Frequency hopping sequences with optimal partial ... - Semantic Scholar
1
Frequency hopping sequences with optimal partial Hamming correlation
arXiv:1511.02924v2 [cs.IT] 11 Nov 2015
Jingjun Bao and Lijun Ji
Abstract—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 FHSs and FHS sets with optimal partial Hamming correlation. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. By virtue of this correspondence, some FHSs and FHS sets with optimal partial Hamming correlation are constructed from various combinatorial structures such as cyclic difference packings, and cyclic relative difference families. We also describe a direct construction and two recursive constructions for FHS sets with optimal partial Hamming correlation. As a consequence, our constructions yield new FHSs and FHS sets with optimal partial Hamming correlation. Index Terms—Frequency hopping sequences (FHSs), partial Hamming correlation, partition-type cyclic difference packings, cyclic relative difference families, cyclotomy.
I. I NTRODUCTION
F
REQUENCY hopping (FH) multiple-access is widely used in the modern communication systems such as ultrawideband (UWB), military communications, Bluetooth and so on, for example, [5], [17], [29]. In FH multipleaccess communication systems, frequency hopping sequences are employed to specify the frequency on which each sender transmits a message at any given time. An important component of FH spread-spectrum systems is a family of sequences having good correlation properties for sequence length over suitable number of available frequencies. The optimality of correlation property is usually measured according to the well-known Lempel-Greenberger bound and Peng-Fan bounds. During these decades, many algebraic or combinatorial constructions for FHSs or FHS sets meeting these bounds have been proposed, see [7]-[11], [14]-[15], [18], [20]-[21], [30][33], and the references therein. Compared with the traditional periodic Hamming correlation, the partial Hamming correlation of FHSs is much less well studied. Nevertheless, FHSs with good partial Hamming correlation properties are important for certain application scenarios where an appropriate window length shorter than the total period of the sequences is chosen to minimize the synchronization time or to reduce the hardware complexity of This work was supported by the NSFC under Grants 11222113, 11431003, and a project funded by the priority academic program development of Jiangsu higher education institutions. J. Bao and L. Ji are with the Department of Mathematics, Soochow University, Suzhou 215006, P. R. China. E-mail: [email protected]; [email protected].
the FH-CDMA receiver [16]. Therefore, for these situations, it is necessary to consider the partial Hamming correlation rather than the full-period Hamming correlation. In 2004, Eun et al. [16] generalized the Lempel-Greenberger bound on the periodic Hamming autocorrelation to the case of partial Hamming autocorrelation, and obtained a class of FHSs with optimal partial autocorrelation [28]. In 2012, Zhou et al. [34] extended the Peng-Fan bounds on the periodic Hamming correlation of FHS sets to the case of partial Hamming correlation. Based on m-sequences, Zhou et al. [34] constructed both individual FHSs and FHS sets with optimal partial Hamming correlation. Very recently, Cai et al. [6] improved lower bounds on partial Hamming correlation of FHSs and FHS sets, and based on generalized cyclotomy, they constructed FHS sets with optimal partial Hamming correlation. In this paper, we present some constructions for FHSs and FHS sets with optimal partial Hamming correlation. First of all, we give combinatorial characterizations of FHSs and FHS sets with optimal partial Hamming correlation. Secondly, by employing partition-type balanced nested cyclic difference packings, cyclic relative difference families, and cyclic relative difference packings, we obtain some FHSs and FHS sets with optimal partial Hamming correlation. Finally, we present two recursive constructions for FHS sets, which increase their lengths and alphabet sizes, and preserve their optimal partial Hamming correlations. Our constructions yield optimal FHSs and FHS sets with new and flexible parameters not covered in the literature. The parameters of FHSs and FHS sets with optimal partial Hamming correlation from the known results and the new ones are listed in the table. The remainder of this paper is organized as follows. Section II introduces the known bounds on the partial Hamming correlation of FHSs and FHS sets. Section III presents combinatorial characterizations of FHSs and FHS sets with optimal partial Hamming correlation. Section IV gives some combinatorial constructions of FHSs and FHS sets with optimal partial Hamming correlation by using partition-type balanced nested cyclic difference packings, cyclic relative difference families and cyclotomic classes. Section V presents a direct construction of FHS sets with optimal partial Hamming correlation. Section VI presents two recursive constructions of FHS sets with optimal partial Hamming correlation. Section VII concludes this paper with some remarks. II. L OWER BOUNDS
H AMMING FHS S AND FHS SETS In this section, we introduce some known lower bounds on the partial Hamming correlation of FHSs and FHS sets. ON THE PARTIAL
CORRELATION OF
2
KNOWN AND NEW FHSs WITH OPTIMAL PARTIAL HAMMING CORRELATION
Length
Alphabet size
q2 − 1
q
qm − 1
q m−1
ev
v
2v
v
2v + 1
v
2v 8v 32v 3v 4v 6v
2v+1 3 8v+1 3 32v+1 3 3v+1 4 4v+2 3
p(p − 1)
vq
l l
p
−1 d
L(q−1) qm −1
L
m
v
L v
vq m−1
ew r
l
1
[34]
1
Theorem 4.1
1
Theorem 4.2
L vw
f
m
(q−1)L v(qm −1)
m
[34] [6]
pm−1
L pm −1
d|(q − 1), gcd(d, m) = 1
f
1 1 1 1 1 1
3v
Source [16]
L Lv 4v L 16v L Lv 2v L
l
m
Constraints
1
d
L
(v − 1)w +
m
L(q−1) qm −1
m
v
2v + 1
m
Number of sequences
L q+1
q m−1
qm −1 d
evw
Hmax over correlation windows of length L m l
d
pi ≡ 1 (mod 12) is a prime pi ≡ 1 (mod 6) is a prime pi ≡ 1 (mod 12) is a prime pi ≡ 1 (mod 4) is a prime pi ≡ 7 (mod 12) is a prime pi ≡ 5 (mod 8) is a prime m≥2 q1 ≥ p1 > 2e, v > e(e − 2) and gcd(w, e) = 1 m > 1, and q m ≤ p1 − 1 for 1 ≤ i ≤ s
qm −1 d |pi
Theorem Theorem Theorem Theorem Theorem Theorem
4.4 4.4 4.4 4.4 6.5 6.5
Theorem 5.1 Corollary 6.6 Corollary 6.11
q is a prime power; ms 1 m2 v is an integer with prime factor decomposition v = pm with p1 < p2 < . . . < ps ; 1 p2 · · · ps e, f are integers such that e > 1 and e|gcd(p1 − 1, p2 − 1, . . . , ps − 1), and f = p1e−1 ; w is an integer with prime factor decomposition w = q1n1 q2n2 · · · qtnt with q1 < q2 < . . . < qt ; r is an integer such that r > 1 and r|gcd(e, q1 − 1, q2 − 1, . . . , qt − 1); p is a prime; d, m are positive integers.
For any positive integer l ≥ 2, let F = {f0 , f1 , . . . , fl−1 } be a set of l available frequencies, also called an alphabet. n−1 A sequence X = {x(t)}t=0 is called a frequency hopping sequence (FHS) of length n over F if x(t) ∈ F for all 0 ≤ t ≤ n−1 n−1 n − 1. For any two FHSs X = {x(t)}t=0 and Y = {y(t)}t=0 of length n over F , the partial Hamming correlation function of X and Y for a correlation window length L starting at j is defined by HX,Y (τ ; j|L) =
j+L−1 X t=j
h[x(t), y(t + τ )], 0 ≤ τ < n,
(1)
where L, j are integers with 1 ≤ L ≤ n, 0 ≤ j < n, h[a, b] = 1 if a = b and 0 otherwise, and the addition is performed modulo n. In particular, if L = n, the partial Hamming correlation function defined in (1) becomes the conventional periodic Hamming correlation [24]. If x(t) = y(t) for all 0 ≤ t ≤ n − 1, i.e., X = Y , we call HX,X (τ ; j|L) the partial Hamming autocorrelation of X; otherwise, we say HX,Y (τ ; j|L) the partial Hamming cross-correlation of X and
Y . For any two distinct sequences X, Y over F and given integer 1 ≤ L ≤ n, we define H(X; L) = max max {HX,X (τ ; j|L)} 0≤j 2e, e ≥ r ≥ 2 and v > e(e − 2), we have v(p1 − 1) > ve + (p1 −2)e(e−r) , which leads to r wve + (p1 −2)we(e−r) −e r 0< < 1. wv(p1 − 1) − 1 m l = e. By Theorem It follows from (7) that 2InM−(I+1)Il (nM−1)M 3.10, S ′ is a strictly optimal (ewv, f, e; (v − 1)w + ew r )-FHS set with respect to the bound (4). This completes the proof.
=
c+L−1 P
c+L−1 P
h[yi (t), yj (t + τ )]
t=c
h[h(ai − aj g τ )tiv , haj g τ τ iv ] · h[xi (htie ), xj (ht + τ ie )]
t=c
P
=
h[xi (ht0 + avie ), xj (ht0 + av + τ ie )]
0≤a<e t0 +av∈{c,c+1,...,c+L−1}
⌊
=
c+L−1−t0 ⌋ v P
a=⌈
h[xi (ht0 + aie ), xj (ht0 + a + τ ie )]
c−t0 ⌉ v
0 ≤ HXi ,Xj (hτ ie ; t0 + ⌈ c−t ⌉|⌈ L ⌉). v v
15
This completes the proof. Theorem 6.8: Let v be an odd integer of the form v = ms 1 m2 pm for s positive integers m1 , m2 , . . . , ms and s 1 p2 · · · ps distinct primes p1 , p2 , . . . , ps . Let e be a positive integer such that e|pi − 1 for 1 ≤ i ≤ s. If there is a strictly optimal (e, λ; l)-FHS with respect to the bound (2) over F such that λ|e and e > l, then the sequence Y in Construction C is a strictly optimal (ve, λ; vl)-FHS with respect to the bound (2). Proof: We first prove that Y in Construction C is a (ve, λ; vl)-FHS. Let X be a strictly optimal (e, λ; l)-FHS, we have λl ≤ e < (λ + 1)l. Let e = λl + ǫ, 0 ≤ ǫ < l, then ve = λ(vl) + vǫ. Since X meets the bound (2), by Lemma 2.1 we get ′ ′ λL L (e − ǫ)(e + ǫ − l) = (8) H(X; L′ ) = e l(e − 1) e for 1 ≤ L′ ≤ e. By Lemma 6.7, we have H(Y, ve) = max {HY,Y (τ ; 0|ve)} ≤ max {HX,X (hτ ie ; 0|e)} = λ.
1≤τ 1 be an integer such that e|pi − 1 for 0 < i ≤ s, and f = min{ pie−1 : 1 ≤ i ≤ s}. Let M be positive integer such that M ≤ f . If there is a strictly optimal (e, M, λ; l)-FHS set with respect to the bound (4) over F , A = {X0 , X1 , . . . , XM−1 }, such that λ|e and e > l ≥ 2, then B={Y0 , Y1 , . . . , YM−1 } generated by Construction C is a strictly optimal (ve, M, λ; vl)-FHS set with respect to the bound (4). Proof: Since A meets the bound (4), A is an optimal FHS set and H(A; e) = λ. By Lemma 6.7, we have H(B; ve) ≤ H(A; e) = λ. By Lemma 2.3, we have that B is a (ve, M, λ; vl)-FHS set. It remains to prove that B is a strictly optimal FHS set with respect to the bound (4). By the definition of I in Lemma 2.4, we have I = eM . Set eM = Il + r where r is the least l
I(r + 1 − l) I + M (Il + r − 1)M
=
L 2I(ve)M − (I + 1)I(vl) H(B; L) ≥ ve ((ve)M − 1)M L Iv(Il + 2r − l) = ve (Ivl + vr − 1)M L I Irv − vIl + I = + ve M (vIl + rv − 1)M I L · = ve M L ·λ , = ve
ve
H(Y ; L) ≤ H(X; ⌈ Lv ⌉) =
≤ 0. Therefore,
I M
(10)
mm
for 1 ≤ L ≤ ve. On the other hand, by Lemma 6.7, equality (8) and the fact that λe is an integer, we have
=
where the last equality holds by the property of ceiling function given in page 71 of [22] since λe is an integer. On the other hand, by Lemma 2.4, we get
By Lemma 3.2, we have H(Y, ve) = λ, i.e, Y is a (ve, λ; vl)FHS. It remains to prove that Y is strictly optimal with respect to the bound (2). On one hand, by Lemma 2.1, we have that mm l l (v(λl+ǫ)−vǫ)(v(λl+ǫ)+vǫ−vl) L H(Y ; L) ≥ ve vl(v(λl+ǫ)−1) l
I(Il + 2r − l) (Il + r − 1)M
On one hand, by Lemma 6.7 and equality (9), we have L λ L L · · λ , (11) )= = H(B; L) ≤ H(A; v e v ve
1≤τ 1 and let q be a prime power such that d|q m − 1. Let m1 , m2 , . . . , ms be s positive integers. Let p1 , p2 , . . . , ps be m s distinct primes such that q d−1 |pi − 1 and q m ≤ pi for all ms 1 m2 1 ≤ i ≤ s. Set vm= pm 1 p2 · · · ps . Then there exists a q −1 q−1 m−1 strictly optimal (v d , d, d ; vq )-FHS set with respect to the bound (4). q−1 d
VII. C ONCLUDING R EMARKS In this paper, a combinatorial characterization of strictly optimal FHSs and FHS set was obtained. Some new individual FHSs and FHS sets having strictly optimal Hamming correlation with respect to the bounds were presented. It would be nice if more individual FHSs and FHS sets whose partial Hamming correlation achieves the lower bounds could be constructed. It may be possible that some lower bounds on the partial Hamming correlation of FHSs could be improved from the combinatorial characterization.
16
R EFERENCES [1] T. M. Apostol, Introduction to Analytic Number Theory. New York, NY, USA: Springer-Verlag, 1976. [2] T. Beth, D. Jungnickel and H. Lenz, Design theory, Cambridge University Press, Cambridge, 1999. [3] M. Buratti and F. Zuanni, ”Explicit constructions for 1rotational Kirkman triple systems,” Utilitas.Math., Vol. 59, pp. 27-30, 2001. [4] M. Buratti and F. Zuanni, ”G-invariantly resolvable Steiner 2-designs arising from 1-rotational difference families,” Bull.Belg.Math.Soc., Vol. 5, pp. 221-235, 1998. [5] Bluetooth Special Interest Group (SIG), Washington, DC, USA. (2003, Nov.). Specification of the Bluetooth SysemsCore [Online]. Available: http://www.bluetooth.org [6] H. Cai, Z. Zhou, Y. Yang, and X. Tang, ”A new construction of frequency hopping sequences with optimal partial Hamming correlation,” IEEE Trans. Inf. Theory, vol. 60, no.9, pp. 5782-5790, 2014. [7] W. Chu and C. J. Colbourn, ”Optimal frequency-hopping sequences via cyclotomy,” IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 1139-1141, Mar. 2005. [8] J.-H. Chung, G. Gong, and K. Yang, ”New classes of optimal frequency-hopping sequences by composite lengths,” IEEE Trans. Inf. Theory, vol. 60, no. 6, pp. 3688-3697, Jun. 2014. [9] J.-H. Chung, Y. K. Han, and K. Yang, ”New classes of optimal frequency-hopping sequences by interleaving techniques,” IEEE Trans. Inf. Theory, vol. 55, no. 12, pp. 5783-5791, Dec. 2009. [10] J.-H. Chung and K. Yang, ”k-fold cyclotomy and its application to frequency-hopping sequences,” IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 2306-2317, Apr. 2011. [11] J.-H. Chung and K. Yang, ”Optimal frequency-hopping sequences with new parameters,” IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1685-1693, Apr. 2010. [12] C. J. Colbourn, Difference matrices, in: The CRC Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz (Editors), CRC Press, Boca Raton, FL, 2007, 411419. [13] C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo, and M. Mishima, ”Sets of frequency hopping sequences: bounds and optimal constructions,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3297-3304, Jul. 2009. [14] C. Ding, M. J. Moisio, and J. Yuan, ”Algebraic constructions of optimal frequency-hopping sequences,” IEEE Trans. Inf. Theory, vol. 53, no. 7, pp. 2606-2610, Jul. 2007. [15] C. Ding and J. Yin, ”Sets of optimal frequency-hopping sequences,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3741-3745, Aug. 2008 [16] Y.-C. Eun, S.-Y. Jin, Y.-P. Hong, and H.-Y. Song, ”Frequency hopping sequences with optimal partial autocorrelation properties,” IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2438-2442, Oct. 2004. [17] P. Fan, and M. Darnell, Sequence Design for Communications Applications. London, U.K.: Wiley, 1996. [18] R. Fuji-Hara, Y. Miao, and M. Mishima, ”Optimal frequency hopping sequences: A combinatorial approach,”
IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2408-2420, Oct. 2004. [19] S. Furino, Y. Miao, and J. Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence. Boca Raton, FL: CRC, 1996. [20] G. Ge, R. Fuji-Hara, and Y. Miao, ”Further combinatorial constructions for optimal frequency-hopping sequences,” J. Combin. Theory Ser. A, vol. 113, no. 8, pp. 1699-1718, Nov. 2006. [21] G. Ge, Y. Miao, and Z. Yao, ”Optimal frequency hopping sequences: Auto- and cross-correlation properties,” IEEE Trans. Inf. Theory, vol. 55, no. 2, pp. 867-879, Feb. 2009. [22] L. Graham, E. Knuth, and O. Patashnik, Concrete mathematics, Addison-Wesley Publishing Company, America, 1994. [23] M. Jimbo and S. A. Vanstone, ”Recursive constructions for resolvable and doubly resolvable 1-rotational Steiner 2-designs,” Utilitas Math., vol. 26 , pp. 45-61, 1984. [24] A. Lempel and H. Greenberger, ”Families of sequences with optimal Hamming-correlation properties,” IEEE Trans. Inf. Theory, vol. 20, no. 1, pp. 90-94, Jan. 1974. [25] E. H. Moore. Tactical Memoranda I-III. Amer. J. Math., vol. 18, pp. 264-303, 1896. [26] D. Peng and P. Fan, ”Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences,” IEEE Trans. Inf. Theory, vol. 50, no. 9, pp. 2149-2154, Sep. 2004. [27] T. Storer, Cyclotomy and Difference Sets, Chicago, IL: Markham, 1967. [28] P. Udaya and M. U. Siddiqi, ”Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,” IEEE Trans. Inf. Theory, vol. 44, no. 4, pp. 1492-1503, Jul. 1998. [29] L. Yang, and G.B. Giannakis, ”Ultra-wideband communications: An idea whose time has come,” IEEE Signal Process. Mag., vol. 21, no. 6, pp. 26-54, Nov. 2004. [30] Y. Yang, X. Tang, P. Udaya, and D. Peng, ”New bound on frequency hopping sequence sets and its optimal constructions,” IEEE Trans. Inf. Theory, vol. 57, no. 11, pp. 7605-7613, Nov. 2011. [31] X. Zeng, H. Cai, X. Tang, and Y. Yang, ”A class of optimal frequency hopping sequences with new parameters,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4899-4907, Jul. 2012. [32] X. Zeng, H. Cai, X. Tang, and Y. Yang, ”Optimal frequency hopping sequences of odd length,” IEEE Trans. Inf. Theory, vol. 59, no. 5, pp. 3237-3248, May 2013. [33] Z. Zhou, X. Tang, D. Peng, and U. Parampalli, ”New constructions for optimal sets of frequency-hopping sequences,” IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3831-3840, Jun. 2011. [34] Z. Zhou, X. Tang, X. Niu, and U. Parampalli, ”New classes of frequency hopping sequences with optimal partial correlation,” IEEE Trans. Inf. Theory, vol. 58, no. 1, pp. 453-458, Jan. 2012.
Frequency hopping sequences with optimal partial Hamming correlation
arXiv:1511.02924v2 [cs.IT] 11 Nov 2015
Jingjun Bao and Lijun Ji
Abstract—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 FHSs and FHS sets with optimal partial Hamming correlation. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. By virtue of this correspondence, some FHSs and FHS sets with optimal partial Hamming correlation are constructed from various combinatorial structures such as cyclic difference packings, and cyclic relative difference families. We also describe a direct construction and two recursive constructions for FHS sets with optimal partial Hamming correlation. As a consequence, our constructions yield new FHSs and FHS sets with optimal partial Hamming correlation. Index Terms—Frequency hopping sequences (FHSs), partial Hamming correlation, partition-type cyclic difference packings, cyclic relative difference families, cyclotomy.
I. I NTRODUCTION
F
REQUENCY hopping (FH) multiple-access is widely used in the modern communication systems such as ultrawideband (UWB), military communications, Bluetooth and so on, for example, [5], [17], [29]. In FH multipleaccess communication systems, frequency hopping sequences are employed to specify the frequency on which each sender transmits a message at any given time. An important component of FH spread-spectrum systems is a family of sequences having good correlation properties for sequence length over suitable number of available frequencies. The optimality of correlation property is usually measured according to the well-known Lempel-Greenberger bound and Peng-Fan bounds. During these decades, many algebraic or combinatorial constructions for FHSs or FHS sets meeting these bounds have been proposed, see [7]-[11], [14]-[15], [18], [20]-[21], [30][33], and the references therein. Compared with the traditional periodic Hamming correlation, the partial Hamming correlation of FHSs is much less well studied. Nevertheless, FHSs with good partial Hamming correlation properties are important for certain application scenarios where an appropriate window length shorter than the total period of the sequences is chosen to minimize the synchronization time or to reduce the hardware complexity of This work was supported by the NSFC under Grants 11222113, 11431003, and a project funded by the priority academic program development of Jiangsu higher education institutions. J. Bao and L. Ji are with the Department of Mathematics, Soochow University, Suzhou 215006, P. R. China. E-mail: [email protected]; [email protected].
the FH-CDMA receiver [16]. Therefore, for these situations, it is necessary to consider the partial Hamming correlation rather than the full-period Hamming correlation. In 2004, Eun et al. [16] generalized the Lempel-Greenberger bound on the periodic Hamming autocorrelation to the case of partial Hamming autocorrelation, and obtained a class of FHSs with optimal partial autocorrelation [28]. In 2012, Zhou et al. [34] extended the Peng-Fan bounds on the periodic Hamming correlation of FHS sets to the case of partial Hamming correlation. Based on m-sequences, Zhou et al. [34] constructed both individual FHSs and FHS sets with optimal partial Hamming correlation. Very recently, Cai et al. [6] improved lower bounds on partial Hamming correlation of FHSs and FHS sets, and based on generalized cyclotomy, they constructed FHS sets with optimal partial Hamming correlation. In this paper, we present some constructions for FHSs and FHS sets with optimal partial Hamming correlation. First of all, we give combinatorial characterizations of FHSs and FHS sets with optimal partial Hamming correlation. Secondly, by employing partition-type balanced nested cyclic difference packings, cyclic relative difference families, and cyclic relative difference packings, we obtain some FHSs and FHS sets with optimal partial Hamming correlation. Finally, we present two recursive constructions for FHS sets, which increase their lengths and alphabet sizes, and preserve their optimal partial Hamming correlations. Our constructions yield optimal FHSs and FHS sets with new and flexible parameters not covered in the literature. The parameters of FHSs and FHS sets with optimal partial Hamming correlation from the known results and the new ones are listed in the table. The remainder of this paper is organized as follows. Section II introduces the known bounds on the partial Hamming correlation of FHSs and FHS sets. Section III presents combinatorial characterizations of FHSs and FHS sets with optimal partial Hamming correlation. Section IV gives some combinatorial constructions of FHSs and FHS sets with optimal partial Hamming correlation by using partition-type balanced nested cyclic difference packings, cyclic relative difference families and cyclotomic classes. Section V presents a direct construction of FHS sets with optimal partial Hamming correlation. Section VI presents two recursive constructions of FHS sets with optimal partial Hamming correlation. Section VII concludes this paper with some remarks. II. L OWER BOUNDS
H AMMING FHS S AND FHS SETS In this section, we introduce some known lower bounds on the partial Hamming correlation of FHSs and FHS sets. ON THE PARTIAL
CORRELATION OF
2
KNOWN AND NEW FHSs WITH OPTIMAL PARTIAL HAMMING CORRELATION
Length
Alphabet size
q2 − 1
q
qm − 1
q m−1
ev
v
2v
v
2v + 1
v
2v 8v 32v 3v 4v 6v
2v+1 3 8v+1 3 32v+1 3 3v+1 4 4v+2 3
p(p − 1)
vq
l l
p
−1 d
L(q−1) qm −1
L
m
v
L v
vq m−1
ew r
l
1
[34]
1
Theorem 4.1
1
Theorem 4.2
L vw
f
m
(q−1)L v(qm −1)
m
[34] [6]
pm−1
L pm −1
d|(q − 1), gcd(d, m) = 1
f
1 1 1 1 1 1
3v
Source [16]
L Lv 4v L 16v L Lv 2v L
l
m
Constraints
1
d
L
(v − 1)w +
m
L(q−1) qm −1
m
v
2v + 1
m
Number of sequences
L q+1
q m−1
qm −1 d
evw
Hmax over correlation windows of length L m l
d
pi ≡ 1 (mod 12) is a prime pi ≡ 1 (mod 6) is a prime pi ≡ 1 (mod 12) is a prime pi ≡ 1 (mod 4) is a prime pi ≡ 7 (mod 12) is a prime pi ≡ 5 (mod 8) is a prime m≥2 q1 ≥ p1 > 2e, v > e(e − 2) and gcd(w, e) = 1 m > 1, and q m ≤ p1 − 1 for 1 ≤ i ≤ s
qm −1 d |pi
Theorem Theorem Theorem Theorem Theorem Theorem
4.4 4.4 4.4 4.4 6.5 6.5
Theorem 5.1 Corollary 6.6 Corollary 6.11
q is a prime power; ms 1 m2 v is an integer with prime factor decomposition v = pm with p1 < p2 < . . . < ps ; 1 p2 · · · ps e, f are integers such that e > 1 and e|gcd(p1 − 1, p2 − 1, . . . , ps − 1), and f = p1e−1 ; w is an integer with prime factor decomposition w = q1n1 q2n2 · · · qtnt with q1 < q2 < . . . < qt ; r is an integer such that r > 1 and r|gcd(e, q1 − 1, q2 − 1, . . . , qt − 1); p is a prime; d, m are positive integers.
For any positive integer l ≥ 2, let F = {f0 , f1 , . . . , fl−1 } be a set of l available frequencies, also called an alphabet. n−1 A sequence X = {x(t)}t=0 is called a frequency hopping sequence (FHS) of length n over F if x(t) ∈ F for all 0 ≤ t ≤ n−1 n−1 n − 1. For any two FHSs X = {x(t)}t=0 and Y = {y(t)}t=0 of length n over F , the partial Hamming correlation function of X and Y for a correlation window length L starting at j is defined by HX,Y (τ ; j|L) =
j+L−1 X t=j
h[x(t), y(t + τ )], 0 ≤ τ < n,
(1)
where L, j are integers with 1 ≤ L ≤ n, 0 ≤ j < n, h[a, b] = 1 if a = b and 0 otherwise, and the addition is performed modulo n. In particular, if L = n, the partial Hamming correlation function defined in (1) becomes the conventional periodic Hamming correlation [24]. If x(t) = y(t) for all 0 ≤ t ≤ n − 1, i.e., X = Y , we call HX,X (τ ; j|L) the partial Hamming autocorrelation of X; otherwise, we say HX,Y (τ ; j|L) the partial Hamming cross-correlation of X and
Y . For any two distinct sequences X, Y over F and given integer 1 ≤ L ≤ n, we define H(X; L) = max max {HX,X (τ ; j|L)} 0≤j 2e, e ≥ r ≥ 2 and v > e(e − 2), we have v(p1 − 1) > ve + (p1 −2)e(e−r) , which leads to r wve + (p1 −2)we(e−r) −e r 0< < 1. wv(p1 − 1) − 1 m l = e. By Theorem It follows from (7) that 2InM−(I+1)Il (nM−1)M 3.10, S ′ is a strictly optimal (ewv, f, e; (v − 1)w + ew r )-FHS set with respect to the bound (4). This completes the proof.
=
c+L−1 P
c+L−1 P
h[yi (t), yj (t + τ )]
t=c
h[h(ai − aj g τ )tiv , haj g τ τ iv ] · h[xi (htie ), xj (ht + τ ie )]
t=c
P
=
h[xi (ht0 + avie ), xj (ht0 + av + τ ie )]
0≤a<e t0 +av∈{c,c+1,...,c+L−1}
⌊
=
c+L−1−t0 ⌋ v P
a=⌈
h[xi (ht0 + aie ), xj (ht0 + a + τ ie )]
c−t0 ⌉ v
0 ≤ HXi ,Xj (hτ ie ; t0 + ⌈ c−t ⌉|⌈ L ⌉). v v
15
This completes the proof. Theorem 6.8: Let v be an odd integer of the form v = ms 1 m2 pm for s positive integers m1 , m2 , . . . , ms and s 1 p2 · · · ps distinct primes p1 , p2 , . . . , ps . Let e be a positive integer such that e|pi − 1 for 1 ≤ i ≤ s. If there is a strictly optimal (e, λ; l)-FHS with respect to the bound (2) over F such that λ|e and e > l, then the sequence Y in Construction C is a strictly optimal (ve, λ; vl)-FHS with respect to the bound (2). Proof: We first prove that Y in Construction C is a (ve, λ; vl)-FHS. Let X be a strictly optimal (e, λ; l)-FHS, we have λl ≤ e < (λ + 1)l. Let e = λl + ǫ, 0 ≤ ǫ < l, then ve = λ(vl) + vǫ. Since X meets the bound (2), by Lemma 2.1 we get ′ ′ λL L (e − ǫ)(e + ǫ − l) = (8) H(X; L′ ) = e l(e − 1) e for 1 ≤ L′ ≤ e. By Lemma 6.7, we have H(Y, ve) = max {HY,Y (τ ; 0|ve)} ≤ max {HX,X (hτ ie ; 0|e)} = λ.
1≤τ 1 be an integer such that e|pi − 1 for 0 < i ≤ s, and f = min{ pie−1 : 1 ≤ i ≤ s}. Let M be positive integer such that M ≤ f . If there is a strictly optimal (e, M, λ; l)-FHS set with respect to the bound (4) over F , A = {X0 , X1 , . . . , XM−1 }, such that λ|e and e > l ≥ 2, then B={Y0 , Y1 , . . . , YM−1 } generated by Construction C is a strictly optimal (ve, M, λ; vl)-FHS set with respect to the bound (4). Proof: Since A meets the bound (4), A is an optimal FHS set and H(A; e) = λ. By Lemma 6.7, we have H(B; ve) ≤ H(A; e) = λ. By Lemma 2.3, we have that B is a (ve, M, λ; vl)-FHS set. It remains to prove that B is a strictly optimal FHS set with respect to the bound (4). By the definition of I in Lemma 2.4, we have I = eM . Set eM = Il + r where r is the least l
I(r + 1 − l) I + M (Il + r − 1)M
=
L 2I(ve)M − (I + 1)I(vl) H(B; L) ≥ ve ((ve)M − 1)M L Iv(Il + 2r − l) = ve (Ivl + vr − 1)M L I Irv − vIl + I = + ve M (vIl + rv − 1)M I L · = ve M L ·λ , = ve
ve
H(Y ; L) ≤ H(X; ⌈ Lv ⌉) =
≤ 0. Therefore,
I M
(10)
mm
for 1 ≤ L ≤ ve. On the other hand, by Lemma 6.7, equality (8) and the fact that λe is an integer, we have
=
where the last equality holds by the property of ceiling function given in page 71 of [22] since λe is an integer. On the other hand, by Lemma 2.4, we get
By Lemma 3.2, we have H(Y, ve) = λ, i.e, Y is a (ve, λ; vl)FHS. It remains to prove that Y is strictly optimal with respect to the bound (2). On one hand, by Lemma 2.1, we have that mm l l (v(λl+ǫ)−vǫ)(v(λl+ǫ)+vǫ−vl) L H(Y ; L) ≥ ve vl(v(λl+ǫ)−1) l
I(Il + 2r − l) (Il + r − 1)M
On one hand, by Lemma 6.7 and equality (9), we have L λ L L · · λ , (11) )= = H(B; L) ≤ H(A; v e v ve
1≤τ 1 and let q be a prime power such that d|q m − 1. Let m1 , m2 , . . . , ms be s positive integers. Let p1 , p2 , . . . , ps be m s distinct primes such that q d−1 |pi − 1 and q m ≤ pi for all ms 1 m2 1 ≤ i ≤ s. Set vm= pm 1 p2 · · · ps . Then there exists a q −1 q−1 m−1 strictly optimal (v d , d, d ; vq )-FHS set with respect to the bound (4). q−1 d
VII. C ONCLUDING R EMARKS In this paper, a combinatorial characterization of strictly optimal FHSs and FHS set was obtained. Some new individual FHSs and FHS sets having strictly optimal Hamming correlation with respect to the bounds were presented. It would be nice if more individual FHSs and FHS sets whose partial Hamming correlation achieves the lower bounds could be constructed. It may be possible that some lower bounds on the partial Hamming correlation of FHSs could be improved from the combinatorial characterization.
16
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