New Sets of Binary and Ternary Sequences with Low Correlation Evgeny I. Krengel1 , Andrew Z. Tirkel2 , and Tom E. Hall2 1
Kedah Electronics Engineering, Zelenograd, Moscow 124498, Russia
[email protected] 2 School of Mathematical Sciences, Monash University, PO Box 28M, Victoria 3800, Australia
[email protected]
Abstract. New binary and ternary sequences with low correlation and simple implementation are presented. The sequences are unfolded from arrays, whose columns are cyclic shifts of a short sequence or constant columns and whose shift sequence (sequence of column shifts) has the distinct difference property. It is known that a binary m-sequence/GMW sequence of length 22m − 1 can be folded row-by-row into an array of 2m − 1 rows of length 2m + 1. We use this to construct new arrays which have at most one column matching for any two dimensional cyclic shift and therefore have low off-peak autocorrelation. The columns of the array can be multiplied by binary orthogonal sequences of commensurate length to produce a set of arrays with low cross-correlation. These arrays are unfolded to produce sequence sets with identical low correlation.
Outline Sequences with low correlation and large linear complexity are widely used in spread spectrum communication systems [1,2]. Here, we present new binary sequences, with low even periodic cross-correlation, low off-peak autocorrelation and simple implementation. This paper generalizes a construction of sequence sets introduced by Gong [7], called interleaved sequences. Such sequence sets are obtained by writing an m/GMW sequence of length pkm − 1 row-by-row into an array, where the columns are cyclic shifts of a single m/GMW sequence of length pm − 1 or constants. Games [4] introduced the shift sequence to describe the sequence of cyclic shifts of the columns. The shift sequence has some remarkable properties, which are crucial to our construction. Where the number of m/GMW columns is commensurate with the length of a pseudonoise sequence, or an integer multiple of periods of a pseudonoise sequence, these columns can be multiplied by cyclic shifts of that sequence. The resultant arrays can be unfolded to produce sequences with identical correlation. The off-peak autocorrelation and cross-correlation of such arrays and sequences is low, because the number of phase-matched columns is constrained by a property of the shift sequence. Our paper is organized in two parts. In Sections 1-4, we find suitable multiplication T. Helleseth et al. (Eds.): SETA 2004, LNCS 3486, pp. 220–235, 2005. c Springer-Verlag Berlin Heidelberg 2005
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sequences, commensurate with m/GMW arrays, as described above. We restrict our results to binary and ternary sequences. In Sections 5-10 we synthesize shift sequences with desirable properties and construct arrays with commensurate pseudonoise column sequences and multiplication sequences. We then unfold the arrays into long sequences with good auto and cross-correlation. We find an abundance of such long sequences and restrict ourselves to purely binary cases.
1
Sequence Construction
Here, we present new binary and ternary sequences of length 2n − 1, with low even periodic cross-correlation, low off-peak autocorrelation and simple implementation. We fold a long m or GMW sequence of length 2n −1, with even n, into an array and multiply the columns of the array by commensurate pseudonoise sequences. When the parent sequence is an m-sequence, the implementation is simple, but linear complexity is low. GMW parent sequences require more complex implementation, but the linear complexity of the resulting sequence is much larger. 1.1
Binary Sequence
Let n be even, T = 2n/2 + 1, and let q = T /3 be a prime of type 4t + 3. Let a= {ai }, i = 0, 1, 2, . . . , 2n − 2, ai ∈ {−1, 1} be a binary m-sequence (or GMW sequence) of length N = 2n −1. Let b= {bi }, i = 0, 1, 2, . . . , q−1, bi ∈ {−1, 1}, be an infinite periodic binary Hall or Legendre sequence with period q [3]. Let b j be the sequence b shifted by j units. We form the sequence c j = bj a, with entries cji = bj+i ai , i = 0, 1, 2, . . . , 2n − 2. We denote the set {c j : j = 0, 1, 2, . . . , q − 1} simply by {cj } and we adjoin a to this set, and denote the final set by {c j ,a}. The number of different sequences in the set is M = q + 1 = (2n/2 + 1)/3 + 1. Example 1. Let n = 6 and a = 1 1 1 1 1-1-1-1-1 1 1-1 1 1-1 1-1 1-1 1 1-1-1 1-1 1 1 1 1-1 1 1 1-1 1-1-1 1-1-1-1-1-1-1 1-1 1-1-1-1 1 1 1-1-1 1 1-1-1-1 1-1-1 and b = 11-1. Sequence a is an m-sequence of length 63 and b is a Legendre sequence of length 3. Then c 0 =b 0 a= 1 1-1 1 1 1-1-1 1 1 1 1 1 1 1 1-1-1-1 1-1-1-1-1-1 1-1 1 1 1 1 1-1-1 1 1-1 1 1-1-1 1-1-1-1-1 1 1-1-1-1 1 1 1-1 1-1-1-1 1 1-1 1 ; c 1 =b 1 a= 1-1 1 1-1-1-1 1-1 1-1-1 1-1-1 1 1 1-1-1 1-1 1 1-1-1 1 1-1-1 1-1 1-1-1-1-1-1-1-1 1-1-1 1 1-1-1-1-1 1 1 1-1-1-1-1 1-1 1-1 1 1-1 ; c 2 =b 2 a= -1 1 1-1 1-1 1-1-1-1 1-1-1 1-1-1-1 1 1 1 1 1-1 1 1 1 1-1 1-1-1 1 1 1 1-1 1 1-1 1-1-1 1-1 1 1 1-1 1-1 1-1 1-1 1 1 1 1-1-1-1-1-1. 1.2
Ternary Sequence Construction
Let n be even, T = 2n/2 + 1 be a prime. New ternary sequences result from the multiplication of a long binary m/GMW sequence a={a i } with ai ∈{-1,1} and length 2n − 1, with 2n/2 − 1 repeats of a short ternary Legendre sequence d = {d i } with di ∈{0,-1,1} and length t = 2n/2 + 1. We apply all shifts of this short sequence and form the ternary sequence t j =d j a, with entries tji = dj+i ai , i = 0, 1, 2, . . . , 2n − 2, j = 0, 1, 2, . . . , 2n/2 . Thus, we obtain a new set
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{t j ,a} consisting of 2n/2 + 1 ternary sequences and the reference long sequence, i.e.M = 2n/2 + 2 distinct sequences. When 2n/2 + 1 is prime, it is called a Fermat prime [2]. Currently five Fermat primes are known for n = 2, 4, 8, 16, 32. Example 2. Let n = 4. a= -1 1 1 1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 is an m-sequence of length 15. d = 0 1-1 -1 1 is a ternary Legendre sequence of length 5. Then t0 t1 t2 t0 t4
=d 0 a= 01 − 1 − 1 − 1011110111 − 1 =d 1 a= −1 − 1 − 1101 − 11 − 10 − 1 − 11 − 10 =d 2 a= 1 − 110 − 1 − 1 − 1 − 1011 − 1 − 10 − 1 =d 0 a= 11011 − 110 − 1 − 1110 − 11 =d 4 a= −101 − 1110 − 11 − 1 − 10 − 111
2 2.1
Correlation Properties Binary Sequence
Now fold the binary sequence c j of Subsection 1.1, row-by-row into a twodimensional array C j of size (2n/2 + 1)(2n/2 − 1) in accordance with Games’ representation [4]. When the sequence c j is shifted by any u (= kT + l) places, each new column is a cyclic shift of a column of C j [4]. We use Games’ representation also to form an array A from a sequence a, of the same size as C j . From [4] the array A contains one column with entries being 1, and the remaining 2n/2 columns are cyclic shifts of a short m-sequence of length 2n/2 − 1. As shown in [4], for any cyclic shift of sequence a, the shifted array agrees with the unshifted array A in exactly one column. The autocorrelation function (AC) of sequence c j =b j a is: ACj (u) =
T −1 T −3
bj+i bj+l+i ai+sT ai+l+(s+k)T ,
(1)
i=0 s=0
where u = kT+l , 0 ≤ k v/2. There are four possible cases of columns matching, when the equation is reduced from modulo v to modulo u. fj+k − fj ≡ l − u fj+k −fj ≡ l+u
for
for v > l > u, 0 > l > −u,
fj+k − fj ≡ l fj+k −fj ≡ l+2u
for
u > l > 0.
(10)
for−v > l > −u. (11)
Therefore, an upper bound on the number of shift matched columns rises to 4. Shortening to less than v/2 makes matters even worse. 8.4
Column Lengthening
Lengthening columns from v to s results in: fj+k − fj ≡ l
for l > 0 or
fj+k − fj ≡ l + s for
0 > l.
(12)
Clearly, if s > 2v, the smallest value of l + s is greater than v, whilst the largest value of l is less than v and thus the two sets of solutions are disjoint and
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hence DDP is preserved. Unfortunately, doubling the column length doubles its contribution to the correlation and therefore, this result is no better than column deletion. However, it is possible for DDP to be preserved for some specific values of v < s < 2v. Clearly, it is desirable to have s as close to v as possible. In addition, if the shift sequence is derived from the quadratic or exponential parent matrix, it is required that gcd(s, T ) = 1, so that a single pass diagonal exists and enables the unfolding of the new matrix into sequences. A method of obtaining low values of s begins with the examination of differences between entries in the shift sequence. The shift sequence has its desirable property when evaluated modulo v. Its entries can be positive or negative numbers modulo v. Now consider what happens if the columns were lengthened to infinity. DDP would remain unaffected. Evaluate the greatest difference between the entries and call it ∆max . Clearly, the largest negative difference is −∆max . Therefore, if the shift sequence is re-expressed modulo s = 2∆max , DDP must be preserved. The objective of the construction is to minimize ∆max . The authors have attempted using a systematic approach, such as a greedy algorithm to compute the smallest ∆max for different starting values of v and T for the parent array, and found this approach unreliable. However, for small arrays, the results are easy to deduce by inspection, and this is what we did to obtain the arrays, whose properties are listed in Tables 3 and 4. Example 9. Consider lengthening the columns of the 7 × 7 quadratic shift array. The shift sequence mod 7 is fj = 0, 1, 4, 2, 2, 4, 1. ∆max = 4, so this shift sequence has DDP for all lengths greater than s = 2∆max + 1 = 9. There exists a binary Legendre sequence of length 11, (1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0), so it can be used to construct the array below (left). Another 7 arrays are produced by multiplying the columns by cyclic shifts of the m-sequence of length 7 : 1, 0, 1, 1, 1, 0, 0. A typical array with multiplied columns is shown on the right (with multiplication sequence above). 1001100 1100001 0111111 1001100 1110011 1111111 0101101 0011110 0010010 1010010 0100001
bj 1 0 1 1 1 0 0 0010000 0111101 1100011 0010000 0101111 0100011 1110001 1000010 1001110 0001110 1111101
7 × 7 Arrays Lengthened to 11 × 7: multiplication sequence bj (left) There are 8 such arrays. They can be unfolded diagonally into into 8 sequences of length 77, with auto and cross correlation values of: +77 (full match for autocorrelation), +13 (one column match, with multiplication sequence agreement), +1
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(0 column match for non-zero horizontal shifts), −7 (0 column match for purely vertical shifts), −11, (one column match, multiplication sequence disagreement).
9
Array Construction
Tables 3 and 4 describe the smallest arrays constructed by the above methods. a denotes column sequence, b j denotes multiplication sequence. These sequences are labeled: m - m-sequence, L - Legendre, H - Hall, PB - Perfect Binary. The types of shift sequences are: e - exponential, m - m-array, l - lengthened quadratic. For symmetric shift sequences only half is shown. Table 3. Shift Sequences with DDP and Array Details T 3 4 6 6 9 7 11 12 19 22 23
33 46 58 70 118 127 129
10
v 4 7 4 7 7 11 15 15
a PB m PB m m L m m
bj m PB 2m(3) 2 m(3) 3 m(3) m L 3PB
Shift Sequence 0,1,1 (l) 0,1,3,2 (ad hoc) 1,2,1,∞,0,2 (m) 3,2,6,4,5,1 (e) ∞,1,2,0,4,4,0,2,1 (m)(d) 0,1,4,2,2,4,1 (l) 0,1,4,13,5,3,3,5,13,4,1 (l) 0,5,3,2,3,0,∞ ,6,0,0,2,5 (m) 31 m/L L 0,1,4,9,28,6,29,11,7,5 ,5,7,11,29,6,28,9,4,1 (l) 23 L 2L(11) Exponential 35 L TP 0,1,4,9,16,2,13,3,18, 12,8,6,6,8,12,18 ,3,13,2,16,9,4,1 (l) 31 m 3L(11) m-array ( TBC) 47 L 2 L(23) Exponential 59 L 2L(29) Exponential 71 L 2TP(35) Exponential 119 L 2L (59) Exponential 127 m/L m/L Cubic Unfold by rows 127 m 3L/H(43) m-array
Large Array
The 127 × 127 array in Tables 3 and 4 is a special case, where the commensurate folding/unfolding rule is violated. 127 is a prime, so it supports m-sequences, binary Legendre sequences, Hall sequences and Baumert-Fredricksen sequences as column and multiplication sequences. Additionally, polynomial shift sequences
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Table 4. Correlation Properties of Unfolded Sequence Sets Length 12 28 24 42 63 77 99 165 180 589 506 805 1023 2162 3244 4970 14042 16129 16383
N Auto 4 +4,0,-4 5 +8,0,-4,-8 4 +4,0,-4 4 +10,+2,-6 4 +11, -1,-5 8 +13,+1,-7,-11 4 +15,+3 ,-9 12 +17,+1,-11,-15 5 +16,+4,0,-12,-16 20 +33,+1,-19,-31 12 +26,+2,-22 24 +37,+1,-23,-35 12 +35,-1,-29 24 +50,+2,-46 30 +62,+2,-58 36 +74,+2,-70 60 +122,+2,-118 2064384
