New Family of -ary Sequences With Optimal Correlation ... - IEEE Xplore
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
[4] S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Aareas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [5] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, Sept. 1999. [6] G. Ganesan and P. Stoica, “Space-time block codes: A maximum SNR approach,” IEEE Trans. Inform. Theory, vol. 47, pp. 1650–1656, May 2001. [7] O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time block codes for complex signal constellations,” IEEE Trans. Inform. Theory, vol. 48, pp. 384–395, Feb. 2002. [8] W. Su and X.-G Xu, “On space-time block codes from complex orthogonal designs,” Wireless Personal Commun., vol. 25, no. 1, pp. 1–26, Apr. 2003. [9] W. Su and X.-G. Xia, “Quasiorthogonal space-time block codes with full diversity,” in Proc. IEEE GLOBECOM’02, vol. 2, 2002, pp. 1098–1102. [10] B. M. Hochwald and W. Sweldens, “Differential unitary space-time modulation,” IEEE Trans. Commun., vol. 48, pp. 2041–2052, Dec. 2000. [11] A. Shokrollahi, B. Hassibi, B. M. Hochwald, and W. Sweldens, “Representation theory for high-rate multiple-antenna code design,” IEEE Trans. Inform. Theory, vol. 47, pp. 2335–2367, Sept. 2001. [12] B. Hassibi and B. M. Hochwald, “Cayley differential unitary space-time codes,” IEEE Trans. Inform. Theory, vol. 48, pp. 1485–1503, June 2002. [13] X.-B. Liang and X.-G. Xia, “Unitary signal constellations for differential space-time modulation with two transmit antennas: Parametric codes, optimal designs and bounds,” IEEE Trans. Inform. Theory, vol. 48, pp. 2291–2322, Aug. 2002. [14] Q. Yan and R. Blum, “Robust space-time block coding for rapid fading channels,” in Proc. IEEE GLOBECOM, vol. 1, 2001, pp. 460–464. [15] S. Zummo and S. Al-Semari, “Space-time coded QPSK for rapid fading channels,” Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications (PIMRC), vol. 1, pp. 504–508, 2000. [16] W. Firmanto, B. Vucetic, and J. Yuan, “Space-time TCM with improved performance on fast fading channels,” IEEE Commun. Letters, vol. 5, pp. 154–156, Apr. 2001. [17] H. Bölcskei and A. Paulraj, “Performance of space-time codes in the presence of spatial fading correlation,” in Proc. Asilomar Conf. Signals, Systems and Computers, vol. 1, 2000, pp. 687–693. [18] M. P. Fitz, J. Grimm, and S. Siwamogsatham, “A new view of performance analysis techniques in correlated Rayleigh fading,” in Proc. IEEE Wireless Communications and Networking Conf. (WCNC), Sept. 1999, pp. 139–144. [19] S. Siwamogsatham, M. P. Fitz, and J. Grimm, “A new view of performance analysis of transmit diversity schemes in correlated Rayleigh fading,” IEEE Trans. Inform. Theory, vol. 48, pp. 950–956, Apr. 2002. [20] S. Siwamogsatham and M. P. Fitz, “Robust space-time codes for correlated Rayleigh fading channels,” IEEE Trans. Signal Processing, vol. 50, pp. 2408–2416, Oct. 2002. [21] Z. Safar and K. J. R. Liu, “Performance analysis of space-time codes over correlated Rayleigh fading channels,” in Proc. IEEE Int. Conf. Communications, vol. 5, Anchorage, AK, May 2003, pp. 3185–3189. [22] H. El Gamal, “On the robustness of space-time coding,” IEEE Trans. Signal Processing, vol. 50, pp. 2417–2428, Oct. 2002. [23] K. Leeuwin-Boulle and J. C. Belfiore, “The cutoff rate of time correlated fading channels,” IEEE Trans. Inform. Theory, vol. 39, pp. 612–617, Mar. 1993. [24] E. Baccarelli, “Performance bounds and cutoff rates for data channels affected by correlated randomly time-variant multipath fading,” IEEE Trans. Commun., vol. 46, pp. 1258–1261, Oct. 1998. [25] O. Y. Takeshita and D. J. Constello Jr, “New classes of algebraic interleavers for turbo-codes,” in Proc. IEEE Int. Symp. Information Theory, MIT, Cambridge, MA, 1998, p. 419. [26] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [27] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1991.
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New Family of -ary Sequences With Optimal Correlation Property and Large Linear Span Ji-Woong Jang, Young-Sik Kim, Jong-Seon No, Member, IEEE, and Tor Helleseth, Fellow, IEEE Abstract—For an odd prime and integers and such that = + 1) , a new family of -ary sequences of period 1 with op(2 timal correlation property is constructed using the -ary Helleseth–Gong sequences with ideal autocorrelation, where the size of the sequence family of all pairs is . That is, the maximum nontrivial correlation value + 1, which means of distinct sequences in the family does not exceed the family has optimal correlation in terms of Welch’s lower bound. The symbol distribution of the sequences in the family is enumerated. It is also + 2) shown that the linear span of the sequences in the family is ( except for the -sequence in the family. Index Terms—Family of sequences, optimal correlation, quences.
-ary se-
I. INTRODUCTION In the wireless communication systems employing code-division multiple-access (CDMA) scheme, a signature sequence is assigned to each user, which makes it possible to distinguish his signal from those of the other users. In design of sequences for CDMA system, the most important properties of the sequences are low periodic correlation between all pairs of distinct sequences and large family size. For an odd prime p, families of p-ary sequences of period pn 0 1 with optimal correlation property have been found, where the optimality of correlation values means that maximum magnitude of out-of-phase autocorrelation and cross-correlation values of any pairs of sequences of period pn 0 1 in the family is upper-bounded by Rmax = p + 1. Sidelnikov constructed a family of p-ary sequences with optimal correlation property and a family of prime-phase sequences with optimal correlation property was introduced by Kumar and Moreno [3]. By extending the alphabet size, Liu and Komo [7] constructed p-ary Kasami sequences. The family of p-ary bent sequences also has the optimal correlation property. Using the p-ary bent functions given by Kumar and Moreno, a family of balanced p-ary sequences with optimal correlation property was constructed by Moriuchi and Imamura [9]. The known families of n p-ary sequences of period p 0 1 with optimal correlation property are listed in Table I. The family size of the sequences due to Sidelnikov and to Kumar and Moreno are larger than that of the others in Table I. But the linear span of the sequences due to Sidelnikov and to Kumar and Moreno are much smaller than those of the others. In this correspondence, for an odd prime p and integers n; m; and k such that n = (2m + 1)k , a new family of p-ary sequences of period n p 0 1 with optimal correlation property is constructed using the p-ary Helleseth–Gong sequences with ideal autocorrelation, where the size of the sequence family is pn . That is, the maximum nontrivial correlation value Rmax of all pairs of distinct sequences in the family does not exceed p + 1, which means the family has optimal correlation with respect to Welch’s lower bound. The symbol distribution of the Manuscript received January 9, 2003; revised February 18, 2004. This work was supported in part by the Korean Ministry of Information and Communications and the Norwegian Research Council. J.-W. Jang, Y.-S. Kim, and J.-S. No are with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]). T. Helleseth is with the Department of Informatics, University of Bergen, N-5020 Bergen, Norway (e-mail: [email protected]). Communicated by K. G. Paterson, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2004.831837
0018-9448/04$20.00 © 2004 IEEE
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FAMILIES
OF
TABLE I SEQUENCES OF PERIOD pn CORRELATION PROPERTY Rmax = p
01
p-ARY
sequences in the family is enumerated. It is also shown that the linear span of the sequences in the family is (m + 2)n except for the m-sequence in the family. II. PRELIMINARIES Let S be the family of M p-ary sequences of period N = pn for an odd prime p given by
01
S = fsi (t) j 0 i M 0 1; 0 t N 0 1g: The correlation function of the sequences si (t) and sj (t) in S is written as
Ri;j ( ) =
N 01 t=0
!s (t+ )0s
(t)
! is a complex pth root of unity, 0 i; j M 0 1, and N 0 1. The maximum magnitude R of the correlation
where 0
max
values is defined as
Rmax = 0i;j Mmax 01;0 N 01 jRi;j ( )j where the maximization excludes the case of in-phase (i = j and = 0) autocorrelation. A family of p-ary sequences of period pn 0 1 is said to have optimal correlation property if Rmax does not exceed p + 1. Let z be an integer and Vzn the n-dimensional vector p space over the set of integers modulo z , Jz . Let !z = ej , j = 01. Let f (x) be a function from Vzn to Jz . The Fourier transform of the function f (x) is defined as
F () =
pz n 1
x2V
!zf (x)01x
2
;
Vzn
!f (x) = p1 n F ()!tr p 2F
Definition 1 (Kumar, Scholtz, and Welch [4]): A function f (x) from Vzn to Jz is said to be a generalized bent function if the Fourier coefficients F () of f (x) all have unit magnitude for any 2 Vzn . In this correspondence, we assume that the integer z is an odd prime p. Thus, Vpn is the n-dimensional vector space over the finite field Fp with p elements and f (x) is a function from Vpn to Fp . Olsen, Scholtz, and Welch introduced the trace transform for a function from F2 to F2 . Their definition can be generalized as follows. Definition 2 (Olsen, Scholtz, and Welch [10]): Let f (x) be a function from Fp to Fp . Then the trace transform of f (x) and its inverse transform are defined by (x)
;
2 Fp
(x)
;
x 2 Fp :
The elements x and in Fp can be related to the elements x and in Vpn as follows:
x= =
n i=1 n i=1
xi i ) x = ( x1 ; x 2 ; x 3 ; . . . ; x n ) i i ) = ( 1 ; 2 ; 3 ; . . . ; n )
where xi and i are in Fp and f 1 ; 2 ; 3 ; . . . ; n g is a basis of Fp over Fp . By replacing x in Fp by x in Vpn , the function f (x) from Fp to Fp makes the corresponding function f (x) from Vpn to Fp . It is known that the set of the trace transform values of the function f (x) is the same as that of the Fourier coefficients of the corresponding function f (x). Therefore, the function f (x) is a generalized bent function from Fp to Fp if and only if the corresponding function f (x) is a generalized bent function from Vpn to Fp . Let n = ek , where e and k are integers. A basis f 1 ; 2 ; 3 ; . . . ; e g of Fp over Fp is said to be trace-orthogonal if n
trk ( i j ) =
ai ; 0;
if i = j otherwise
where ai 2 Fp3 . It is known that for any positive integer e and odd prime p, there exists a trace-orthogonal basis of Fp over Fp [12]. Suppose elements x and in Fp can be related to the elements x and in the e-dimensional vector space Vpe over Fp as follows:
x= =
where xT denotes the transpose of x.
F () = p1 n !f (x)0tr p x2F
WITH OPTIMAL
+1
e i=1 e i=1
xi i ) x = ( x1 ; x 2 ; x 3 ; . . . ; x e )
(1)
i i ) = ( 1 ; 2 ; 3 ; . . . ; e )
(2)
where xi and i are in Fp . Then, if we choose the basis to be traceorthogonal, we have the relation e n trk (x) = ai i xi : (3) i=1
Let 0i = ai i for 1 i e and 0 = (01 ; 02 ; 03 ; . . . ; 0e ). Then the relation in (3) can be rewritten as e n trk (x) = 0i xi = 0 1 xT : i=1 Suppose that using (1) and (2), a function f (x) from Fp to Fp is related to the corresponding function f (x) from Vpe to Fp . Then the trace transform in Definition 2 can be modified into the trace transform defined in the intermediate field as follows.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
Definition 3: Let n = ek . Let f (x) be a function from Vpe to Fp . Then the trace transform of trk1 (f (x)) and its inverse transform are defined as
FM () = p
1
!tr
(f (x))
=
p1pn
(f (x))
!tr
pn x2V
0tr (1x ) ;
2 Vpe
FM ()!tr (1x ) ;
2V
x 2 Vpe :
F () = FM (0 ):
Q(x) =
i=1 j =1
bij xi xj
(4)
where bij 2 Fp . It is known from Dickson [1] that for an odd integer , any quadratic form with rank can be transformed by linear transformations into a canonical form
Q(x) =
i=1
rxi2
(5)
where e and r = 1 or a quadratic nonresidue in Fp . It is clear that Q(x) and the corresponding function Q(x) in (4) and (5) have the same rank. From Definition 3, we easily derive the following lemma. Lemma 4: tr1k (Q(x)) is a quadratic p-ary bent function from Vpe to Fp if and only if the quadratic function Q(x) from Vpe to Fp has full rank e. Proof: It is straightforward from Deligne’s theorem [3] that a p-ary quadratic function with full rank is a bent function. To prove the converse, let tr1k (Q(x)) be a p-ary quadratic bent function, where Q(x) has rank , < e. From Dickson [1, p. 157], k tr1 (Q(x)) can be rewritten as k
k
k
tr1 (Q(x)) = tr1 (Q(x1 ; x2 )) = tr1
i=1
ai x2i
(6)
where ai 2 Fp3 , x1 is a -tuple vector on Vp , and x2 is an (e 0 )tuple vector on Vpe0 . Then the trace transform of (6) is given as
QM () =
p1pn
=
p1pn
p1pn
=
=
x
p1pn
x
x2V x2V
2V 2V
0tr
wtr
(Q(x))
wtr
(
wtr
(
)
0tr
ax
)
x
(
(
1x
(
1x
)
x
2V
wtr
w0tr
(
2V ax
k
i=1
ai x2i :
k 6 pppn 0;
2V
x
6 pk e0 ; (
)
w0tr
(
1x
)
if 2 6= 0 if 2 = 0,
(8)
which means that tr1k (Q(x)) is not a bent function on Vpe . Thus. every quadratic bent function has full rank. Helleseth and Gong introduced new p-ary sequences with ideal autocorrelation, which are referred to as Helleseth–Gong (HG) sequences [2]. Theorem 5 (Helleseth and Gong [2] ): Let be a primitive element of Fp . Let n = (2m + 1)k and let s, 1 s 2m be an integer such that gcd(s; 2m + 1) = 1. Define b0 = 1, bis = (01)i , and bi = b2m+10i for i = 1; 2; . . . ; m. Let u0 = b0 =2 = (p + 1)=2 and ui = b2i for i = 1; 2; . . . ; m where all indexes of the bi ’s are taken k n mod 2m + 1 and q = p . Then the HG sequence of period p 0 1 is given by m
s(t) = tr1n
l=0
t
ul
:
(9)
The HG sequence has an ideal two-level autocorrelation. We provide an example of HG sequence as follows. Example 6: For p = 3, let m = 2, k = 2, and n = (2m +1)k = 10. Let s = 2 so gcd(s; 2m + 1) = 1. Then the parameters bi and ui are given as follows:
b0 = 1; b1 = b6 = (01)3 = 01; b2 = (01)1 = 01 b3 = b8 = (01)4 = 1; b4 = (01)2 = 1 u0 = b0 = 2 ; u 1 = b2 = 2 ; u 2 = b4 = 1 : 2
Thus, the HG sequence of period 310 0 1 is given as
s(t) = tr1n( t ) + tr1n( 41t ) 0 tr1n( 3281t )
where is a primitive element in F3 . Let h(x) be the HG polynomial defined by m
l=0
ul x
;
x 2 Fp :
Then the HG sequence in (9) can be rewritten as
)
)
x
w0tr
QM () =
h (x ) =
)
0tr
ax
(
1x
0
k
tr1 (Q (x1 )) = tr1
=
That is, the set of the trace transform values of the function tr1k (f (x)) is the same as that of the corresponding function tr1k (f (x)). Therefore, if the trace transform of the function tr1k (f (x)) or tr1k (f (x)) only takes values of unit magnitude, the functions tr1k (f (x)) and tr1k (f (x)) become generalized bent functions. Let Q(x) be a quadratic form from Fp to Fp . Using (1), the quadratic form Q(x) can be expressed as e
Let
Then the inner summation in (7) is the trace transform of tr1k (Q0 (x1 )). Clearly, Q0 (x1 ) has full rank on Vp and thus tr1k (Q0 (x1 )) is a bent function on Vp . Therefore, the inner summation of (7) is equal to 6 pk . Thus, we can rewrite (7) as follows:
It is clear that the trace transform of a function tr1k (f (x)) from Vpe to Fp is related to the trace transform F () of the corresponding function k tr1 (f (x)) from Fp to Fp as follows:
e
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0tr
)
(
(
1x
)
1x ) : (7)
s(t) = tr1n(h( t )); 0 t pn 0 2: We can construct a new p-ary sequence s for each 2 Fp
using the
HG sequence as follows:
s (t) = tr1n( t ) + tr1n(h( 2t )) n
= tr1
t +
m
l=0
ul
( q
+1)t
It is clear that the sequences in (10) have period pn 0 1.
:
(10)
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
III. NEW CONSTRUCTION OF A FAMILY OF p-ARY SEQUENCES Using the new p-ary sequence defined in (10), a family of p-ary sequences with family size pn and optimal correlation property can be constructed as follows. Theorem 7: Let s (t) be the p-ary sequence defined in (10). Then the family of p-ary sequences given by
S = f s ( t) j 2 F p ;
Rij ( ) + 1 =
02
t=0
=
w
s
wtr
x2F
0s
(t+ )
(t)
(cx+h(
c x
c = 1 , i
Case (iii)
x
x2F
w
y2F
w w
y2F
(h(r
wtr
tr (h(r
Using the property that for r
=
m l=0
aql
0h(r
a c x u y
)
0h(r
)
y
h(u y
tr (tr (r
0r
h(y
)
h(u y
0r
)
m l=0
aql
6
01)x)
))+tr ((c
y)
))+tr (
m
n
= rj trk
ul [ri uq
l=0 m
+1
ul [ ri uq rj l=0
+1
+1
= 0:
m +
l=0
al z q
+1
=0
m
m l=0
+
l=0
al z q
al z q
=0
(13)
0 1 zq
=0
= 0:
+1
y)
)))+tr (
l=0
ul
ri u(q rj
0 1 zq
+1)q
ri u(q rj
+
+1)
and thus we have 2m
bl ri u(q rj l=0
0 1 zq
+1)
=0
where u0 = b2 and ul = b2l = b2m+102l . To prove that the rank of Q(y) is = 2m + 1, we have to show that the equation q
+1
0 1 zq
=0
has z = 0 as its only solution for any rr ( aa c)2 = 6 1. This is already proved in [2]. From the condition, it is clear that
y)
))+tr (
0 r j ]y q
+1
al z q
al z q
l=0
bl ri ai c rj aj l=0
:
(12)
ri ai c rj aj
2
=
bi c2 bj
6= 1
and thus we have proved the theorem. Clearly, the sequence in (10) becomes a p-ary m-sequence when = 0. For 6= 0, (10) can be transformed into the form
Q(y) = trkn (ri h(u2 y2 ) 0 rj h(y2 )) n
l=0
+
2m
:
Let Q(y ) be the quadratic function defined by
= trk
m )+
zq
and
m
a x
h(y
l=0
al y q
0 1]. Then
Equations (13) can be rewritten as
2 Fp , h(rx) = rh(x), we have
tr (r
= trk
m
zq
n
6
y2F
l=0
al (yq z + yz q
trk
2
Rij ( ) + 1 =
y
(11)
Since nk is an odd integer, it is clear that a quadratic nonresidue in Fp is also a quadratic nonresidue in Fp . Thus, i and j can be Fp and ri expressed as i = ri a2i and j = rj a2j , where ai ; aj and rj are 1 or quadratic nonresidues in Fp . We assume that j = 0 and thus aj = 0. Let u = aa c and y = aj x. Then the cross-correlation
=
m
=
function in (11) can be rewritten as
m
n
+1
+1
which is equivalent to
))
The condition excludes the case of i = j = 0. We will prove it assuming j 6= 0; the proof in the case i 6= 0 is similar.
Rij ( ) + 1 =
n
+1
0x0h(
a l (y + z ) q
uq
Raising the first term to the q 2m+102l power, we have trk
j : It is clear that Rij ( ) = pn 0 1. c 6= 1, i c2 = j : It is also clear that Rij ( ) = 01. i c2 6= j :
Case (ii)
l=0
n
)
. Let al = ul [ rr
can be modified into trk
where c = 2 Fp3 . Then the proof can be classified into the following three cases. Case (i)
m
n
trk
t pn 0 2g
0
has the optimal correlation property with Rmax = p + 1. Proof: The cross-correlation function of two p-ary sequences s (x) and s (x) in S can be rewritten as p
Q(y + z ) = Q(y) for all y 2 Fp the condition
+1
0 1]yq
+1
s i (t) = tr1n t+i + :
Then (12) corresponds to the trace transform of the quadratic function tr1k (Q(y )) from Fp to Fp as in Definition 2. If the quadratic function tr1k (Q(y )) is a p-ary bent function, then we have jRij ( ) + 1j = p . From Definition 3 and Lemma 4, if Q(y) has full rank, then k tr1 (Q(y )) is bent. Thus, it is sufficient to show that Q(y ) has full rank 2m + 1. It is already known that the rank of the quadratic function Q(y ) is equal to if q 2m+10 is the number of elements z 2 Fp satisfying
m l=0
ul (q
+1)t
(14)
where 0 i p 201 0 1 and is 1 or a quadratic nonresidue in Fp . Then the family of p-ary sequences defined in Theorem 7 can be rewritten as
S= si (t) j 0 i p
n
0 1 0 1; 0 t p n 0 2
2
Let W be a subset of Vpn and g (x) a function from W we define the notation
wc = j fx j g(x) = c; x 2 W g j;
n
t
tr1 ( )
:
(15) to Fp . Then
c 2 Fp :
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
Then the symbol distribution of g (x) is defined as the ordered p-tuple (w0 ; w1 ; w2 ; . . . ; wp01 ). It is interesting to find the symbol distribution of the sequences in the family given in (15). Using the result in Section II, it is easy to derive that the quadratic part of the sequence in (14) can be transformed into the quadratic form
f (x) = a1 x21 + a2 x22 + 1 1 1 + an x2n
(16)
mapping from Vpn nf0g to Fp . It is clear that the quadratic form in (14) is bent and thus f (x) has full rank, that is, ai 2 Fp3 . Let (x) denote the quadratic character of x defined by +1; 0;
(x ) =
0 1;
if x is a quadratic residue if x = 0 if x is a quadratic nonresidue.
Dickson [1] derived the number of solutions in Vpn to quadratic equations over Fp . Theorem 8 (Dickson [1, pp. 47–48]): Let p be an odd prime and n = (2m + 1)k = 2h be an even integer. Then the number of solutions n (x1 ; x2 ; . . . ; xn ) in Vp of the quadratic equation
a1 x21 + a2 x22 + 1 1 1 + an x2n = c
2 Fp3 and c 2 Fp , is given by 2h01 h h01 vc = p2h01 + (ph010 p ); p 0 p ; where = ((01)h a1 a2 1 1 1 an ).
where aj
if c = 0 if c 6= 0
(17)
Theorem 9 (Dickson [1, pp. 47–48]): Let p be an odd prime and = (2m + 1)k = 2h + 1 be an odd integer. Then the number of solutions (x1 ; x2 ; . . . ; xn ) in Vpn to the quadratic equation
n
a1 x21 + a2 x22 + 1 1 1 + an x2n = c
where aj
2 Fp3 and c 2 Fp , is given as
vc = p2h + ph
(18)
where = ((01)h a1 a2 1 1 1 an c). It is clear that the new sequence family includes one p-ary quence, which has the symbol distribution (p
n01
m-se-
0 1; pn01 ; pn01 ; . . . ; pn01 ):
We denote this distribution by D1 = 1. The distribution of Dc for the new sequence family defined in (15) is given in the following theorem. Theorem 10: Let Dc be the number of sequences with symbol distribution (vc 0 1; vc+1 ; . . . ; vp01 ; v0 ; v1 ; . . . ; vc01 ) in the sequence family in (15). Then
Dc =
1;
01 2 ; 2;
v v
if c = 1 if c = 0 if c 2 Fp3
where vc is defined in (17) and (18). Proof: From the balance property of a p-ary m-sequence, it is clear that D1 = 1. Since gcd(q 2l + 1; pn 0 1) = 2 for all l, the period of the quadratic part in (14) is p 201 . As i varies over 0 i pn 0 2 in (14), the p 201 different sequences of period pn 0 1 can be generated, where each sequence occurs exactly twice. Using (16) and the transformation of the linear part tr1n ( t+i ), s i (t) in (14) can be transformed into the function from Vpn nf0g to Fp given by
a1 x21 + a2 x22 + 1 1 1 + an x2n + d1 x1 + d2 x2 + 1 1 1 + dn xn
(19)
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where ai
2 Fp3 and di 2 nFp . As i varies over 0 i pn 0 2, every
(d1 ; d2 ; d3 ; . . . ; dn )
in Vp occurs exactly once except for 0. We can
modify (19) into
2 2 2 a 1 x 1 + d 1 + a 2 x 2 + d 2 + 1 1 1 + an x n + d n 0 c 2a 1 2a 2 2a n d 0 and putting xi = xi + 2a , we have a1 x0 2 + a2 x0 2 + 1 1 1 + an x0 2 0 c (20) 1
where
2 c = d1 4a 1
2
+
n
d22 4a 2
+
2
1 1 1 + 4dann
is in Fp . From (17) and (18), the symbol distribution of f (x) in (16) is given as (v0 0 1; v1 ; v2 ; . . . ; vp01 ), where excluding x = 0 makes v0 0 1 instead of v0 . Then for a fixed c in Fp , the symbol distribution of (20) is given as (vc 0 1; vc+1 ; . . . ; vp01 ; v0 ; v1 ; . . . ; vc01 ) (21) where for x = 0, (19) takes the value 0 and thus excluding x = 0 in (19) makes vc 0 1 instead of vc . In order to find the number of sequences with the symbol distribution in (21) in the sequence family defined in (15), we have to find the number of solutions (d1 ; d2 ; d3 ; . . . ; dn ) in Vpn nf0g satisfying the quadratic equation
d12 + d22 + 1 1 1 + dn2 = c 4a 1 4a 2 4a n as i varies over 0 i pn 02. In fact, this corresponds to 2Dc because each sequence occurs exactly twice as i varies over 0 i pn 0 2, which is already calculated in (17) and (18). We have to exclude the solution 0 for c = 0 because (d1 ; d2 ; d3 ; . . . ; dn ) is in Vpn nf0g. Thus, we have D0 = v 201 and Dc = v2 for c 2 Fp3 .
Kumar and Moreno [3] calculated the correlation distribution of the sequences in their sequence family by finding the correlation values of two sequences in the family, which corresponds to Fourier coefficients of the quadratic forms with full rank. In their paper, the correlation distribution was derived by using the full rank property of the quadratic function. Therefore, the correlation distribution of the sequences in the new family defined in Theorem 7 is the same as that of the sequence family by Kumar and Moreno. The linear span of the new p-ary sequences s (t) in Theorem 7 is derived as in the following theorem. Theorem 11: The linear span of the sequence s (t) defined in Theorem 7 is equal to (m + 2)n. Proof : For di = q 2i + 1; 0 i m in (10), gcd(di ; pn 0 1) = d 2. It is clear that does not belong to any subfield of Fp . Thus, the 2 coset of in Fp has size n and so does that of each element d . Therefore, the linear span of s (x) is (m + 2)n. REFERENCES [1] L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory. New York: Dover, 1958. [2] T. Helleseth and G. Gong, “New nonbinary sequences with ideal twolevel autocorrelation function,” IEEE Trans. Inform. Theory, vol. 48, pp. 2868–2872, Nov. 2002. [3] P. V. Kumar and O. Moreno, “Prime-phase sequences with periodic correlation properties better than binary sequences,” IEEE Trans. Inform. Theory, vol. 37, pp. 603–616, May 1991. [4] P. V. Kumar, R. A. Scholtz, and L. R. Welch, “Generalized bent functions and their properties,” J. Combin. Theory, ser. A, vol. 40, pp. 90–107, 1985. [5] A. Lempel and M. Cohn, “Maximal families of bent sequences,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 865–868, Nov. 1982. [6] R. Lidl and H. Niederreiter, Finite Fields, ser. Encyclopedia of Mathematics and Its Applications. Reading, MA: Addison-Wesley, 1983, vol. 20.
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[7] S.-C. Liu and J. F. Komo, “Nonbinary Kasami sequences over GF( ),” IEEE Trans. Inform. Theory, vol. 38, pp. 1409–1412, July 1992. [8] F. J. McWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [9] T. Moriuchi and K. Imamura, “Balanced nonbinary sequences with good periodic correlation properties obtained from modified Kumar–Moreno sequences,” IEEE Trans. Inform. Theory, vol. 41, pp. 572–576, Mar. 1995. [10] J. D. Olsen, R. A. Scholtz, and L. R. Welch, “Bent-function sequences,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 858–864, Nov. 1982. [11] O. S. Rothaus, “On bent functions,” J. Combin. Theory, ser. A, vol. 20, pp. 300–305, 1976. [12] G. Seroussi and A. Lempel, “Factorization of symmetric matrices and trace-orthogonal bases in finite fields,” SIAM J. Comput., vol. 9, no. 4, pp. 758–767, Nov. 1980. [13] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications. Rockville, MD: Computer Sci., 1985, vol. 1. [14] L. R. Welch, “Lower bounds on the maximal cross correlation of signals,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 396–399, May 1976.
II. PRELIMINARIES Let R = GR (2 ; m) denote the Galois ring of characteristic with 2lm elements. Let be an element in GR (2l ; m) that generates the Teichmüller set T of GR (2l ; m). Specifically, let T = f0; 1; ; 2 ; . . . ; 2 02g and T 3 = f1; ; 2 ; . . . ; 2 02 g. The l 2-adic expansion of x 2 GR (2 ; m) is given by l
2
l
x = x0 + 2x1 + 1 1 1 + 2 01 x 01 where x0 ; x1 ; . . . ; x 01 2 T . The Frobenius operator F is defined for such an x as F (x0 + 2x1 + 1 1 1 + 2 01 x 01 ) = x20 + 2x21 + 1 1 1 + 2 01 x201 and the trace Tr, from GR (2 ; m) down to 2 , as 01 Tr(x) := F (x): l
l
l
l
l
m
j
j =0
We also define another trace tr from 2
S
Let MSB : n 2
Index Terms—Autocorrelation, Galois rings, maximum-length sequences over rings, most significant bit.
l;m
:=
Maximum-length (ML) sequences over the rings 2 were introduced by Dai [1], motivated by cryptographic applications. Using the 2-adic expansion in the local ring 2 , various binary sequences can be defined from a given ML sequence over 2 . Of particular interest, from the nonlinearity viewpoint, is the most significant bit [6]. In a cryptographic usage such as, for instance, a key in a one-time-pad system, such a deterministic sequence should look as random as possible. It should be as balanced as possible the frequency of each of the two symbols 0, 1 should be close to 1=2. It should be uncorrelated with itself: low inner products with its successive shifts. Recently some estimates were given for the autocorrelation and the imbalance of the binary sequences in the title [4], using the Galois rings character sum estimates of [7]. The aim of this note is, still using [7], to sharpen the autocorrelation and imbalance estimate of [4] by a factor exponential in l. Our approach uses the Discrete Fourier Transform techniques of [8], [9]. Manuscript received July 30, 2003; revised February 11, 2004. The material in this correspondence was presented in part at the Kodierungtheorie, Oberwolfach, Germany, December 2003. P. Solé is with the CNRS-I3S, ESSI, 06 903 Sophia Antipolis, France (e-mail: [email protected]). D. Zinoviev is with the CNRS-I3S, ESSI, 06 903 Sophia Antipolis, France, and with the Institute for Problems of Information Transmission, Russian Academy of Sciences, GSP-4, Moscow, 101447, Russia (e-mail: zinoviev@ essi.fr, [email protected]). Communicated by K. G. Patterson, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2004.831858
j =0
down to 2 as
x2 :
(Tr(
t N 01 t=0
))
j 2 R3 :
(1)
! n be the most significant bit (MSB) map, i.e., MSB(x + 2x + 1 1 1 + 2l0 xl0 ) := xl0 : 2
0
1
1
1
1
III. IMBALANCE AND AUTOCORRELATION PROPERTIES Let l be a positive integer (without loss of generality we assume that l 4) and ! = e2i=2 be a primitive 2l th root of 1 in . Let k be the additive character of
I. INTRODUCTION
01
m
Throughout this note, we let n = 2m and R3 = R n 2R. Let = (1 + 2) 2 R, where 2 T and 2 R3 . Assume 1 + 2 is of order 2l01 . Since is of order 2m 0 1 then is an element of order N = 2l01 (2m 0 1). Following [4, Lemma 2], we define the sequence
Patrick Solé, Member, IEEE, and Dmitrii Zinoviev Abstract—The imbalance and the autocorrelation of the binary sequences in the title are explored by combining the local Weil bound with spectral analysis. Recent estimates on these quantities are improved by a factor of order 2 for large .
l
l
tr(x) :=
The Most Significant Bit of Maximum-Length Sequences : Autocorrelation and Imbalance Over
l
2
such that
(x) = !kx : Let : ! f61g be the mapping (t) = (01)c , where c is the , i.e., it maps 0; 1; . . . ; 2l0 0 1 to +1 most significant bit of t 2 l 0 l 0 l and 2 ; 2 + 1; . . . ; 2 0 1 to 01. Our goal is to express this map k
2
1
1
2
1
as a linear combination of characters. Recall the Fourier transformation formula on 2
=
2
01
j =0
j j ;
where j
0 = 21l (x) j (0x): 2
1
(2)
x=0
2 R = GR (2l; m), we denote by 9 the character 9 : R ! 3 ; x 7! ! x : Note that for the previously defined k and 9 we have k (Tr( x)) = 9 k (x): For all
Tr(
)
(3)
The following lemma follows from [7].
2 R, 6= 0, we have p 9 (x) (2l0 0 1) 2m :
Lemma 3.1: For all
1
x2T
Proof: We restate [7, Theorem 1] for the special Galois ring of concern here. Let f (X ) denote a polynomial in R[X ] and let
f (x) = F0 (x) + 2F1 (x) + 1 1 1 + 2l01 Fl01 (x)
0018-9448/04$20.00 © 2004 IEEE
[4] S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Aareas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [5] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, Sept. 1999. [6] G. Ganesan and P. Stoica, “Space-time block codes: A maximum SNR approach,” IEEE Trans. Inform. Theory, vol. 47, pp. 1650–1656, May 2001. [7] O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time block codes for complex signal constellations,” IEEE Trans. Inform. Theory, vol. 48, pp. 384–395, Feb. 2002. [8] W. Su and X.-G Xu, “On space-time block codes from complex orthogonal designs,” Wireless Personal Commun., vol. 25, no. 1, pp. 1–26, Apr. 2003. [9] W. Su and X.-G. Xia, “Quasiorthogonal space-time block codes with full diversity,” in Proc. IEEE GLOBECOM’02, vol. 2, 2002, pp. 1098–1102. [10] B. M. Hochwald and W. Sweldens, “Differential unitary space-time modulation,” IEEE Trans. Commun., vol. 48, pp. 2041–2052, Dec. 2000. [11] A. Shokrollahi, B. Hassibi, B. M. Hochwald, and W. Sweldens, “Representation theory for high-rate multiple-antenna code design,” IEEE Trans. Inform. Theory, vol. 47, pp. 2335–2367, Sept. 2001. [12] B. Hassibi and B. M. Hochwald, “Cayley differential unitary space-time codes,” IEEE Trans. Inform. Theory, vol. 48, pp. 1485–1503, June 2002. [13] X.-B. Liang and X.-G. Xia, “Unitary signal constellations for differential space-time modulation with two transmit antennas: Parametric codes, optimal designs and bounds,” IEEE Trans. Inform. Theory, vol. 48, pp. 2291–2322, Aug. 2002. [14] Q. Yan and R. Blum, “Robust space-time block coding for rapid fading channels,” in Proc. IEEE GLOBECOM, vol. 1, 2001, pp. 460–464. [15] S. Zummo and S. Al-Semari, “Space-time coded QPSK for rapid fading channels,” Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications (PIMRC), vol. 1, pp. 504–508, 2000. [16] W. Firmanto, B. Vucetic, and J. Yuan, “Space-time TCM with improved performance on fast fading channels,” IEEE Commun. Letters, vol. 5, pp. 154–156, Apr. 2001. [17] H. Bölcskei and A. Paulraj, “Performance of space-time codes in the presence of spatial fading correlation,” in Proc. Asilomar Conf. Signals, Systems and Computers, vol. 1, 2000, pp. 687–693. [18] M. P. Fitz, J. Grimm, and S. Siwamogsatham, “A new view of performance analysis techniques in correlated Rayleigh fading,” in Proc. IEEE Wireless Communications and Networking Conf. (WCNC), Sept. 1999, pp. 139–144. [19] S. Siwamogsatham, M. P. Fitz, and J. Grimm, “A new view of performance analysis of transmit diversity schemes in correlated Rayleigh fading,” IEEE Trans. Inform. Theory, vol. 48, pp. 950–956, Apr. 2002. [20] S. Siwamogsatham and M. P. Fitz, “Robust space-time codes for correlated Rayleigh fading channels,” IEEE Trans. Signal Processing, vol. 50, pp. 2408–2416, Oct. 2002. [21] Z. Safar and K. J. R. Liu, “Performance analysis of space-time codes over correlated Rayleigh fading channels,” in Proc. IEEE Int. Conf. Communications, vol. 5, Anchorage, AK, May 2003, pp. 3185–3189. [22] H. El Gamal, “On the robustness of space-time coding,” IEEE Trans. Signal Processing, vol. 50, pp. 2417–2428, Oct. 2002. [23] K. Leeuwin-Boulle and J. C. Belfiore, “The cutoff rate of time correlated fading channels,” IEEE Trans. Inform. Theory, vol. 39, pp. 612–617, Mar. 1993. [24] E. Baccarelli, “Performance bounds and cutoff rates for data channels affected by correlated randomly time-variant multipath fading,” IEEE Trans. Commun., vol. 46, pp. 1258–1261, Oct. 1998. [25] O. Y. Takeshita and D. J. Constello Jr, “New classes of algebraic interleavers for turbo-codes,” in Proc. IEEE Int. Symp. Information Theory, MIT, Cambridge, MA, 1998, p. 419. [26] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [27] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1991.
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New Family of -ary Sequences With Optimal Correlation Property and Large Linear Span Ji-Woong Jang, Young-Sik Kim, Jong-Seon No, Member, IEEE, and Tor Helleseth, Fellow, IEEE Abstract—For an odd prime and integers and such that = + 1) , a new family of -ary sequences of period 1 with op(2 timal correlation property is constructed using the -ary Helleseth–Gong sequences with ideal autocorrelation, where the size of the sequence family of all pairs is . That is, the maximum nontrivial correlation value + 1, which means of distinct sequences in the family does not exceed the family has optimal correlation in terms of Welch’s lower bound. The symbol distribution of the sequences in the family is enumerated. It is also + 2) shown that the linear span of the sequences in the family is ( except for the -sequence in the family. Index Terms—Family of sequences, optimal correlation, quences.
-ary se-
I. INTRODUCTION In the wireless communication systems employing code-division multiple-access (CDMA) scheme, a signature sequence is assigned to each user, which makes it possible to distinguish his signal from those of the other users. In design of sequences for CDMA system, the most important properties of the sequences are low periodic correlation between all pairs of distinct sequences and large family size. For an odd prime p, families of p-ary sequences of period pn 0 1 with optimal correlation property have been found, where the optimality of correlation values means that maximum magnitude of out-of-phase autocorrelation and cross-correlation values of any pairs of sequences of period pn 0 1 in the family is upper-bounded by Rmax = p + 1. Sidelnikov constructed a family of p-ary sequences with optimal correlation property and a family of prime-phase sequences with optimal correlation property was introduced by Kumar and Moreno [3]. By extending the alphabet size, Liu and Komo [7] constructed p-ary Kasami sequences. The family of p-ary bent sequences also has the optimal correlation property. Using the p-ary bent functions given by Kumar and Moreno, a family of balanced p-ary sequences with optimal correlation property was constructed by Moriuchi and Imamura [9]. The known families of n p-ary sequences of period p 0 1 with optimal correlation property are listed in Table I. The family size of the sequences due to Sidelnikov and to Kumar and Moreno are larger than that of the others in Table I. But the linear span of the sequences due to Sidelnikov and to Kumar and Moreno are much smaller than those of the others. In this correspondence, for an odd prime p and integers n; m; and k such that n = (2m + 1)k , a new family of p-ary sequences of period n p 0 1 with optimal correlation property is constructed using the p-ary Helleseth–Gong sequences with ideal autocorrelation, where the size of the sequence family is pn . That is, the maximum nontrivial correlation value Rmax of all pairs of distinct sequences in the family does not exceed p + 1, which means the family has optimal correlation with respect to Welch’s lower bound. The symbol distribution of the Manuscript received January 9, 2003; revised February 18, 2004. This work was supported in part by the Korean Ministry of Information and Communications and the Norwegian Research Council. J.-W. Jang, Y.-S. Kim, and J.-S. No are with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]). T. Helleseth is with the Department of Informatics, University of Bergen, N-5020 Bergen, Norway (e-mail: [email protected]). Communicated by K. G. Paterson, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2004.831837
0018-9448/04$20.00 © 2004 IEEE
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FAMILIES
OF
TABLE I SEQUENCES OF PERIOD pn CORRELATION PROPERTY Rmax = p
01
p-ARY
sequences in the family is enumerated. It is also shown that the linear span of the sequences in the family is (m + 2)n except for the m-sequence in the family. II. PRELIMINARIES Let S be the family of M p-ary sequences of period N = pn for an odd prime p given by
01
S = fsi (t) j 0 i M 0 1; 0 t N 0 1g: The correlation function of the sequences si (t) and sj (t) in S is written as
Ri;j ( ) =
N 01 t=0
!s (t+ )0s
(t)
! is a complex pth root of unity, 0 i; j M 0 1, and N 0 1. The maximum magnitude R of the correlation
where 0
max
values is defined as
Rmax = 0i;j Mmax 01;0 N 01 jRi;j ( )j where the maximization excludes the case of in-phase (i = j and = 0) autocorrelation. A family of p-ary sequences of period pn 0 1 is said to have optimal correlation property if Rmax does not exceed p + 1. Let z be an integer and Vzn the n-dimensional vector p space over the set of integers modulo z , Jz . Let !z = ej , j = 01. Let f (x) be a function from Vzn to Jz . The Fourier transform of the function f (x) is defined as
F () =
pz n 1
x2V
!zf (x)01x
2
;
Vzn
!f (x) = p1 n F ()!tr p 2F
Definition 1 (Kumar, Scholtz, and Welch [4]): A function f (x) from Vzn to Jz is said to be a generalized bent function if the Fourier coefficients F () of f (x) all have unit magnitude for any 2 Vzn . In this correspondence, we assume that the integer z is an odd prime p. Thus, Vpn is the n-dimensional vector space over the finite field Fp with p elements and f (x) is a function from Vpn to Fp . Olsen, Scholtz, and Welch introduced the trace transform for a function from F2 to F2 . Their definition can be generalized as follows. Definition 2 (Olsen, Scholtz, and Welch [10]): Let f (x) be a function from Fp to Fp . Then the trace transform of f (x) and its inverse transform are defined by (x)
;
2 Fp
(x)
;
x 2 Fp :
The elements x and in Fp can be related to the elements x and in Vpn as follows:
x= =
n i=1 n i=1
xi i ) x = ( x1 ; x 2 ; x 3 ; . . . ; x n ) i i ) = ( 1 ; 2 ; 3 ; . . . ; n )
where xi and i are in Fp and f 1 ; 2 ; 3 ; . . . ; n g is a basis of Fp over Fp . By replacing x in Fp by x in Vpn , the function f (x) from Fp to Fp makes the corresponding function f (x) from Vpn to Fp . It is known that the set of the trace transform values of the function f (x) is the same as that of the Fourier coefficients of the corresponding function f (x). Therefore, the function f (x) is a generalized bent function from Fp to Fp if and only if the corresponding function f (x) is a generalized bent function from Vpn to Fp . Let n = ek , where e and k are integers. A basis f 1 ; 2 ; 3 ; . . . ; e g of Fp over Fp is said to be trace-orthogonal if n
trk ( i j ) =
ai ; 0;
if i = j otherwise
where ai 2 Fp3 . It is known that for any positive integer e and odd prime p, there exists a trace-orthogonal basis of Fp over Fp [12]. Suppose elements x and in Fp can be related to the elements x and in the e-dimensional vector space Vpe over Fp as follows:
x= =
where xT denotes the transpose of x.
F () = p1 n !f (x)0tr p x2F
WITH OPTIMAL
+1
e i=1 e i=1
xi i ) x = ( x1 ; x 2 ; x 3 ; . . . ; x e )
(1)
i i ) = ( 1 ; 2 ; 3 ; . . . ; e )
(2)
where xi and i are in Fp . Then, if we choose the basis to be traceorthogonal, we have the relation e n trk (x) = ai i xi : (3) i=1
Let 0i = ai i for 1 i e and 0 = (01 ; 02 ; 03 ; . . . ; 0e ). Then the relation in (3) can be rewritten as e n trk (x) = 0i xi = 0 1 xT : i=1 Suppose that using (1) and (2), a function f (x) from Fp to Fp is related to the corresponding function f (x) from Vpe to Fp . Then the trace transform in Definition 2 can be modified into the trace transform defined in the intermediate field as follows.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
Definition 3: Let n = ek . Let f (x) be a function from Vpe to Fp . Then the trace transform of trk1 (f (x)) and its inverse transform are defined as
FM () = p
1
!tr
(f (x))
=
p1pn
(f (x))
!tr
pn x2V
0tr (1x ) ;
2 Vpe
FM ()!tr (1x ) ;
2V
x 2 Vpe :
F () = FM (0 ):
Q(x) =
i=1 j =1
bij xi xj
(4)
where bij 2 Fp . It is known from Dickson [1] that for an odd integer , any quadratic form with rank can be transformed by linear transformations into a canonical form
Q(x) =
i=1
rxi2
(5)
where e and r = 1 or a quadratic nonresidue in Fp . It is clear that Q(x) and the corresponding function Q(x) in (4) and (5) have the same rank. From Definition 3, we easily derive the following lemma. Lemma 4: tr1k (Q(x)) is a quadratic p-ary bent function from Vpe to Fp if and only if the quadratic function Q(x) from Vpe to Fp has full rank e. Proof: It is straightforward from Deligne’s theorem [3] that a p-ary quadratic function with full rank is a bent function. To prove the converse, let tr1k (Q(x)) be a p-ary quadratic bent function, where Q(x) has rank , < e. From Dickson [1, p. 157], k tr1 (Q(x)) can be rewritten as k
k
k
tr1 (Q(x)) = tr1 (Q(x1 ; x2 )) = tr1
i=1
ai x2i
(6)
where ai 2 Fp3 , x1 is a -tuple vector on Vp , and x2 is an (e 0 )tuple vector on Vpe0 . Then the trace transform of (6) is given as
QM () =
p1pn
=
p1pn
p1pn
=
=
x
p1pn
x
x2V x2V
2V 2V
0tr
wtr
(Q(x))
wtr
(
wtr
(
)
0tr
ax
)
x
(
(
1x
(
1x
)
x
2V
wtr
w0tr
(
2V ax
k
i=1
ai x2i :
k 6 pppn 0;
2V
x
6 pk e0 ; (
)
w0tr
(
1x
)
if 2 6= 0 if 2 = 0,
(8)
which means that tr1k (Q(x)) is not a bent function on Vpe . Thus. every quadratic bent function has full rank. Helleseth and Gong introduced new p-ary sequences with ideal autocorrelation, which are referred to as Helleseth–Gong (HG) sequences [2]. Theorem 5 (Helleseth and Gong [2] ): Let be a primitive element of Fp . Let n = (2m + 1)k and let s, 1 s 2m be an integer such that gcd(s; 2m + 1) = 1. Define b0 = 1, bis = (01)i , and bi = b2m+10i for i = 1; 2; . . . ; m. Let u0 = b0 =2 = (p + 1)=2 and ui = b2i for i = 1; 2; . . . ; m where all indexes of the bi ’s are taken k n mod 2m + 1 and q = p . Then the HG sequence of period p 0 1 is given by m
s(t) = tr1n
l=0
t
ul
:
(9)
The HG sequence has an ideal two-level autocorrelation. We provide an example of HG sequence as follows. Example 6: For p = 3, let m = 2, k = 2, and n = (2m +1)k = 10. Let s = 2 so gcd(s; 2m + 1) = 1. Then the parameters bi and ui are given as follows:
b0 = 1; b1 = b6 = (01)3 = 01; b2 = (01)1 = 01 b3 = b8 = (01)4 = 1; b4 = (01)2 = 1 u0 = b0 = 2 ; u 1 = b2 = 2 ; u 2 = b4 = 1 : 2
Thus, the HG sequence of period 310 0 1 is given as
s(t) = tr1n( t ) + tr1n( 41t ) 0 tr1n( 3281t )
where is a primitive element in F3 . Let h(x) be the HG polynomial defined by m
l=0
ul x
;
x 2 Fp :
Then the HG sequence in (9) can be rewritten as
)
)
x
w0tr
QM () =
h (x ) =
)
0tr
ax
(
1x
0
k
tr1 (Q (x1 )) = tr1
=
That is, the set of the trace transform values of the function tr1k (f (x)) is the same as that of the corresponding function tr1k (f (x)). Therefore, if the trace transform of the function tr1k (f (x)) or tr1k (f (x)) only takes values of unit magnitude, the functions tr1k (f (x)) and tr1k (f (x)) become generalized bent functions. Let Q(x) be a quadratic form from Fp to Fp . Using (1), the quadratic form Q(x) can be expressed as e
Let
Then the inner summation in (7) is the trace transform of tr1k (Q0 (x1 )). Clearly, Q0 (x1 ) has full rank on Vp and thus tr1k (Q0 (x1 )) is a bent function on Vp . Therefore, the inner summation of (7) is equal to 6 pk . Thus, we can rewrite (7) as follows:
It is clear that the trace transform of a function tr1k (f (x)) from Vpe to Fp is related to the trace transform F () of the corresponding function k tr1 (f (x)) from Fp to Fp as follows:
e
1841
0tr
)
(
(
1x
)
1x ) : (7)
s(t) = tr1n(h( t )); 0 t pn 0 2: We can construct a new p-ary sequence s for each 2 Fp
using the
HG sequence as follows:
s (t) = tr1n( t ) + tr1n(h( 2t )) n
= tr1
t +
m
l=0
ul
( q
+1)t
It is clear that the sequences in (10) have period pn 0 1.
:
(10)
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III. NEW CONSTRUCTION OF A FAMILY OF p-ARY SEQUENCES Using the new p-ary sequence defined in (10), a family of p-ary sequences with family size pn and optimal correlation property can be constructed as follows. Theorem 7: Let s (t) be the p-ary sequence defined in (10). Then the family of p-ary sequences given by
S = f s ( t) j 2 F p ;
Rij ( ) + 1 =
02
t=0
=
w
s
wtr
x2F
0s
(t+ )
(t)
(cx+h(
c x
c = 1 , i
Case (iii)
x
x2F
w
y2F
w w
y2F
(h(r
wtr
tr (h(r
Using the property that for r
=
m l=0
aql
0h(r
a c x u y
)
0h(r
)
y
h(u y
tr (tr (r
0r
h(y
)
h(u y
0r
)
m l=0
aql
6
01)x)
))+tr ((c
y)
))+tr (
m
n
= rj trk
ul [ri uq
l=0 m
+1
ul [ ri uq rj l=0
+1
+1
= 0:
m +
l=0
al z q
+1
=0
m
m l=0
+
l=0
al z q
al z q
=0
(13)
0 1 zq
=0
= 0:
+1
y)
)))+tr (
l=0
ul
ri u(q rj
0 1 zq
+1)q
ri u(q rj
+
+1)
and thus we have 2m
bl ri u(q rj l=0
0 1 zq
+1)
=0
where u0 = b2 and ul = b2l = b2m+102l . To prove that the rank of Q(y) is = 2m + 1, we have to show that the equation q
+1
0 1 zq
=0
has z = 0 as its only solution for any rr ( aa c)2 = 6 1. This is already proved in [2]. From the condition, it is clear that
y)
))+tr (
0 r j ]y q
+1
al z q
al z q
l=0
bl ri ai c rj aj l=0
:
(12)
ri ai c rj aj
2
=
bi c2 bj
6= 1
and thus we have proved the theorem. Clearly, the sequence in (10) becomes a p-ary m-sequence when = 0. For 6= 0, (10) can be transformed into the form
Q(y) = trkn (ri h(u2 y2 ) 0 rj h(y2 )) n
l=0
+
2m
:
Let Q(y ) be the quadratic function defined by
= trk
m )+
zq
and
m
a x
h(y
l=0
al y q
0 1]. Then
Equations (13) can be rewritten as
2 Fp , h(rx) = rh(x), we have
tr (r
= trk
m
zq
n
6
y2F
l=0
al (yq z + yz q
trk
2
Rij ( ) + 1 =
y
(11)
Since nk is an odd integer, it is clear that a quadratic nonresidue in Fp is also a quadratic nonresidue in Fp . Thus, i and j can be Fp and ri expressed as i = ri a2i and j = rj a2j , where ai ; aj and rj are 1 or quadratic nonresidues in Fp . We assume that j = 0 and thus aj = 0. Let u = aa c and y = aj x. Then the cross-correlation
=
m
=
function in (11) can be rewritten as
m
n
+1
+1
which is equivalent to
))
The condition excludes the case of i = j = 0. We will prove it assuming j 6= 0; the proof in the case i 6= 0 is similar.
Rij ( ) + 1 =
n
+1
0x0h(
a l (y + z ) q
uq
Raising the first term to the q 2m+102l power, we have trk
j : It is clear that Rij ( ) = pn 0 1. c 6= 1, i c2 = j : It is also clear that Rij ( ) = 01. i c2 6= j :
Case (ii)
l=0
n
)
. Let al = ul [ rr
can be modified into trk
where c = 2 Fp3 . Then the proof can be classified into the following three cases. Case (i)
m
n
trk
t pn 0 2g
0
has the optimal correlation property with Rmax = p + 1. Proof: The cross-correlation function of two p-ary sequences s (x) and s (x) in S can be rewritten as p
Q(y + z ) = Q(y) for all y 2 Fp the condition
+1
0 1]yq
+1
s i (t) = tr1n t+i + :
Then (12) corresponds to the trace transform of the quadratic function tr1k (Q(y )) from Fp to Fp as in Definition 2. If the quadratic function tr1k (Q(y )) is a p-ary bent function, then we have jRij ( ) + 1j = p . From Definition 3 and Lemma 4, if Q(y) has full rank, then k tr1 (Q(y )) is bent. Thus, it is sufficient to show that Q(y ) has full rank 2m + 1. It is already known that the rank of the quadratic function Q(y ) is equal to if q 2m+10 is the number of elements z 2 Fp satisfying
m l=0
ul (q
+1)t
(14)
where 0 i p 201 0 1 and is 1 or a quadratic nonresidue in Fp . Then the family of p-ary sequences defined in Theorem 7 can be rewritten as
S= si (t) j 0 i p
n
0 1 0 1; 0 t p n 0 2
2
Let W be a subset of Vpn and g (x) a function from W we define the notation
wc = j fx j g(x) = c; x 2 W g j;
n
t
tr1 ( )
:
(15) to Fp . Then
c 2 Fp :
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
Then the symbol distribution of g (x) is defined as the ordered p-tuple (w0 ; w1 ; w2 ; . . . ; wp01 ). It is interesting to find the symbol distribution of the sequences in the family given in (15). Using the result in Section II, it is easy to derive that the quadratic part of the sequence in (14) can be transformed into the quadratic form
f (x) = a1 x21 + a2 x22 + 1 1 1 + an x2n
(16)
mapping from Vpn nf0g to Fp . It is clear that the quadratic form in (14) is bent and thus f (x) has full rank, that is, ai 2 Fp3 . Let (x) denote the quadratic character of x defined by +1; 0;
(x ) =
0 1;
if x is a quadratic residue if x = 0 if x is a quadratic nonresidue.
Dickson [1] derived the number of solutions in Vpn to quadratic equations over Fp . Theorem 8 (Dickson [1, pp. 47–48]): Let p be an odd prime and n = (2m + 1)k = 2h be an even integer. Then the number of solutions n (x1 ; x2 ; . . . ; xn ) in Vp of the quadratic equation
a1 x21 + a2 x22 + 1 1 1 + an x2n = c
2 Fp3 and c 2 Fp , is given by 2h01 h h01 vc = p2h01 + (ph010 p ); p 0 p ; where = ((01)h a1 a2 1 1 1 an ).
where aj
if c = 0 if c 6= 0
(17)
Theorem 9 (Dickson [1, pp. 47–48]): Let p be an odd prime and = (2m + 1)k = 2h + 1 be an odd integer. Then the number of solutions (x1 ; x2 ; . . . ; xn ) in Vpn to the quadratic equation
n
a1 x21 + a2 x22 + 1 1 1 + an x2n = c
where aj
2 Fp3 and c 2 Fp , is given as
vc = p2h + ph
(18)
where = ((01)h a1 a2 1 1 1 an c). It is clear that the new sequence family includes one p-ary quence, which has the symbol distribution (p
n01
m-se-
0 1; pn01 ; pn01 ; . . . ; pn01 ):
We denote this distribution by D1 = 1. The distribution of Dc for the new sequence family defined in (15) is given in the following theorem. Theorem 10: Let Dc be the number of sequences with symbol distribution (vc 0 1; vc+1 ; . . . ; vp01 ; v0 ; v1 ; . . . ; vc01 ) in the sequence family in (15). Then
Dc =
1;
01 2 ; 2;
v v
if c = 1 if c = 0 if c 2 Fp3
where vc is defined in (17) and (18). Proof: From the balance property of a p-ary m-sequence, it is clear that D1 = 1. Since gcd(q 2l + 1; pn 0 1) = 2 for all l, the period of the quadratic part in (14) is p 201 . As i varies over 0 i pn 0 2 in (14), the p 201 different sequences of period pn 0 1 can be generated, where each sequence occurs exactly twice. Using (16) and the transformation of the linear part tr1n ( t+i ), s i (t) in (14) can be transformed into the function from Vpn nf0g to Fp given by
a1 x21 + a2 x22 + 1 1 1 + an x2n + d1 x1 + d2 x2 + 1 1 1 + dn xn
(19)
1843
where ai
2 Fp3 and di 2 nFp . As i varies over 0 i pn 0 2, every
(d1 ; d2 ; d3 ; . . . ; dn )
in Vp occurs exactly once except for 0. We can
modify (19) into
2 2 2 a 1 x 1 + d 1 + a 2 x 2 + d 2 + 1 1 1 + an x n + d n 0 c 2a 1 2a 2 2a n d 0 and putting xi = xi + 2a , we have a1 x0 2 + a2 x0 2 + 1 1 1 + an x0 2 0 c (20) 1
where
2 c = d1 4a 1
2
+
n
d22 4a 2
+
2
1 1 1 + 4dann
is in Fp . From (17) and (18), the symbol distribution of f (x) in (16) is given as (v0 0 1; v1 ; v2 ; . . . ; vp01 ), where excluding x = 0 makes v0 0 1 instead of v0 . Then for a fixed c in Fp , the symbol distribution of (20) is given as (vc 0 1; vc+1 ; . . . ; vp01 ; v0 ; v1 ; . . . ; vc01 ) (21) where for x = 0, (19) takes the value 0 and thus excluding x = 0 in (19) makes vc 0 1 instead of vc . In order to find the number of sequences with the symbol distribution in (21) in the sequence family defined in (15), we have to find the number of solutions (d1 ; d2 ; d3 ; . . . ; dn ) in Vpn nf0g satisfying the quadratic equation
d12 + d22 + 1 1 1 + dn2 = c 4a 1 4a 2 4a n as i varies over 0 i pn 02. In fact, this corresponds to 2Dc because each sequence occurs exactly twice as i varies over 0 i pn 0 2, which is already calculated in (17) and (18). We have to exclude the solution 0 for c = 0 because (d1 ; d2 ; d3 ; . . . ; dn ) is in Vpn nf0g. Thus, we have D0 = v 201 and Dc = v2 for c 2 Fp3 .
Kumar and Moreno [3] calculated the correlation distribution of the sequences in their sequence family by finding the correlation values of two sequences in the family, which corresponds to Fourier coefficients of the quadratic forms with full rank. In their paper, the correlation distribution was derived by using the full rank property of the quadratic function. Therefore, the correlation distribution of the sequences in the new family defined in Theorem 7 is the same as that of the sequence family by Kumar and Moreno. The linear span of the new p-ary sequences s (t) in Theorem 7 is derived as in the following theorem. Theorem 11: The linear span of the sequence s (t) defined in Theorem 7 is equal to (m + 2)n. Proof : For di = q 2i + 1; 0 i m in (10), gcd(di ; pn 0 1) = d 2. It is clear that does not belong to any subfield of Fp . Thus, the 2 coset of in Fp has size n and so does that of each element d . Therefore, the linear span of s (x) is (m + 2)n. REFERENCES [1] L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory. New York: Dover, 1958. [2] T. Helleseth and G. Gong, “New nonbinary sequences with ideal twolevel autocorrelation function,” IEEE Trans. Inform. Theory, vol. 48, pp. 2868–2872, Nov. 2002. [3] P. V. Kumar and O. Moreno, “Prime-phase sequences with periodic correlation properties better than binary sequences,” IEEE Trans. Inform. Theory, vol. 37, pp. 603–616, May 1991. [4] P. V. Kumar, R. A. Scholtz, and L. R. Welch, “Generalized bent functions and their properties,” J. Combin. Theory, ser. A, vol. 40, pp. 90–107, 1985. [5] A. Lempel and M. Cohn, “Maximal families of bent sequences,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 865–868, Nov. 1982. [6] R. Lidl and H. Niederreiter, Finite Fields, ser. Encyclopedia of Mathematics and Its Applications. Reading, MA: Addison-Wesley, 1983, vol. 20.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
[7] S.-C. Liu and J. F. Komo, “Nonbinary Kasami sequences over GF( ),” IEEE Trans. Inform. Theory, vol. 38, pp. 1409–1412, July 1992. [8] F. J. McWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [9] T. Moriuchi and K. Imamura, “Balanced nonbinary sequences with good periodic correlation properties obtained from modified Kumar–Moreno sequences,” IEEE Trans. Inform. Theory, vol. 41, pp. 572–576, Mar. 1995. [10] J. D. Olsen, R. A. Scholtz, and L. R. Welch, “Bent-function sequences,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 858–864, Nov. 1982. [11] O. S. Rothaus, “On bent functions,” J. Combin. Theory, ser. A, vol. 20, pp. 300–305, 1976. [12] G. Seroussi and A. Lempel, “Factorization of symmetric matrices and trace-orthogonal bases in finite fields,” SIAM J. Comput., vol. 9, no. 4, pp. 758–767, Nov. 1980. [13] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications. Rockville, MD: Computer Sci., 1985, vol. 1. [14] L. R. Welch, “Lower bounds on the maximal cross correlation of signals,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 396–399, May 1976.
II. PRELIMINARIES Let R = GR (2 ; m) denote the Galois ring of characteristic with 2lm elements. Let be an element in GR (2l ; m) that generates the Teichmüller set T of GR (2l ; m). Specifically, let T = f0; 1; ; 2 ; . . . ; 2 02g and T 3 = f1; ; 2 ; . . . ; 2 02 g. The l 2-adic expansion of x 2 GR (2 ; m) is given by l
2
l
x = x0 + 2x1 + 1 1 1 + 2 01 x 01 where x0 ; x1 ; . . . ; x 01 2 T . The Frobenius operator F is defined for such an x as F (x0 + 2x1 + 1 1 1 + 2 01 x 01 ) = x20 + 2x21 + 1 1 1 + 2 01 x201 and the trace Tr, from GR (2 ; m) down to 2 , as 01 Tr(x) := F (x): l
l
l
l
l
m
j
j =0
We also define another trace tr from 2
S
Let MSB : n 2
Index Terms—Autocorrelation, Galois rings, maximum-length sequences over rings, most significant bit.
l;m
:=
Maximum-length (ML) sequences over the rings 2 were introduced by Dai [1], motivated by cryptographic applications. Using the 2-adic expansion in the local ring 2 , various binary sequences can be defined from a given ML sequence over 2 . Of particular interest, from the nonlinearity viewpoint, is the most significant bit [6]. In a cryptographic usage such as, for instance, a key in a one-time-pad system, such a deterministic sequence should look as random as possible. It should be as balanced as possible the frequency of each of the two symbols 0, 1 should be close to 1=2. It should be uncorrelated with itself: low inner products with its successive shifts. Recently some estimates were given for the autocorrelation and the imbalance of the binary sequences in the title [4], using the Galois rings character sum estimates of [7]. The aim of this note is, still using [7], to sharpen the autocorrelation and imbalance estimate of [4] by a factor exponential in l. Our approach uses the Discrete Fourier Transform techniques of [8], [9]. Manuscript received July 30, 2003; revised February 11, 2004. The material in this correspondence was presented in part at the Kodierungtheorie, Oberwolfach, Germany, December 2003. P. Solé is with the CNRS-I3S, ESSI, 06 903 Sophia Antipolis, France (e-mail: [email protected]). D. Zinoviev is with the CNRS-I3S, ESSI, 06 903 Sophia Antipolis, France, and with the Institute for Problems of Information Transmission, Russian Academy of Sciences, GSP-4, Moscow, 101447, Russia (e-mail: zinoviev@ essi.fr, [email protected]). Communicated by K. G. Patterson, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2004.831858
j =0
down to 2 as
x2 :
(Tr(
t N 01 t=0
))
j 2 R3 :
(1)
! n be the most significant bit (MSB) map, i.e., MSB(x + 2x + 1 1 1 + 2l0 xl0 ) := xl0 : 2
0
1
1
1
1
III. IMBALANCE AND AUTOCORRELATION PROPERTIES Let l be a positive integer (without loss of generality we assume that l 4) and ! = e2i=2 be a primitive 2l th root of 1 in . Let k be the additive character of
I. INTRODUCTION
01
m
Throughout this note, we let n = 2m and R3 = R n 2R. Let = (1 + 2) 2 R, where 2 T and 2 R3 . Assume 1 + 2 is of order 2l01 . Since is of order 2m 0 1 then is an element of order N = 2l01 (2m 0 1). Following [4, Lemma 2], we define the sequence
Patrick Solé, Member, IEEE, and Dmitrii Zinoviev Abstract—The imbalance and the autocorrelation of the binary sequences in the title are explored by combining the local Weil bound with spectral analysis. Recent estimates on these quantities are improved by a factor of order 2 for large .
l
l
tr(x) :=
The Most Significant Bit of Maximum-Length Sequences : Autocorrelation and Imbalance Over
l
2
such that
(x) = !kx : Let : ! f61g be the mapping (t) = (01)c , where c is the , i.e., it maps 0; 1; . . . ; 2l0 0 1 to +1 most significant bit of t 2 l 0 l 0 l and 2 ; 2 + 1; . . . ; 2 0 1 to 01. Our goal is to express this map k
2
1
1
2
1
as a linear combination of characters. Recall the Fourier transformation formula on 2
=
2
01
j =0
j j ;
where j
0 = 21l (x) j (0x): 2
1
(2)
x=0
2 R = GR (2l; m), we denote by 9 the character 9 : R ! 3 ; x 7! ! x : Note that for the previously defined k and 9 we have k (Tr( x)) = 9 k (x): For all
Tr(
)
(3)
The following lemma follows from [7].
2 R, 6= 0, we have p 9 (x) (2l0 0 1) 2m :
Lemma 3.1: For all
1
x2T
Proof: We restate [7, Theorem 1] for the special Galois ring of concern here. Let f (X ) denote a polynomial in R[X ] and let
f (x) = F0 (x) + 2F1 (x) + 1 1 1 + 2l01 Fl01 (x)
0018-9448/04$20.00 © 2004 IEEE
