The 2-adic complexity of Yu-Gong sequences with interleaved structure and optimal autocorrelation magnitude
aa r X i v : . [ c s . I T ] S e p Noname manuscript No. (will be inserted by the editor)
The 2-adic complexity of Yu-Gong sequences withinterleaved structure and optimal autocorrelationmagnitude
Yuhua Sun ∗ · Tongjiang Yan · QiuyanWang
Received: date / Accepted: date
Abstract
In 2008, a class of binary sequences of period N = 4(2 k − k +1) with optimal autocorrelation magnitude has been presented by Yu andGong based on an m -sequence, the perfect sequence (0 , , ,
1) of period 4 andinterleaving technique. In this paper, we study the 2-adic complexity of thesesequences. Our result shows that it is larger than N − ⌈ log N ⌉ + 4 (whichis far larger than N/
2) and could attain the maximum value N if suitableparameters are chosen, i.e., the 2-adic complexity of this class of interleavedsequences is large enough to resist the Rational Approximation Algorithm. Keywords m -sequence · interleaved sequence · optimal autocorrelationmagnitude · Since the interleaved structure of sequences was introduced by Gong in [3],several classes of binary sequences with this form have been constructed andwere proved to have so many good pseudo-random properties, such as low auto-correlation, large linear complexity. For example, in 2010, Tang and Gong con-structed three classes of sequences with optimal autocorrelation value/magnitude
This work is financially supported by the National Natural Science Foundation of China (No.61902429), the Fundamental Research Funds for the Central Universities (No. 20CX05012A),and the Major Scientific and Technological Projects of CNPC under Grant ZD2019-183-008. ∗ The corresponding authorYuhua Sun, Tongjiang YanCollege of Sciences, China University of Petroleum, Qingdao 266555, Shandong, China;Provincial Key Laboratory of Applied Mathematics, Putian University, Putian, Fujian351100, China;Qiuyan WangSchool of Computer Science and Technology,Tiangong University, Tianjin 300387,China Yuhua Sun ∗ et al. using Legendre sequences, twin-prime sequences and a generalized GMW se-quence, respectively [13], which were showed by Li and Tang to have largelinear complexity [7]. In quick succession, Tang and Ding presented two moregeneral constructions which include constructions in [13] as special cases andgave more sequences with optimal autocorrelation and large linear complexity[12]. Later, Yan et al. also gave a generalized version for the constructions in[13] and put forward a sufficient and necessary condition for an interleaved se-quence to have optimal autocorrelation [16]. What’s more exciting is that thesesequences have also been proved to have large 2-adic complexity by Xiong etal. [14,15] and Hu [5] using different methods respectively. Moreover, Su etal. constructed another class of sequences with optimal autocorrelation mag-nitude combining interleaved structure and Ding-Helleseth-Lam sequences [9]and these sequences have also been shown to have large linear complexity byFan [2] and large 2-adic complexity by Sun et al. [10] and Yang et al. [17].Note that each of the above mentioned sequences can be described as aninterleaved form s = I ( s , s , s , s ), i.e., the sequence s is obtained by con-catenating the successive rows of the matrix I ( s , s , s , s ), in which eachcolumn is a periodic sequence s i , 1 ≤ i ≤
4. In fact, Yu and Gong also pre-sented another description method of an interleaved structure by an indicatorsequence (see Construction 1) [18]. Using this description, Yu and Gong rep-resented an ADS (almost difference set) sequence of period 4 v in [1] as a v × v in [8] as a 4 × v interleaved structure respectively, which provides us a new understanding forthe two sequence structures. Not only that, they also discovered another newclasses of sequences with optimal autocorrelation magnitude and large linearcomplexity using binary m -sequences as the indictor sequences, which we callYu-Gong sequences. However, the 2-adic complexity of this class of sequenceshas not been studied yet as far as we know.In this paper, using the method of Hu [5], we investigate the 2-adic com-plexity of a Yu-Gong sequence with an m -sequence as its indicator sequence,which is proved to be lower bounded by N − ⌈ log N ⌉ + 4 ( ≫ N ) and couldattain the maximum value N if suitable parameters are chosen, where N isthe period of the sequence.The rest of the paper is organized as follows. Some notations and definitionsare introduced in section 2. We describe the generalized construction and thedefinition of a Yu-Gong sequence in Section 3. In Section 4, we point out a veryinteresting and useful law of the autocorrelation values of a Yu-Gong sequence.Using the method of Hu and the law of the autocorrelation distribution of aYu-Gong sequence, we derive a lower bound on the 2-adic complexity of thissequence in Section 5. The following symbols will be used throughout the whole paper.(1) Z N is a ring of integers modulo N and Z + N = { t ∈ Z N | t = 0 } . itle Suppressed Due to Excessive Length 3 (2) F q is a finite field with q elements.(3) For positive integers n and m satisfying m | n , the trace function T r nm ( x )from F n to F m is defined by T r nm ( x ) = x + x m + · · · + x m ( nm − , x ∈ F n . Let s = ( s , s , · · · , s N − ) be a binary sequence of period N . Then theautocorrelation of s is given by AC s ( τ ) = N − X t =0 ( − s t + s t + τ , ≤ τ ≤ N − , (1)where τ is called a phase shift of the sequence s and t + τ is computed modulo N . The sequence s is called to have optimal autocorrelation if AC s ( τ ) satisfiesthe following:(1) AC s ( τ ) ∈ { N, , − } for N ≡ AC s ( τ ) ∈ { N, , − } for N ≡ AC s ( τ ) ∈ { N, − } for N ≡ AC s ( τ ) ∈ { N, , − } or { N, , } for N ≡ τ ’s. Specially, the case (3) is called to have ideal two-level autocor-relation. Additionally, for N ≡ τ ’s, it is called to haveperfect autocorrelation if AC s ( τ ) ∈ { N, } and optimal autocorrelation mag-nitude if AC s ( τ ) ∈ { N, , , − } . So far, the sequence (0 , , ,
1) of period 4 isa uniquely known binary sequence with perfect autocorrelation in the senseof cyclic equivalence. Hence, it is often used to construct new sequences withgood correlation and Yu-Gong sequence discussed in this paper is one of theapplications.Denote S ( x ) = N − P i =0 s ( i ) x i ∈ Z [ x ] and suppose S (2)2 N − N − P i =0 s ( i )2 i N − ef , ≤ e ≤ f, gcd( e, f ) = 1 . (2)Then the integer ⌊ log ( f + 1) ⌋ is called the 2-adic complexity of the sequence s and is denoted as Φ ( s ), i.e., Φ ( s ) = (cid:22) log (cid:18) N − N − , S (2)) + 1 (cid:19)(cid:23) , (3)where ⌊ z ⌋ is the largest integer that is less than or equal to z .It is well known that the 2-adic complexity of a binary sequence s withperiod N should be larger than N to resist the Rational Approximation Al-gorithm by Klapper et al. [6]. Yuhua Sun ∗ et al. m -sequence and Yu-GongsequenceConstruction 1 [18]: Let each column of a v × w matrix C = ( C i,j ) be givenby C ( i, j ) = c j ( i ) and c j = ( c j (0) , c j (1) , · · · , c j ( v − ≤ j ≤ w −
1, i.e., thematrix C can be expressed as C = c (0) c (0) · · · c w − (0) c (1) c (1) · · · c w − (1)... ... . . . ... c ( v − c ( v − · · · c w − ( v − . (4)If each sequence c j is either a cyclic shift of a binary sequence a = ( a , a , · · · , a v − )of period v or a zero sequence and the sequence u = { u t } is obtained by con-catenating the successive rows of the above matrix C , then u is called a ( v, w )interleaved sequence. By the definition, c j = L e j ( a ), 0 ≤ j ≤ w −
1, here L e j is a cyclic e j left shift operation, e j ∈ Z v or e j = ∞ if c j is a zero sequence.Adding a binary sequence b = ( b , b , · · · , b w − ) of period w to the sequence u , a new sequence s will be produced, which is denoted s := I ( a , e ) + b ,where u := I ( a , e ) and e = ( e , e , · · · , e w − ), and it still preserves the ( v, w )interleaved structure. We call a , e and b the base, the shift and the indicatorsequences of s , respectively.For a positive integer k >
1, let b be a binary m -sequence of period 2 k − w = 2 k −
1, where b t = T r k ( α t ) and α is a primitive element of thefinite field F k , 0 ≤ t ≤ k −
2. It is well known that b can be expressedas a (2 k − , k + 1) interleaved sequence [3], i.e., b = I ( a ′ , e ′ ), where thebase sequence a ′ = ( a ′ , a ′ , · · · , a ′ k − ) is a binary m -sequence of period 2 k − a ′ i = T r k ( β i ), 0 ≤ i ≤ k − β = α k +1 is a primitive elementof F k , and the shift sequence e ′ = ( e ′ , e ′ , · · · , e ′ k ) is given by e ′ = ∞ and β e ′ j = T r kk ( α j ) for 1 ≤ j ≤ k .A Yu-Gong sequence s = I ( a , e ) + b of period N = 4(2 k − × (2 k −
1) interleaved structure, where:(1) a = (0 , , ,
1) is the perfect binary sequence;(2) b is the binary m -sequence defined as above;(3) e is a sequence over Z represented as a (2 k − × (2 k + 1) interleavedstructure by a matrix E = ( e i,j ), where e i,j = (cid:26) i + δ (mod 4) , if j = 0;3( i + j ) (mod 4) , if 1 ≤ j ≤ k (5)for 0 ≤ i ≤ k − δ = 1 or − itle Suppressed Due to Excessive Length 5 In order to analyze the 2-adic complexity of the Yu-Gong sequence s , we needthe exact order according to the value τ of the autocorrelation value AC s ( τ )of s , which can be given by the following two results. Theorem 1 [18] Let s = I ( a , e ) + b be a Yu-Gong sequence of period N =4(2 k − , k > . Then, it has the four-valued optimal autocorrelation of AC s ( τ ) ∈ { N, , ± } for any τ . Precisely, its complete autocorrelation isgiven by AC s ( τ ) = N, if τ = 00 , if ( τ = 0 and x = 0)or ( x, y, v ) = ( σ, , v )or ( y, v ) = ( ψ, , − , if ( x, y, v ) = ( σ, , y, v ) = ( ψ, y, v ) = ( ψ, , +4 , if ( y, v ) = ( ψ, , (6) where x ≡ τ (mod 2 k − , y ≡ τ (mod 2 k + 1) , and v ≡ τ (mod 4) . Also, σ ∈ Z +2 k − , ψ ∈ Z +2 k +1 , v ∈ Z +4 , and Z + h = { , , · · · , h − } for a positiveinteger h . Now, the order of the autocorrelation value of s can be described as follows. Corollary 1
Let the symbols be the same as those in Theorem 1. Then thefollowing results hold:(1) Suppose ≤ τ , τ ≤ N − and τ ≡ τ (mod 4(2 k + 1)) . Then theautocorrelation function of the sequence s satisfies AC s ( τ ) = AC s ( τ ) .Particularly, AC s (4(2 k + 1) i ) = − , i = 1 , , · · · , k − ;(2) For ≤ τ ≤ k + 1) , if we divide the set of the autocorrelation values { AC s ( τ ) } k +1) τ =1 of s into k + 1 subsets S j ’s, j = 1 , , · · · , k + 1 , accord-ing to the order of τ and each subset contains four elements, i.e., S j = { AC s (4( j −
1) + 1) , AC s (4( j −
1) + 2) , AC s (4( j −
1) + 3) , AC s (4( j −
1) + 4) } ,Then S = S = · · · = S k − = {− , , − , } ; (7) S k − +1 = { , , − , } ; (8) S k − +2 = S k − +3 = · · · = S × k − = {− , , − , } ; (9) S × k − +1 = {− , , , } ; (10) S × k − +2 = S × k − +3 = · · · = S k = {− , , − , } ; (11) S k +1 = {− , , − , − } . (12) It should be pointed out that there are k − sets in Eq. (7), k − − sets inEq. (9), and k − − sets in Eq.(11). Yuhua Sun ∗ et al. Table 1: The autocorrelation of Yu-Gong sequence for k = 2 τ AC s ( τ )1 −
20 ( S − S − , , − , , , − , ; − , , − , − , , , ; − , , − , − ;21 − − , , − , , , − , ; − , , − , − , , , ; − , , − , − ;41 − − , , − , , , − , ; − , , − , − , , , ; − , , − k = 3 τ AC s ( τ )1 −
36 ( S − S − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − ;37 − − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − ;73 − − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − ;109 − − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − ;145 − − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − ;181 − − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − ;217 − − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − ; Proof
Note that τ ≡ τ (mod 4(2 k + 1)) implies τ ≡ τ (mod 2 k + 1) and τ ≡ τ (mod 4). Then, from Theorem 1, we find that τ and τ induce thesame pair ( y, v ), which leads to AC s ( τ ) = AC s ( τ ). Other results can also bedirectly verified by Theorem 1. (cid:3) Example 1
By direct computation using Matlab programs, the autocorrelationdistributions of Yu-Gong sequences for k = 2 , , S k − +1 , S × k − +1 , S k +1 inred. Especially, the number of sets in Eq. (11) is 0 since 2 k − − k = 2and the autocorrelation distribution is the set {− , , , , − , , − , , , , − } for k = 1 and τ = 1 , , , , , , , , , , In order to derive a lower bound on the 2-adic complexity of the Yu-Gongsequence s , we need employ the method of Hu [5]. It can be described as thefollowing Lemma 1, which have also been used in several other references [4,10,11,15]. Lemma 1 [5] Let s = (cid:0) s , s , · · · , s N − (cid:1) be a binary sequence of period N , S ( x ) = N − P i =0 s i x i ∈ Z [ x ] and T ( x ) = N − P i =0 ( − s i x i ∈ Z [ x ] . Then − S ( x ) T (cid:0) x − (cid:1) ≡ N + N − X τ =1 AC s ( τ ) x τ − T (cid:0) x − (cid:1) N − X i =0 x i ! (mod x N − . (cid:3) The following Lemma 2 is also important to prove our main result.
Lemma 2
Let k be a positive integer. Then the following results hold: itle Suppressed Due to Excessive Length 7 Table 3: The autocorrelation of Yu-Gong sequence for k = 4 τ AC s ( τ )1 −
68 ( S − S − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;69 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;137 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;205 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;273 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;341 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;409 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;477 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;545 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;613 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;681 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;749 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;817 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;885 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − , − ;953 − − , , − , − , , − , − , , − , − , , − , , , − , ; − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , − , − , , , ; − , , − , − , , − , − , , − , − , , − ; (1) For k ≡ , we have | gcd (cid:16) k − − k + 1 , k +1) + 15 (cid:17) . (13) But for k ≡ , we have ∤ gcd (cid:16) k − − k + 1 , k +1) + 15 (cid:17) . (14) (2) gcd (cid:16) k − − k + 1 , k +1) +15 (cid:17) = 1 , if 2 k − − k + 1 is aprime for k ≡ ,< k − , otherwise . Yuhua Sun ∗ et al. (3) gcd (cid:16) k − − , k +1 + 1 (cid:17) (cid:26) = 1 , if k is even ,< k − , otherwise . Proof (1) Suppose k = 4 t + 2. Note that the multiplicative order of 2 modular5 is 4. On one hand, we have2 k − − k + 1 = 2 t +3 − t +2 + 1 ≡ − + 1 ≡ , i.e., 5 | (2 k − − k + 1) , (15)on the other hand, since 5 | (2 t − | [4(2 t − | (2 k − k + 1 ≡ k + 1) ≡
10 (mod 20). It is easyto know that the multiplicative order of 2 modular 25 is 20. Then we have2 k +1) + 1 ≡ + 1 ≡ | k +1) + 15 . (16)Combining (15) and (16) we know that (13) holds. But, if k = 4 t , we have2 k − − k + 1 ≡ − ≡ k − − k + 1 is a prime. Then by little Fermat Theorem wehave (2 k − − k + 1) | (2 k − − k − . (17)Note that2 k +1) + 15 | (2 k +1) + 1) , (2 k +1) + 1) | (2 k +1) − . (18)Furthermore, we havegcd (cid:16) k − − k − , k +1) − (cid:17) = 2 gcd (cid:0) k − − k , k +1) (cid:1) −
1= 2 gcd (cid:0) k (2 k − − , k +1) (cid:1) − . (19)Without loss of generality, let k ≥ k = 1, the conclusion is trivial).Then gcd (cid:16) k (2 k − − , k + 1) (cid:17) = 4gcd (cid:16) k − − , k + 1 (cid:17) = 4gcd (cid:0) k − − , (cid:1) (20)= (cid:26) , if k is even , , otherwise , (21)where (20) holds because 2 k + 1 = 2(2 k − −
1) + 3. Thus, by (19), (21), weget gcd (cid:16) k − − k − , k +1) − (cid:17) = (cid:26) − , if k is even , − , otherwise , (22) itle Suppressed Due to Excessive Length 9 We know that 2 k +1) − k +1) − k +1) + 1) , (23)gcd(2 k +1) − , k +1) + 1) = 1 , (24)2 − − + 1) = 3 × , (25)3 | (2 k +1) − , ∤ (2 k +1) + 1) , (26)5 | (2 k +1) + 1) . (27)Therefore, by (22)-(27), we havegcd (cid:16) k − − k − , k +1) + 1 (cid:17) = 5 for an even k. (28)Finally, combining (13)-(14), (17)-(18) and (28), the result follows.(3) It is easy to known that2 k +1) − k +1 + 1)(2 k +1 − , gcd (cid:16) k +1 + 1 , k +1 − (cid:17) = 1 , gcd (cid:16) k − − , k +1) − (cid:17) = gcd (cid:16) k − − , k +1 + 1 (cid:17) × gcd (cid:16) k − − , k +1 − (cid:17) , (29)= 2 gcd (cid:0) k − , k +1) (cid:1) − , (30)gcd (cid:16) k − − , k +1 − (cid:17) = 2 gcd (cid:0) k − , k +1 (cid:1) − . (31)For an even k , k − (cid:0) k − , k + 1) (cid:1) = gcd (cid:0) k − , k + 1 (cid:1) , which results ingcd (cid:16) k − − , k +1) − (cid:17) = gcd (cid:16) k − − , k +1 − (cid:17) (32)by (30)-(32). Furthermore, combining (29) and (32), we can get the desiredresult. Example 2
By direct computation using Mathematica programs, we find thatthe smallest two positive integers k ’s such that 2 k − − k + 1 are primes and k ≡ ∗ et al. Theorem 2
Let s = I ( a , e ) + b be the Yu-Gong sequence of period N =4(2 k − , k > , introduced in Section 3. Then the 2-adic complexity Φ ( s ) of s satisfies the following lower bound Φ ( s ) = N, if k ≡ k − − k + 1) is a prime ,> N − log N + 1 , if k is even ,> N − N + 4 , otherwise , i.e., the 2-adic complexity of s far outweight one half of the period. (cid:3) Proof
Above all, by Lemma 1, we know that S (2) T (2 − ) ≡ − k − − X τ =1 AC s ( τ )2 τ − k − (cid:16) mod 2 k − − (cid:17) From Corollary 1 (1), if we add the value − { AC s ( τ ) } k − τ =1 , we will get a sequence segment consisting 2 k − { AC s ( τ ) } k +1) τ =1 . Therefore, we have S (2) T (2 − ) ≡ − n k − − X τ =1 AC s ( τ )2 τ + ( − × k − − ( − × k − o − k − (cid:16) mod 2 k − − (cid:17) = − n(cid:16) k − X i =0 k +1) i (cid:17)(cid:16) k +1) X τ =1 AC s ( τ )2 τ (cid:17) + 4 × k − o − k − (cid:16) mod 2 k − − (cid:17) ≡ − n(cid:16) k − − k +1) − (cid:17)(cid:16) k +1) X τ =1 AC s ( τ )2 τ (cid:17) + 4 o − k − (cid:16) mod 2 k − − (cid:17) = − (cid:16) k − − k +1) − (cid:17)(cid:16) k +1) X τ =1 AC s ( τ )2 τ (cid:17) − k +1 (cid:16) mod 2 k − − (cid:17) (33)From Corollary 1 (2), we have k +1) X τ =1 AC s ( τ )2 τ = (cid:16) k X i =0 i (cid:17)(cid:16) ( − × × + ( − × + 4 × (cid:17) − × k − (cid:16) ( − × × + ( − × + 4 × (cid:17) itle Suppressed Due to Excessive Length 11 +2 × k − (cid:16) × × + ( − × + 4 × (cid:17) − × × k − (cid:16) ( − × × + ( − × + 4 × (cid:17) +2 × × k − (cid:16) ( − × × + 0 × + 4 × (cid:17) − × k (cid:16) ( − × × + ( − × + 4 × (cid:17) +2 × k (cid:16) ( − × × + ( − × + ( − × (cid:17) = 4 × × k +1) − − × k +1 + 4 × × (2 k +1) − × × (2 k +1) = 8 n k +1) − − k + 2 × k +2 − × (2 k +1) o . (34)Bringing (34) into (33) and simplifying it, we get S (2) T (2 − ) ≡ − ( k − − k +1) − h k +1) −
15 + 2 k + 2 × k +2 − × (2 k +1) i + 2 k − ) (cid:16) mod 2 k − − (cid:17) . (35)On one hand, it is obvious that 2 − × k − −
1, onthe other hand, we can derive S (2) T (2 − ) ≡
13 (mod 15) , if k ≡ , , if k ≡ ,
10 (mod 15) , if k ≡ , , if k ≡ (cid:16) S (2) T (2 − ) , k − − (cid:17) > k ≡ , , . (37)In order to obtain a more exact lower bound on the 2-adic complexity of Yu-Gong sequence, we need to give some more detailed computation. Again from(35), we get S (2) T (2 − ) ≡ − k +1 mod 2 k − − k +1) − ! , (38) S (2) T (2 − ) ≡ − ( (2 k − h k +1) −
15 + 2 k + 2 × k +2 − i +2 k − ) (cid:16) mod 2 k +1) − (cid:17) , (39) ∗ et al. where (39) comes from the following congruence2 k − − k +1) − k +1)(2 k − − k +1) − ≡ k − (cid:16) mod 2 k +1) − (cid:17) . Furthermore, since 2 k +1) − × k +1) +15 × (2 k +1 + 1) × (2 k +1 − S (2) T (2 − ) ≡ − k − − k + 1) (cid:16) mod 2 k +1) + 15 (cid:17) , (40) S (2) T (2 − ) ≡ − k − − (cid:16) mod 2 k +1 + 1 (cid:17) , (41) S (2) T (2 − ) ≡ − k +1 (cid:16) mod 2 k +1 − (cid:17) . (42)Combining the results in Lemma 2, the proof is finished. Example 3
To ensure the correctness of our main result, at the same time,in order to compare the actual values with the lower bounds of the 2-adiccomplexity of Yu-Gong sequences obtained in this paper, we have done thefollowing verification work by combining Matlab and Mathematica programs:(1) For k = 1 , , , , , , ,
8, the correctness of the congruences (40)-(42) havebeen verified using the direct definitions of Yu-Gong sequences and themathematical expression S (2) T (2 − ).(2) For k = 1 , , , , , , ,
8, the actual values of the 2-adic complexity of Yu-Gong sequences have been determined by determining the correspondinggcd( S (2) , N −
1) in the definition of 2-adic complexity of binary sequences.And we list a table to compare the actual values and the lower bounds ofthe 2-adic complexity of Yu-Gong sequences (Please see Table 4).(3) From the process of determining the lower bound of the 2-adic complexityof Yu-Gong sequence, the value of gcd(2 k − − k + 1 , k +1) +15 ) is a keyfactor, which especially affects the cases of the maximal values of the 2-adiccomplexity. In Lemma 2, we proved gcd(2 k − − k + 1 , k +1) +15 ) = 1 onlywhen k ≡ k − − k + 1 is a prime. In fact, we findgcd (cid:16) k − − k + 1 , k +1) + 15 (cid:17) = 1 (43)for each k ∈ { , , , , , , , } by Mathematica programs and wecan not determine it for the cases of k ≡ k ≥
36 because ofthe limitation of computer performance. So we guess that (43) maybe holdfor all k ≡ itle Suppressed Due to Excessive Length 13 Table 4: A comparison between the actual values andthe lower bounds of the 2-adic complexity of Yu-Gong sequencesThe value of k The period N ofYu-Gong sequence The actual value ofthe 2-adic complexityof Yu-Gong sequence The lower bound ofthe 2-adic complexityof Yu-Gong sequenceobtained in this paper1 12 8 62 60 60 553 252 250 2404 1020 1020 10205 4092 4082 40726 16380 16380 163677 65532 65530 655048 262140 262140 262123 References
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