# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021071
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## Constructions of optimal low hit zone frequency hopping sequence sets with large family size

 School of Computer and Software Engineering, Xihua University, Chengdu, Sichuan 610039, China

* Corresponding author: Xianhua Niu

Received  July 2021 Revised  November 2021 Early access January 2022

Frequency hopping sequences with low hit zone is significant for application in quasi synchronous multiple-access systems. In this paper, we obtained two constructions of optimal frequency hopping sequence sets with low hit zone based on interleaving techniques. The presented low hit zone frequency hopping sequence sets are with new and flexible parameters and large family size which can meet the needs of the practical applications. Moreover, all the sequences in the proposed sets are cyclically inequivalent. Some low hit zone frequency hopping sequence sets constructed in literatures are included in our family. The proposed frequency hopping sequence sets with low hit zone are contributed for quasi-synchronous frequency hopping multiple access system to reduce or eliminate multiple-access interference.

Citation: Xiujie Zhang, Xianhua Niu, Xin Tan. Constructions of optimal low hit zone frequency hopping sequence sets with large family size. Advances in Mathematics of Communications, doi: 10.3934/amc.2021071
##### References:
 [1] J.-H. Chung and K. Yang, New classes of optimal low-hit-zone frequency-hopping sequence sets by cartesian product, IEEE Trans. Inf. Theory, 59 (2013), 726-732.  doi: 10.1109/TIT.2012.2213065. [2] P. Fan and M. Darnell, Sequence Design for Communications Applications, RSP-John Wiley Sons Inc., London, 1996. [3] G. Gong, Theory and applications of q-ary interleaved sequences, IEEE Trans. Inform. Theory, 41 (1995), 400-411.  doi: 10.1109/18.370141. [4] G. Gong, New designs for signal sets with low cross correlation, balance property, and large linear span: Gf(p) case, IEEE Trans. Inf. Theory, 48 (2002), 2847-2867.  doi: 10.1109/TIT.2002.804044. [5] H. Han, D. Peng and U. Parampalli, New sets of optimal low-hit-zone frequency-hopping sequences based on m-sequences, Cryptogr. Commun., 9 (2017), 511-522.  doi: 10.1007/s12095-016-0192-7. [6] L. Ling, X. Niu, B. Zeng and X. Liu, New classes of optimal low hit zone frequency hopping sequence set with large family size, IEICE Trans. Fundam. Electron., Commun. Comput. Sci., E101.A (2018), 2213-2216. [7] X. Liu, S. Qin and Z. Qi, Low-hit-zone frequency/time hopping sequence sets with large family size, IEEE Access, 7 (2019), 181733-181739.  doi: 10.1109/ACCESS.2019.2959718. [8] W. Ma and S. Sun, New designs of frequency hopping sequences with low hit zone, Des. Codes Cryptogr., 60 (2011), 145-153.  doi: 10.1007/s10623-010-9422-8. [9] X. Niu, D. Peng and Z. Zhou, New classes of optimal low hit zone frequency hopping sequences with new parameters by interleaving technique, IEICE Trans. Fundam. Electron., Commun. Comput Sci., E95-A (2012), 1835-1842. [10] X. Niu, D. Peng and Z. Zhou, New classes of optimal frequency hopping sequences with low hit zone, Adv. Math. Commun., 7 (2013), 293-310.  doi: 10.3934/amc.2013.7.293. [11] X. Niu, C. Xing, Y. Liu and L. Zhou, A construction of optimal frequency hopping sequence set via combination of multiplicative and additive groups of finite fields, IEEE Trans. Inform. Theory, 66 (2020), 5310-5315.  doi: 10.1109/TIT.2020.2972388. [12] X. Niu and Z. Zhou, New classes of optimal low hit zone frequency hopping sequences with new parameters, IEICE Trans. Fundam., E97-A (2014), 2567-2571.  doi: 10.1587/transfun.E97.A.2567. [13] D. Peng and P. Fan, Lower bounds on the hamming auto- and cross correlations of frequency-hopping sequences, IEEE Trans. Inform. Theory, 50 (2004), 2149-2154.  doi: 10.1109/TIT.2004.833362. [14] D. Peng, P. Fan and M. H. Lee, Lower bounds on the periodic hamming correlations of frequency hopping sequences with low hit zone, Sci. China Ser., 49 (2006), 208-218.  doi: 10.1007/s11432-006-0208-6. [15] C. U. Press, Signal Design for Good Correlation for Wireless Communication, Cryptography, and Radar, Signal design for good correlation for wireless communication, cryptography, and radar, 2005. [16] C. Wang, D. Peng and L. Zhou, New constructions of optimal frequency-hopping sequence sets with low-hit-zone, Int. J. Found. Comput. Sci., 27 (2016), 53-66.  doi: 10.1142/S0129054116500040. [17] X. Wang and P. Fan, A class of frequency hopping sequences with no hit zone, International Conference on Parallel and Distributed Computing, (2003), 896–898. [18] X. Ye and P. Fan, Two classes of frequency-hopping sequences with no hit zone, International Conference on Parallel and Distributed Computing, (2003), 304–306. [19] W. Yin, C. Xiang and F. W. Fu, Two constructions of low-hit-zone frequency-hopping sequence sets, Advances in Mathematics of Communications. doi: 10.3934/amc.2020110. [20] L. Zhou, D. Peng, H. Liang, C. Wang and Z. Ma, Constructions of optimal low-hit-zone frequency hopping sequence sets, Des. Codes Cryptogr., 85 (2017), 219-232.  doi: 10.1007/s10623-016-0299-z. [21] L. Zhou, D. Peng, H. Liang, C. Wang and Z. Ma, Generalized methods to construct low-hit-zone frequency-hopping sequence sets and optimal constructions, Crytogr. Commun., 9 (2017), 707-728.  doi: 10.1007/s12095-017-0211-3. [22] Z. Zhou, X. Tang, D. Peng and U. Parampalli, New constructions for optimal sets of frequency-hopping sequences, IEEE Trans. Inform. Theory, 57 (2011), 3831-3840.  doi: 10.1109/TIT.2011.2137290.

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##### References:
 [1] J.-H. Chung and K. Yang, New classes of optimal low-hit-zone frequency-hopping sequence sets by cartesian product, IEEE Trans. Inf. Theory, 59 (2013), 726-732.  doi: 10.1109/TIT.2012.2213065. [2] P. Fan and M. Darnell, Sequence Design for Communications Applications, RSP-John Wiley Sons Inc., London, 1996. [3] G. Gong, Theory and applications of q-ary interleaved sequences, IEEE Trans. Inform. Theory, 41 (1995), 400-411.  doi: 10.1109/18.370141. [4] G. Gong, New designs for signal sets with low cross correlation, balance property, and large linear span: Gf(p) case, IEEE Trans. Inf. Theory, 48 (2002), 2847-2867.  doi: 10.1109/TIT.2002.804044. [5] H. Han, D. Peng and U. Parampalli, New sets of optimal low-hit-zone frequency-hopping sequences based on m-sequences, Cryptogr. Commun., 9 (2017), 511-522.  doi: 10.1007/s12095-016-0192-7. [6] L. Ling, X. Niu, B. Zeng and X. Liu, New classes of optimal low hit zone frequency hopping sequence set with large family size, IEICE Trans. Fundam. Electron., Commun. Comput. Sci., E101.A (2018), 2213-2216. [7] X. Liu, S. Qin and Z. Qi, Low-hit-zone frequency/time hopping sequence sets with large family size, IEEE Access, 7 (2019), 181733-181739.  doi: 10.1109/ACCESS.2019.2959718. [8] W. Ma and S. Sun, New designs of frequency hopping sequences with low hit zone, Des. Codes Cryptogr., 60 (2011), 145-153.  doi: 10.1007/s10623-010-9422-8. [9] X. Niu, D. Peng and Z. Zhou, New classes of optimal low hit zone frequency hopping sequences with new parameters by interleaving technique, IEICE Trans. Fundam. Electron., Commun. Comput Sci., E95-A (2012), 1835-1842. [10] X. Niu, D. Peng and Z. Zhou, New classes of optimal frequency hopping sequences with low hit zone, Adv. Math. Commun., 7 (2013), 293-310.  doi: 10.3934/amc.2013.7.293. [11] X. Niu, C. Xing, Y. Liu and L. Zhou, A construction of optimal frequency hopping sequence set via combination of multiplicative and additive groups of finite fields, IEEE Trans. Inform. Theory, 66 (2020), 5310-5315.  doi: 10.1109/TIT.2020.2972388. [12] X. Niu and Z. Zhou, New classes of optimal low hit zone frequency hopping sequences with new parameters, IEICE Trans. Fundam., E97-A (2014), 2567-2571.  doi: 10.1587/transfun.E97.A.2567. [13] D. Peng and P. Fan, Lower bounds on the hamming auto- and cross correlations of frequency-hopping sequences, IEEE Trans. Inform. Theory, 50 (2004), 2149-2154.  doi: 10.1109/TIT.2004.833362. [14] D. Peng, P. Fan and M. H. Lee, Lower bounds on the periodic hamming correlations of frequency hopping sequences with low hit zone, Sci. China Ser., 49 (2006), 208-218.  doi: 10.1007/s11432-006-0208-6. [15] C. U. Press, Signal Design for Good Correlation for Wireless Communication, Cryptography, and Radar, Signal design for good correlation for wireless communication, cryptography, and radar, 2005. [16] C. Wang, D. Peng and L. Zhou, New constructions of optimal frequency-hopping sequence sets with low-hit-zone, Int. J. Found. Comput. Sci., 27 (2016), 53-66.  doi: 10.1142/S0129054116500040. [17] X. Wang and P. Fan, A class of frequency hopping sequences with no hit zone, International Conference on Parallel and Distributed Computing, (2003), 896–898. [18] X. Ye and P. Fan, Two classes of frequency-hopping sequences with no hit zone, International Conference on Parallel and Distributed Computing, (2003), 304–306. [19] W. Yin, C. Xiang and F. W. Fu, Two constructions of low-hit-zone frequency-hopping sequence sets, Advances in Mathematics of Communications. doi: 10.3934/amc.2020110. [20] L. Zhou, D. Peng, H. Liang, C. Wang and Z. Ma, Constructions of optimal low-hit-zone frequency hopping sequence sets, Des. Codes Cryptogr., 85 (2017), 219-232.  doi: 10.1007/s10623-016-0299-z. [21] L. Zhou, D. Peng, H. Liang, C. Wang and Z. Ma, Generalized methods to construct low-hit-zone frequency-hopping sequence sets and optimal constructions, Crytogr. Commun., 9 (2017), 707-728.  doi: 10.1007/s12095-017-0211-3. [22] Z. Zhou, X. Tang, D. Peng and U. Parampalli, New constructions for optimal sets of frequency-hopping sequences, IEEE Trans. Inform. Theory, 57 (2011), 3831-3840.  doi: 10.1109/TIT.2011.2137290.
The Maximum Periodic Hamming Correlation of $S$
The Maximum Periodic Hamming Correlation of $S$
SOME OPTIMAL LHZ FHS SETS WITH OPTIMAL HAMMING CORRELATION
 Parameters $(N, q, l, L_H, H_m)$ Constraints According to the bound(3) According to the bound(4) Cyclical equivalence Ref. $(TN, q, Ml, \omega-1, TH_m)$ $M=\lceil \frac{N}{\omega}\rceil$, $gcd(l, N)=1$, $T=\lambda\omega+1$, $\lambda\ge1, T < lN.$ Optimal Not optimal family size Inequivalent [9] $(TN, q, M, \omega-1, TH_a)$ $M\omega=N, T\ge2$, $gcd(s, N)=1.$ Optimal Not optimal family size Inequivalent [10] $(lN, q, M, \omega l-1, lH_m)$ $M=\lceil \frac{N}{\omega}\rceil, gcd(l, N)=1.$ Optimal Not optimal family size Inequivalent [12] $(lN, q$, $M\left[\omega-(x-1)(l-1)\right]$, $l-1$, $lH_m)$ $x\ne1, 02l$, $xl-x2$, $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Thm.1 $(lN, q, Ml+nM(\omega-{xl}+x-1), l-n, {lH}_m)$ $02, \omega>2l, M=\lceil\frac{N}{\omega}\rceil$, $02$, $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Thm.3 $(s(q^m-1), q, l, L_H, s(q^{m-1}-1))$ $m\ge1, q^m-1=(L_H+1)l$, $gcd(s, q^m-1)=1, s\leq l$. Optimal Not optimal family size Equivalent [8] $(p^2(q-1), pq, pq, min\left\{p^2-1, q-2\right\}, p)$ $gcd(p, q-1)=1, 2p\leq q-1.$ Optimal Not optimal family size Inequivalent [1] $(p^2(p^2-1), p^2, p, p^2-2, p(p-1))$ $gcd(p^2, p^2-1)=1.$ Optimal Not optimal family size Inequivalent [21] $\big(q^k(q^m-1), q^k, nM(\omega-1), q^k-n, q^m\big)$ $1\leq k\leq m$, $02q^k$, $M=\lceil\frac{q^m-1}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.1 $(q^m-1, q^k, Tq^k, L_H, q^{m-k})$ $m\ge1$, $02l$, $M=\lceil\frac{q^m-1}{l\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.2 $(\frac{q^m-1}{l}, q^k, T, L_H, \frac{q^{m-k}-1}{l})$ $m\ge1, 0\frac{2e'}{l}$, $M=\lceil\frac{q^m-1}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.3 $p$ is a prime, $q$ is a prime power and $lpf(y)$ denotes the least prime factor of an integer $y>1$.
 Parameters $(N, q, l, L_H, H_m)$ Constraints According to the bound(3) According to the bound(4) Cyclical equivalence Ref. $(TN, q, Ml, \omega-1, TH_m)$ $M=\lceil \frac{N}{\omega}\rceil$, $gcd(l, N)=1$, $T=\lambda\omega+1$, $\lambda\ge1, T < lN.$ Optimal Not optimal family size Inequivalent [9] $(TN, q, M, \omega-1, TH_a)$ $M\omega=N, T\ge2$, $gcd(s, N)=1.$ Optimal Not optimal family size Inequivalent [10] $(lN, q, M, \omega l-1, lH_m)$ $M=\lceil \frac{N}{\omega}\rceil, gcd(l, N)=1.$ Optimal Not optimal family size Inequivalent [12] $(lN, q$, $M\left[\omega-(x-1)(l-1)\right]$, $l-1$, $lH_m)$ $x\ne1, 02l$, $xl-x2$, $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Thm.1 $(lN, q, Ml+nM(\omega-{xl}+x-1), l-n, {lH}_m)$ $02, \omega>2l, M=\lceil\frac{N}{\omega}\rceil$, $02$, $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Thm.3 $(s(q^m-1), q, l, L_H, s(q^{m-1}-1))$ $m\ge1, q^m-1=(L_H+1)l$, $gcd(s, q^m-1)=1, s\leq l$. Optimal Not optimal family size Equivalent [8] $(p^2(q-1), pq, pq, min\left\{p^2-1, q-2\right\}, p)$ $gcd(p, q-1)=1, 2p\leq q-1.$ Optimal Not optimal family size Inequivalent [1] $(p^2(p^2-1), p^2, p, p^2-2, p(p-1))$ $gcd(p^2, p^2-1)=1.$ Optimal Not optimal family size Inequivalent [21] $\big(q^k(q^m-1), q^k, nM(\omega-1), q^k-n, q^m\big)$ $1\leq k\leq m$, $02q^k$, $M=\lceil\frac{q^m-1}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.1 $(q^m-1, q^k, Tq^k, L_H, q^{m-k})$ $m\ge1$, $02l$, $M=\lceil\frac{q^m-1}{l\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.2 $(\frac{q^m-1}{l}, q^k, T, L_H, \frac{q^{m-k}-1}{l})$ $m\ge1, 0\frac{2e'}{l}$, $M=\lceil\frac{q^m-1}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.3 $p$ is a prime, $q$ is a prime power and $lpf(y)$ denotes the least prime factor of an integer $y>1$.
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