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Large families of sequences for CDMA, frequency hopping, and UWB

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This paper generalizes three constructions of families of sequences with bounded off peak correlation with application to Code Division Multiple Access (CDMA), frequency hopping , and Ultra Wide Band (UWB). These new families present flexible family
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  Cryptography and Communicationshttps://doi.org/10.1007/s12095-019-00399-x LargefamiliesofsequencesforCDMA,frequencyhopping,andUWB DomingoG´omez-P´erez 1 · AnaI.G´omez 1 · AndrewTirkel 2 Received: 31 January 2019 / Accepted: 3 September 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract This paper generalizes three constructions of families of sequences with bounded off peak correlation with application to Code Division Multiple Access (CDMA), frequency hop-ping, and Ultra Wide Band (UWB). These new families present flexible family sizes andsequence lengths, making them well suited to wireless communications and Multiple InputMultiple Output (MIMO) radar. In particular, we show that the generalized Kamaletdinov Iconstruction offers the lowest off-peak correlation for a given family size. For the twosmallest family sizes, this construction asymptotically matches the performance of the smallKasami set and Gold codes, respectively. Among other properties, it has one of the slow-est off-peak correlation growth of all known constructions, increasing proportionally to thelogarithm of the family size and generates frequency and time hopping sequences. Keywords  Binary sequences · Array · Off-peak correlation · Polynomials · Discriminationfactor MathematicsSubjectClassification(2010)  94A55 · 94A55 · 68P30 The research of the first and second author was supported by the Spanish Ministerio de Econom´ıa yCompetitividad research project MTM2014-55421-P.This article is part of the Topical Collection on  Special Issue on Sequences and Their Applications   Domingo G´omez-P´erezdomingo.gomez@unican.esAna I. G´omezgomezanai@unican.esAndrew Tirkelatirkel@bigpond.au 1 Department of Mathematics, Statistics and Computer Science, University of Cantabria,Santander, Spain 2 Scientific Technology, 8 Cecil Street, Brighton East 3187, Australia  Cryptography and Communications 1 Introduction Families of sequences with low off-peak auto-correlation and cross-correlation are essentialin multiuser wireless communications [6]. They are required in countless applications likeCode-Division Multiple Access (CDMA), frequency hopping, and Ultra Wide Band (UWB)[14].Binary sequences are preferred for CDMA because of their easy implementation andtheir full efficiency (highest peak autocorrelation) and resistance to channel noise andhardware imperfections. As previously stated, this concept relies on a family of binarysequenceswithlowoff-peakauto-correlationandcross-correlation,alsoreferredasoff-peak correlation for short.Another important area of application is multiple-input multiple-output (MIMO) radar.Such radars are now being considered as future replacements for classical beamformingradars and mechanically scanned radars [1]. MIMO radar consists of arrays of transmitter and receiver elements. Each transmitterelement sends out a signal encoded with a unique signature sequence. Then, each receiverelement processes each transmitted sequence. In the far field this generates a virtual arrayof transceivers with better resolution and lower range and cross-range sidelobes (aliasingartefacts) than physical arrays. The far field is radiation from all array elements, consideringan error within the phase less than say one degree [24]. In order to select an appropriate family of sequences for a MIMO radar, the averageand peak correlation must comply with a figure of merit and provide good discriminationbetween sequences and shifts of other sequences of the family. For this, it is necessaryto correctly select the parameters of the family, i.e. sequence length and family size. Thesequence length is determined by the maximum unambiguous radar range and signal tonoise ratio considerations. The family size must match or exceed the number of transmitterelements, that may be determined by cost, power consumption, layout complexity, or localoscillator feed distribution. Usually, the desirable family size is a power of 2, 3 or a productof these for one-dimensional arrays; or a power of 4 in the case of two-dimensional arrays.Detailed requirements of the sequence families, and signal waveforms for MIMO radarsare analysed in [11, 20]. The absence of suitable sequences has resulted in practical imple- mentations which deploy time domain multiplexing instead of CDMA [8]. This requiressequential pulsing of each transmitter. The low duty cycle of each transmitter reduces theSNR, and the sequential pulsing of transmitters requires the target to be stationary during animage acquisition. This limits the applications to short range, slow moving targets, such asimaging for security screening. By contrast, CDMA permits 100% duty cycle, and removesthese restrictions.In order to meet these competing requirements, one straightforward solution is to gen-erate families of random binary sequences. Recently, M´erai [12] has studied the off-peak  correlation of families of random binary sequences taking only into account the family sizeand sequence length. Despite being of theoretical interest, the bound on the off-peak corre-lation provides only a probabilistic estimate and checking is required after the family hasbeen generated. Random sequences in general do not have useable bounds on the off peak correlation, which can result in ambiguous detection, false detection or erroneous decodingin applications such as radar.By contrast, the algebraic constructions presented in this work present an average off peak correlation similar to random sequences but a fixed bound is proven for the off peak correlation. In fact, for one of the families, the bound is close to the lowest possible orlowest known value for a given sequence length and family size. Important measures in  Cryptography and Communications applications are the ratio between the peak correlation and the largest off peak value, whichis known as the discrimination factor and the ratio of the peak correlation to the average off peak value, called a figure of merit [13]. Only algebraic instructions deliver guaranteed highdiscrimination factors, therefore are preferred in applications.Classical constructions, such as Kasami sets and Gold codes, provide a relatively bigfamily size with small off peak correlation bound, approximately equal to the square root of the sequence length. Tone et al. [26] proposed also the use of families of GMW sequences, which is very plausible, but one of the major drawbacks to using GMW is again the fam-ily parameters. For a nice compendium of these constructions and properties, see the book by Golomb and Gong [3]. While these constructions are optimal with respect to the off- peak correlation, they may not meet the design requirements of MIMO radar, because of thestrong restrictions on the family size and sequence length. Similarly, in wireless communi-cations the number of active subscribers and desired sequence length may forbid their use.This has led authors like Yu and Gong [29], Zhou and Tang [30], Parampalli [18] and Tirkel [9, 17] among others to investigate new families for certain lengths and family sizes. Sequence designs with larger and flexible family size for a fixed sequence length havebeen described by Gong and Yu [29] and have all been found to be sub-optimal with respect to the Sidelnikov bound [22], diverging further from the bound as the family size increases. Also the sequence length is restricted to, approximately, powers of two. Although, Param-palli, Tang and Botzas [18] showed bigger families, which include those found by Gong and Yu, sequence length has to be 2 m − 1. This is too restrictive in the design of CDMA systems.The method of composition, that generates families of sequences, has proven to be flex-ible in terms of family parameters. The idea behind this strategy starts with two inputsequences, referred to as base sequence and shift sequence, and outputs an array, con-structed by copying shifted version of the base sequence. The method of composition wasfirst sucessfully applied by Tirkel, Osborne and Hall [25] and generalized later by Moreno and Tirkel [16]. Also, two near-optimal constructions of families of sequences found byKamaletdinov [7] are naturally explained using the method of composition [16] and classi- cal finite field theory. In this case, the shift sequences are based on fractional functions andtrace maps.We remark that the method of composition is far more general and can compose arraysin the dimension of the desired output. For simplicity, in this article we focus on sequences,i.e. one-dimensional arrays.It is known that the best shift selection and order is provided by Costas-like arrays of commensurate size, i.e. those with the lowest number of auto- and cross-hits. These areusableintheirownrightasfrequencyhoppingortimehoppingpatterns.Therearealsosomeknown sequence families which, when converted into arrays are also composed of shifts of the same pseudo-noise sequence or blanks. The small Kasami set is an example. Similarly,the selected shifts to generate the small Kasami set can also be converted into frequencyhopping and time hopping sequences.In this paper, we propose three new families of shift sequences, that combining themethod ofcomposition with theLegendresequence,yield newfamilies ofbinarysequences.The new constructions are based on adding new sequences to certain known families, sothey can be seen as generalizations of the Kamaletdinov and Extended Rational Cycle Con-structions [9]. The resulting binary families are available for sequence lengths of   N(N   + 1 ) and  N(N   −  1 )  where  N   is a prime of the type 4 k  −  1, consequently these constructionsprovide many more suitable sequence lengths than powers of two. For primes of the type4 k  + 1, the result is almost binary sequences with  N   zeros in arrays of size order  N  2 , thusbecoming less efficient .  Cryptography and Communications There are two other advantages of these new constructions. First, they can also be inter-preted as families of arrays as explained for the parent constructions by Moreno and Tirkel[17]. Consequently, they provide frequency hopping or UWB families.Second, as families of two and three dimensional binary arrays with good correlation,they are also useful in watermarking of images and video respectively. Such family shouldmeet the criteria standardized by the Society of Motion Picture / Television Engineers(SMPTE) watermarking standard [23].From the three constructions studied, we show that the one extending Kamaletdinov Ifamily has the lowest off-peak correlation for a given family size. We also show that for thetwo smallest family sizes asymptotically matches the small Kasami set and the Gold codesand begins to diverge from the Sidelnikov bound when the family size approaches of thelarge Kasami set.For larger family size, the families have lower off-peak correlation than the bestconstruction of Gong and Yu [29] and Parampalli, Tang and Botzas [18]. Finally the family size is dependent on the factorization of   N   − 1 so, in general, this hasto be calculated when the correlation bound is given. However, for selected primes, we givethe exact family size and the distribution of such primes.The paper is organized as follows: In Section 2, we introduce the necessary preliminaries to understand the paper. Section 3 presents the main theorem for generating new sequencefamilies and then, apply it to three known constructions. These new families are comparedwith other families in Section 4 and the properties of the underlying shift sequences arestudied in Section 5. We finish with the conclusions in Section 6. 2 Preliminaries For a prime  N  , let F N  s  denote the finite field of   N  s elements. In the case  s  =  1, we identifythe elements of   F N   with the integers in the range  { 0 ,...,N   −  1 } .  F N  [ X ]  and  F N  [ X,Y  ] denote the rings of univariate and bivariate polynomials with coefficients in  F N  .In this work, sequences are either finite lists of zeros and ones or elements of   F N  . In thefirst case, the sequences are binary sequences, while in the second case the sequences are N  -ary sequences. Although a sequence  u(k)  is a finite list of length  L , it can be extendedperiodically for all indexes, i.e.  u(k + L)  =  u(k) .The cross-correlation function with shift  d   of two binary sequences ( u  and  v ) of length L  is defined by C d  (u,v)  = L − 1  k = 0 ( − 1 ) u(k) − v(k + d) .For a set of sequences,  u 0 ,...,u M  − 1 , the off-peak correlation is the maximum of the cross-correlation function among pairs of family sequences with non-trivial shifts, i.e. C max  = { max | C d  (u i ,u j  ) | :  0  ≤  i < M,  0  ≤  j < M,  0  ≤  d < L } , where  d   =  0 if   i  =  j  .The method of composition builds on two sequences: a binary sequence  s  of length  N  and a shift sequence  y  with elements in  { 0 ,...,N   − 1 }  and of length  L , coprime with  N  .The composition of   s  and  y  is the sequence of length  NL  defined by S(k)  =  s(k + y(k)), k  =  0 ,...,NL − 1.  Cryptography and Communications Fig. 1  Graphical representation of the arrays involved in Example 1. White squares represent 0, while 1 isrepresented as black squares. The indexing on  s(k)  is from  k  =  0 (bottom) to 2 (top). For  y(k) , each columnrepresents an element of the shift sequence, the first column represents the first value and so on. Finally, on S(k)  the values are arranged diagonally, that is starting from the lower left and wrapping around as if on atorus, i.e. identifying oposite sides. In the case of the shift sequence  y(k)  the position of the black squaresrepresents the shifts of the base sequence  Example 1  Consider the binary sequence  s  = [ 0 , 1 , 1 ]  (with length  N   =  3) and the shiftsequence  y  = [ 2 , 1 ] . The composition of both is defined in the following way: S( 0 )  =  s( 0 + y( 0 ))  =  s( 2 )  =  1 ,S( 1 )  =  s( 1 + y( 1 ))  =  s( 2 )  =  1 ,S( 2 )  =  s( 2 + y( 0 ))  =  s( 1 )  =  1 ,S( 3 )  =  s( 3 + y( 1 ))  =  s( 1 )  =  1 ,S( 4 )  =  s( 4 + y( 0 ))  =  s( 0 )  =  0 ,S( 5 )  =  s( 5 + y( 1 ))  =  s( 0 )  =  0.Figure 1 contains the graphical representation of the shift and base sequence with the resultof the method of composition.Moreno and Tirkel [17] applied this method to the binary Legendre sequence with  N  prime of the type 4 k − 1 to define several families of sequences. To prove properties on thecorrelation of these families of sequences, we need to recall the following definition. Definition 1  The difference function of two  N  -ary sequences ( y 1  and  y 2 ) of length  L  isgiven by (y 2 ∆y 1 )(k  :  t,d)  =  y 2 (k + t) + d   − y 1 (k)  for 0  ≤  k < L .This function associates a sequence to a pair of parameters  (t,d)  with 0  ≤  t < L  and0  ≤  d < N,  named the vertical and horizontal shifts, respectively. The hit array of twosequences  y 1  and  y 2  is an  L × N   array whose entry with index  (t,d)  is the number of values k  such that  (y 2 ∆y 1 )(k  :  t,d)  =  0. The hit array is called auto-hit array if   y 1  =  y 2  andcross-hit array otherwise. The maximum element in the hit array is an important parametercalled  number of hits .  Example 2  Take  N   =  3 and these two sequences  y 1  = [ 1 , 2 ]  and  y 2  = [ 2 , 0 ] , In this case, (y 2 ∆y 1 )(k  :  0 , 0 )  = [ 1 , 1 ] , the 2 × 3 hit array is  0 0 21 1 0 
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