=cd$, $c\geq p(d+1)$, gcd$(p, q-1)=1$[25]
$q^m-1$ $q^u$ $q^{m-u}$ $q^u$ $m > u \geq 1$[31]
$\frac{q^m-1}{d}$ $d$ $\frac{q^{m-u}-1}{d}$ $q^u$ $d|q-1$, $ m > u \geq 1$, gcd$(d, m)=1$[31]
$\frac{q+1}{d}$ $d(q-1)$ $1$ $q$ $q+1\equiv d \pmod {2d}$[15]
$2^{2u}+1$ $2^{2u}-1$ $2^u+1$ $2^u$[13]
$up^2$ $\left\lfloor \frac{p}{u}\right\rfloor$ $up$ $p$ $ p>u\geq 2$[9]
$u\frac{q^m-1}{d}$ $\left\lfloor \frac{d}{u}\right\rfloor$ $u\frac{q^{m-1}-1}{d}$ $q$ $ d\geq u \geq 2$, $d|q-1$, gcd$(m, d)=1$[9]
$v$ $f$ $e$ $\frac{v-1}{e}+1$ $v$ is not a prime or $v$ is a prime with $f\geq e >1$[30]
$\frac{w(q^m-1)}{d}$ $d$ $\frac{q^{m-u}-1}{d}$ $wq^u$ $d|q-1$, gcd$(d, m)=1$, $q_1>q^{m-u}$, $m > u \geq 1$[2]
$vw$ $f$ $e$ $\frac{v-1}{e}w+\frac{w-1}{e^{'}}+1$ $e\geq e'\geq 2$, $v\geq e^2$, $q_1\geq p_1> 2e$[2]
$vwu(q-1)$ $\lfloor \frac{c}{u}\rfloor$ $ud$ $cvw+\frac{(v-1)w}{e}+\frac{w-1}{e^{'}}+1$ $q-1=cd$, $c\geq u(d+1)$, $d\geq e\geq e'\geq 2$, gcd$(u, q-1)=1$, $q_1 \geq p_1> q-1$[2]
$vu(q-1)$ $\lfloor\frac{f}{u}\rfloor$ $eu$ $\frac{vq-q}{e}+c+1$ $q-1=cd$, $c \geq u(d+1)$, gcd$(u, q-1)=1$, $q\geq \max\{p_1-1, 4ue\}$, $e\geq d$, $p_1> 2ue$, $v \geq 4e^2u$[2]
$m, u, c$ and $d$ are positive integers;
$p$ is a prime and $q$ is a prime power;
$v$ is an integer with prime factor decomposition $v=p_1^{m_1}p_2^{m_2}\cdots p_s^{m_s}$ with $p_1 <p_2<\ldots<p_s$;
$e$ is an integer such that $e|gcd(p_1-1, p_2-1, \ldots, p_s-1)$, and $f=\frac{p_1-1}{e}$;
$w$ is an integer with prime factor decomposition $w=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}$ with $q_1<q_2<\ldots<q_t$;
$e'$ is an integer such that $e'|gcd(q_1-1, q_2-1, \ldots, q_t-1)$
Table Ⅱ.  SOME KNOWN FHS SETS WITH OPTIMAL PARTIAL HAMMING CORRELATION
LengthAlphabet size $H_{max}$ over correlation window of length LNumber of sequencesConstraintsSource
$\frac{q^m-1}{d}$ $q^{m-1}$ $\left\lceil \frac{L(q-1)}{q^m-1}\right\rceil$ $d$ $d|(q-1)$, gcd$(d, m)=1$[32]
$ev$ $v$ $\left\lceil \frac{L}{v}\right\rceil$ $f$[4]
$ p(p^m-1)$ $p^{m}$ $\left\lceil\frac{L}{p^{m}-1}\right\rceil$ $p^{m-1}$ $m\geq 2$[5]
$evw$ $(v-1)w+\frac{ew}{r}$ $\left\lceil \frac{L}{vw}\right\rceil$ $f$ $q_1\geq p_1>2e$, $r|e$ $v\geq \frac{p_1e}{r}$ and $gcd(w, e)=1$[1]
$v\frac{q^m-1}{d}$ $vq^{m-1} $ $\left\lceil \frac{(q-1)L}{v(q^{m}-1)}\right\rceil$ $d$ $m>1, $ $q^m \leq p_1$ and $gcd(d, m)=1, $ $d|q-1$, $\frac{q^m-1}{d}|p_i-1$ for $1\leq i \leq s$[1]
$q$ is a prime power and $p$ is a prime;
$v$ is an integer with prime factor decomposition $v=p_1^{m_1}p_2^{m_2}\cdots p_s^{m_s}$ with $p_1 < p_2<\cdots <p_s$;
$e, f$ are integers such that $e>1$ and $e|gcd(p_1-1, p_2-1, \ldots, p_s-1)$, and $f=\frac{p_1-1}{e}$;
$w$ is an integer with prime factor decomposition $w=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}$ with $q_1<q_2<\cdots <q_t$;
$r$ is an integer such that $r>1$ and $r|gcd(q_1-1, q_2-1, \ldots, q_t-1)$;
$d, m$ are positive integers.
LengthAlphabet size $H_{max}$ over correlation window of length LNumber of sequencesConstraintsSource
$\frac{q^m-1}{d}$ $q^{m-1}$ $\left\lceil \frac{L(q-1)}{q^m-1}\right\rceil$ $d$ $d|(q-1)$, gcd$(d, m)=1$[32]
$ev$ $v$ $\left\lceil \frac{L}{v}\right\rceil$ $f$[4]
$ p(p^m-1)$ $p^{m}$ $\left\lceil\frac{L}{p^{m}-1}\right\rceil$ $p^{m-1}$ $m\geq 2$[5]
$evw$ $(v-1)w+\frac{ew}{r}$ $\left\lceil \frac{L}{vw}\right\rceil$ $f$ $q_1\geq p_1>2e$, $r|e$ $v\geq \frac{p_1e}{r}$ and $gcd(w, e)=1$[1]
$v\frac{q^m-1}{d}$ $vq^{m-1} $ $\left\lceil \frac{(q-1)L}{v(q^{m}-1)}\right\rceil$ $d$ $m>1, $ $q^m \leq p_1$ and $gcd(d, m)=1, $ $d|q-1$, $\frac{q^m-1}{d}|p_i-1$ for $1\leq i \leq s$[1]
$q$ is a prime power and $p$ is a prime;
$v$ is an integer with prime factor decomposition $v=p_1^{m_1}p_2^{m_2}\cdots p_s^{m_s}$ with $p_1 < p_2<\cdots <p_s$;
$e, f$ are integers such that $e>1$ and $e|gcd(p_1-1, p_2-1, \ldots, p_s-1)$, and $f=\frac{p_1-1}{e}$;
$w$ is an integer with prime factor decomposition $w=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}$ with $q_1<q_2<\cdots <q_t$;
$r$ is an integer such that $r>1$ and $r|gcd(q_1-1, q_2-1, \ldots, q_t-1)$;
$d, m$ are positive integers.
Table Ⅲ.  NEW FHS SETS WITH OPTIMAL PARTIAL HAMMING CORRELATION
LengthAlphabet size $H_{max}$Number of sequencesConstraintsReference
$\frac{w(q^m-1)}{d}$ $(q^{m-1}-1+\frac{q-1}{d})w$ $\frac{q-1}{d}$ $d$ $m\geq 3$, $d|q-1$, gcd$(m, d)=1$, $q_1>q, $Corollary 6
$wp(p^m-1)$ $p^{m}w$ $p$ $p^{m-1}$ $m>1, \ $ and $q_1>p^m, $Corollary 7
$ewv$ $(v-1+e)w$ $e$ $f$ $q_1>p_1-1$, If $v$ is not a prime with $f>1$ or $v$ is a prime with $f\geq e$, Corollary 8
$pv(p^m-1)$ $vp^m$ $p$ $f_1$ $p|p_i-1$, $f_1=\frac{p_1-1}{p}\geq 2$, $p^m >p_1-1$, $p^m>2(1+p)$Corollary 9
$(q'-1)\frac{q^m-1}{d}$ $(q^{m-1}-1+\frac{q-1}{d})q'$ $\frac{q-1}{d}$ $d$ $d|q-1$, gcd$(m, d)=1$, $d\geq 2$ $gcd(q'-1, \frac{q-1}{d})=1$, $q'>q+1$Corollary 10
$(q-1)p(p^m-1)$ $p^{m}q$ $p$ $p^{m-1}$ $p^{m}-3p\geq 1$, $q\geq p^m$, $gcd(q-1, p)=1$, $m>1$Corollary 11
$ev(q-1)$ $(v-1+e)q$ $e$ $f$ $q> p_1-1$, $v\geq e^3f^2$, $q\geq 2e+5$, $gcd(q-1, e)=1$, $f>1$Corollary 12
$q, q'$ are prime powers and $p$ is a prime;
$v$ is an integer with prime factor decomposition $v=p_1^{m_1}p_2^{m_2}\cdots p_s^{m_s}$ with $p_1 < p_2<\ldots<p_s$;
$e$ is an integer such that $e| gcd(p_1-1, p_2-1, \ldots, p_s-1)$, and $f=\frac{p_1-1}{e}$;
$w$ is any an integer with prime factor decomposition $w=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}$ with $q_1<q_2<\ldots<q_t$;
$d, m$ are positive integers.
LengthAlphabet size $H_{max}$Number of sequencesConstraintsReference
$\frac{w(q^m-1)}{d}$ $(q^{m-1}-1+\frac{q-1}{d})w$ $\frac{q-1}{d}$ $d$ $m\geq 3$, $d|q-1$, gcd$(m, d)=1$, $q_1>q, $Corollary 6
$wp(p^m-1)$ $p^{m}w$ $p$ $p^{m-1}$ $m>1, \ $ and $q_1>p^m, $Corollary 7
$ewv$ $(v-1+e)w$ $e$ $f$ $q_1>p_1-1$, If $v$ is not a prime with $f>1$ or $v$ is a prime with $f\geq e$, Corollary 8
$pv(p^m-1)$ $vp^m$ $p$ $f_1$ $p|p_i-1$, $f_1=\frac{p_1-1}{p}\geq 2$, $p^m >p_1-1$, $p^m>2(1+p)$Corollary 9
$(q'-1)\frac{q^m-1}{d}$ $(q^{m-1}-1+\frac{q-1}{d})q'$ $\frac{q-1}{d}$ $d$ $d|q-1$, gcd$(m, d)=1$, $d\geq 2$ $gcd(q'-1, \frac{q-1}{d})=1$, $q'>q+1$Corollary 10
$(q-1)p(p^m-1)$ $p^{m}q$ $p$ $p^{m-1}$ $p^{m}-3p\geq 1$, $q\geq p^m$, $gcd(q-1, p)=1$, $m>1$Corollary 11
$ev(q-1)$ $(v-1+e)q$ $e$ $f$ $q> p_1-1$, $v\geq e^3f^2$, $q\geq 2e+5$, $gcd(q-1, e)=1$, $f>1$Corollary 12
$q, q'$ are prime powers and $p$ is a prime;
$v$ is an integer with prime factor decomposition $v=p_1^{m_1}p_2^{m_2}\cdots p_s^{m_s}$ with $p_1 < p_2<\ldots<p_s$;
$e$ is an integer such that $e| gcd(p_1-1, p_2-1, \ldots, p_s-1)$, and $f=\frac{p_1-1}{e}$;
$w$ is any an integer with prime factor decomposition $w=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}$ with $q_1<q_2<\ldots<q_t$;
$d, m$ are positive integers.
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