A New Method for Constructing Large Size WBE Codes with Low Complexity ML Decoder

Abstract — In this paper w e wish to introduce a method to reconstruct large size Welch Bound Equality (WBE) codes from small siz e WBE codes. T he advantage of these codes is t ha t the implementation of M L decoder for the large size codes is reduced to implementation of ML decoder for the co re codes. This le ads to a drastic reduction of t he computational cost of ML decoder. Our method can also be used for constructing large B inary WBE (BWBE) codes f rom s maller o nes . Additionally, w e explain that although WBE codes are maximizing the sum channel capacity when the inputs are real valu ed, they are not nec essarily appropriate when t he input alphabet is binary. The discussion shows tha t w hen the input alphabet is binary, the Total Squared Correlation (TSC) of codes is not a proper figure of merit. I. I NTRODUCTI ON In Direct Sequen ce Code Division Multiple A ccess (DS- CDMA) each user is assigned a signature vector for transmitting the data through a co mm on channel ( in time and frequency). As proved in [1] the s um c hannel capacity of such a syste m is maximized when the T otal Squared Correlation (TSC) o f the signature vectors meets its lower bound; i.e. the Welch bound [2]. T hese signature sets are called Welch Bound Equality (WBE) cod es [3]. It is w orth mentionin g t hat the channel capacity is maximized when the input distr ibution of the user data are Gaussian. Some methods for constructing WBE code s are proposed in [4-6]. For practical co nditions, it is favorable to use binary antipodal signatures. Limiting the signat ures to binar y antipodal vectors, signature s ets that meet the Welc h bound may not exist. For these codes, the lower bounds of TSC fo r different spreading factors ( ܮ ) and d ifferent number o f u sers ( ܭ ) are derived in [7 -8]. T hese bou nds are k nown as Karystinos-Pad os (KP) bounds and are presented in T abs. 1 and 2. Notice that in the under-loaded case wh en ܮ is a multiple of 4 , the KP bo und is equal to th e Welch bound. Furthermore, in the o ver-loaded case the KP bound is e qual to the Welch bo und when ܭ is a multiple of 4 . Some method s for constructing binar y a ntipodal signat ures that meet KP b ounds are propo sed in [3], [7-10]. Clearly, to have hi gh perfor mance dec oders for the over- loaded systems w e should p erform M ulti User Detectio n (MUD) [11-13] . So me exa mples o f the su ggested d ecoding methods are P IC [14-15] , SIC [1 6], and Iterative interference cancelation [17-18]. These methods are s omewhat heuristic and none of them is p roved to be optimum. P ractical realization of opti mum decoders has not been propo sed yet. In this paper, first we in troduce a meth od for constructin g large W BE co des b y e nlarging t he smaller ones. We also prove that our method can b e used for enlarging t he binar y antipodal codes when the KP bound meets the Welc h bou nd. These codes have the advantage of lo w co m putationa l complexity optimum decod er. In fact, their dec oding p roblem can be reduced to decoding pr oblems of the core codes. This means that the Maxi mum Likelihood (ML) d ecoding of the smaller codes p rovide ML decoding o f the large codes. T his leads to a dr amatic decrease in computatio nal co mplexity. For example, using the WBE codes that ar e co nstructed by t he proposed method, we simulate a CDMA syste m with 64 chips and 96 u sers w ith b inary a ntipodal input alp habet and ML decoding. It is noticeable that the ML decod er o f such s ystem was not pra ctically imple mentable until no w (notice t hat binary input alphabet is enco uraging fro m the pr actical point of view). In the next section we propo se a method for constructi ng large W BE co des fro m small er ones. Section III includes the method of decoding the proposed codes. Special discussions about CDM A syste ms with binary antip odal input alphabets are done in section IV. Simulation results are discussed in section V. Sectio n VI consists of conclusion a nd future works. Processing Gain Number of Users Lower Bound on TSC ܮ ≡ 0 ( mod 4 ) Any ܭ ܭ ܮ ≡ 2 ( mod 4 ) ܭ ≡ 0 ( mod 2 ) ܭ + 2 ௄ ( ௄ ି ଶ ) ௅ మ ܭ ≡ 1 ( mod 2 ) ܭ + 2 ቀ ௄ ି ଵ ௅ ቁ ଶ ܮ ≡ 1 ( mod 2 ) Any ܭ ܭ + ௄ ሺ ௄ ି ଵ ሻ ௅ మ Tab. 1. Under-loaded DS-CDMA System ( ܭ ≤ ܮ ) [ 7] Number of Users Processing Ga in Lower Bound on TSC ܭ ≡ 0 ( mod 4 ) Any ܮ ௄ మ ௅ ܭ ≡ 2 ( mod 4 ) ܮ ≡ 0 ( mod 2 ) ௄ మ ௅ + 2 ௅ ି ଶ ௅ ܮ ≡ 1 ( mod 2 ) ௄ మ ௅ + 2 ቀ ௅ ି ଵ ௅ ቁ ଶ ܭ ≡ 1 ( mod 2 ) Any ܮ ௄ మ ௅ + ௅ ି ଵ ௅ Tab. 2. Over-loaded DS-CDMA System ( ܭ ≥ ܮ ) [7] Mohamm ad Javad Faraji, Pedram Pad, and Farokh M arv asti Advanced Comm unications Resea rch Institute (ACRI ) Department of E lectrical Eng ineering, Sharif Un iversity of Technolog y, Tehran, I ran Email: {faraji, ped ram_pad}@ ee.sharif.edu, m arvasti@sharif.edu A New Method for Constructing Lar ge Size WBE Codes with Low Complexity ML Decoder id11484762 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com II. N EW M ET HOD FOR C ONST RUCTING WBE C ODES A CDMA syste m in an AWGN channel can be modeled as ܻ = ۱ܺ + ܰ (1) where ۱ is the ܮ × ܭ code matrix (eac h of its columns is the signature of eac h user), ܺ is the ܭ × 1 vector of the user data, ܰ is a Gaussian noise vector with zero mean a nd auto- covariance matrix o f ߪ ଶ ۷ ௅ (where ۷ ௅ is the ܮ × ܮ identity matrix). For maximizing the su m channel cap acity of such a system, we should fi nd codes which are as orthogonal a s possible according to the T SC criterion [1 ]. As defined in [1] TSC ሺ ۱ ሻ = ෍ ෍หܥ ௜ ୌ ∙ ܥ ௝ ห ଶ ௄ ௝ ୀଵ ௄ ௜ ୀଵ ( 2) where ܥ ௜ is the ݅ th column of ۱ which is nor malized. As proved in [2 ] TSC ሺ ۱ ሻ ≥ ൜ ܭ ܭ ≤ ܮ ܭ ଶ ܮ Τ ܭ ≥ ܮ = (3) The c odes that their TSC meet this lo w er bo und are called WBE code s [3]. Using the obviou s expression that TSC ሺ ۱ ሻ = ෍ ෍ห݀ ௜௝ ห ଶ ௄ ௝ ୀଵ ௄ ௜ ୀଵ ( 4) where ۲ = ൣ݀ ௜௝ ൧ = ۱ ୌ ۱ , w e co nstruct large W BE codes fro m smaller ones in the following theore m . Theorem 1 If ۱ is a ܮ × ܭ W BE matri x and ۿ is a ݀ × ݀ unitary matrix, the n ۿ ⊗ ۱ is a ݀ܮ × ݀ܭ WBE matrix, where ⨂ denotes the Kro necker prod uct. Proof : Let ۲ = ൣ ݀ ௜௝ ൧ = ۱ ୌ ۱ . We have ۳ = ൣ݁ ௜௝ ൧ = ሺ ۿ ⊗ ۱ ሻ ୌ ⋅ ሺ ۿ ⊗ ۱ ሻ = ۷ ௗ ⨂ ۲ (5 ) According to (4 ) TSC ሺ ۿ ⊗ ۱ ሻ = ෍ ෍ห݁ ௜௝ ห ଶ ௗ௄ ௝ୀ ଵ ௗ௄ ௜ ୀଵ ( 6) Using (5) and ( 6), we have TSC ሺ ۿ ⊗ ۱ ሻ = ݀ ∙ ෍ ෍ห݀ ௜௝ ห ଶ ௄ ௝ୀ ଵ ௄ ௜ ୀଵ = ݀ ∙ TSC ሺ ۱ ሻ (7) According to (7 ), if ܭ ≤ ܮ , th en TSC ሺ ۱ ሻ = ܭ and TSC ሺ ۿ ⊗ ۱ ሻ = ݀ܭ which i s equal to the We lch bound for the ݀ܮ × ݀ܭ matrix. If ܭ ≥ ܮ , then TSC ሺ ۱ ሻ = ܭ ଶ ܮ Τ and TSC ሺ ۿ ⊗ ۱ ሻ = ݀ ܭ ଶ ܮ Τ = ሺ ݀ܭ ሻ ଶ ሺ ݀ܮ ሻ Τ which is equal to the Welch bound for the ݀ܮ × ݀ܭ matrix. T herefore, ۿ ⊗ ۱ is a WBE matrix. ∎ This t heorem provides a systematic w ay for cons tructing large WBE codes from smaller ones. Example 1 In Theorem 1, if ۿ is the identity matrix, the n the code matrix ۿ ⊗ ۱ is equi valent to a multiple a ccess channel with ݀ T DMA channels, each of these channels consisting o f ܭ CDMA channels. Corollary 1 I n T heorem 1, if ۱ is a binary antipodal m atrix and ܮ and ܭ are in the mode that t he KP bound eq uals the Welch bound, then ቀ ଵ ξ ௗ ۶ ௗ ቁ ⊗ ۱ is a ݀ܮ × ݀ܭ binary antipodal m atrix where ۶ ௗ is a ݀ × ݀ Hada mard matrix. In the next section we will p ropose a method for re ducing the decoding proble m o f the large codes constructed in Theorem 1 to the d ecoding problem of the smal ler codes. III. T HE D ECODI NG OF THE P ROPOSED C O DES In this sectio n we use the special struct ure of the pro posed large codes to reduce their decoding proble m to the deco ding problem of the s maller codes. Suppose ۱ is a ܮ × ܭ matrix and ۿ is a ݀ × ݀ unita ry matrix. Si milar to (1) in an AWGN c hannel the received vector is ܻ = ሺ ۿ ⊗ ۱ ሻ ܺ + ܰ (8) In whi ch ܺ , ܰ and ܻ are ݀ܭ × 1 , ݀ܮ × 1 a nd ݀ܮ × 1 vectors, respectively. Multip lying both sides b y ۿ ୌ ⊗ ۷ ௅ , we have ܻ ᇱ = ሺ ۿ ୌ ⊗ ۷ ௅ ሻ ܻ = ሺ ۿ ୌ ⊗ ۷ ௅ ሻሺ ۿ ⊗ ۱ ሻ ܺ + ܰ ′ = ሺ ۷ ௗ ⨂ ۱ ሻ ܺ + ܰ ′ (9) where the subscrip t of ۷ determines the d imension o f the identity matrix and ܰ ᇱ = ሺ ۿ ୌ ⊗ ۷ ௅ ሻ ܰ . Obviously, ሺ ۿ ୌ ⊗ ۷ ௅ ሻ ୌ ∙ ሺ ۿ ୌ ⊗ ۷ ௅ ሻ = ሺ ۿ ⊗ ۷ ௅ ሻ ∙ ሺ ۿ ୌ ⊗ ۷ ௅ ሻ = ۷ ௗ௅ (10) and thus ۿ ୌ ⊗ ۷ ௅ is a u nitary matrix. Since ۿ ୌ ⊗ ۷ ௅ is a unitary matri x and ܰ is a Gaussia n random vector with z ero mean and auto-covariance matrix ߪ ଶ ۷ ௗ௅ , ܰ ′ is a random vector with prop erties identical to ܰ . In other words, the entries o f ܰ ′ are independent Gaussian ran dom variable with zero mean and variance ߪ ଶ . Hence, the ML e xtraction of the ܺ vector fro m ܻ is equivalent to M L extraction of the ܺ vector from ܻ ᇱ . Rewriting (9), w e have ܻ ᇱ = ൥ ۱ ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ ۱ ൩ ܺ + ܰ ′ (11) This means that the first ܮ entries of ܻ ′ depend only on t he first ܭ entries of ܺ and first ܮ e ntries of ܰ ′ , the second ܮ entries of ܻ ′ depend only on the second ܭ e ntries of ܺ and second ܮ entries of ܰ ′ , and so o n. In other w ords, the prob lem of detecting ܺ from the received vector is decoupled to ݀ smaller problems. This lea ds to a dramatic reduction in computational co mplexity in overload ed systems ( ܭ ≥ ܮ ). In th e remaining of this paper, w e f ocus on the CDMA systems with binar y antipoda l input alp habet i.e., from now o n we assu me that the data vector ܺ in (1) belongs to the set ሼ − 1, +1 ሽ ௄ . Corollary 2 Suppose that we have a CDM A system with ݀ܮ chips, ݀ܭ users and WBE signatures. For direct implementation of the M L decod er, we need to calculate a bout 2 ௗ௄ Euclidea n dista nces. But if we used the WBE signatures that have been pro posed in the previous section, for the decoding problem we need to calculate o nly ݀ × 2 ௄ Euclidean distance. This means a tre mendous reduction in the complexity of t he decoder. Some similar s ystems are simulated in sectio n V. IV. S PECIAL D ISCUSSIONS ABOUT C DMA S YSTEMS WITH B INARY A NTIPO DAL I NPUT A LPHABET We k now that the direct imple m entation of ML decoder from Bit Error Rate (BER) point o f vie w needs a large amount of co mputational co mplexity. In the following, we desire to propose a decoding method with very lo w computatio nal complexity which is, under some conditions, M L fr om Symbol Error Rate (SE R) point of view. We call this method SER Almost ML ( AML) d ecoding. Althou gh in the o ver- loaded case the SE R ML decoding i s not e quivalent to BER ML deco ding, the SER o ptimum dec oder is some what suboptimum from BER point of view. It will be seen i n the next section, despite the f act that SER AML decoding has significantly lower co mputational complexity t han BER ML, they have very si milar performance in slightl y high ܧ ௕ ܰ ଴ Τ . Performing the ML dec oder for an AWGN cha nnel from SER point of vie w, we have ܺ ෠ = argmin ௑ ∈ ሼ ିଵ , ାଵ ሽ ಼ ԡ ܻ − ۱ܺ ԡ (12 ) where ԡ ԡ indicates the E uclidean norm. Direct implementation of t his m ethod needs about 2 ௄ searches which is not prac tical for usual ܭ ’ s . No w s uppose that ۱ is full rank. B y per muting the co lumns o f ۱ we can write ۱ = ሾ = ۯ ȁ ۰ ሿ wh ere ۯ in an ܮ × ܮ invertible matrix. Rewriting (1), w e have ܻ = ۱ܺ + ܰ = ۯܺ ଵ + ۰ܺ ଶ + ܰ (13) where ܺ ଵ and ܺ ଶ are ܮ × 1 a nd ( ܭ − ܮ ) × 1 vector s, respectively. Now multiplying both sides b y ۯ ିଵ , we arrive at ۯ ିଵ ܻ = ܺ ଵ + ۯ ିଵ ۰ܺ ଶ + ܰ ′ (14) where ܰ ᇱ = ۯ ିଵ ܰ . Similar to (12 ), we are looking for ܺ ෘ = ቈ ܺ ෘ ଵ ܺ ෘ ଶ ቉ = argmin ௑ భ ∈ ሼ ିଵ , ାଵ ሽ ಽ ௑ మ ∈ ሼ ିଵ , ାଵ ሽ ಼షಽ ԡ ۯ ିଵ ܻ − ܺ ଵ − ۯ ିଵ ۰ܺ ଶ ԡ (15) But it is clear that t he nearest ሼ − 1, + 1 ሽ -vector to a vector ܼ is ݏ݅݃݊ ሺ ܼ ሻ which is obtained by s ubstituting the positive entries of ܼ with + 1 and its negative entries w ith − 1. Using this fact, we find that ܺ ෘ ଶ = argmin ௑ మ ∈ ሼ ିଵ , ାଵ ሽ ಼షಽ ԡሺ ۯ ିଵ ܻ − ۯ ିଵ ۰ܺ ଶ ሻ − ݏ݅݃݊ ሺ ۯ ିଵ ܻ − ۯ ିଵ ۰ܺ ଶ ሻ ԡ (16) and ܺ ෘ ଵ = ݏ݅݃݊ ൫ۯ ିଵ ܻ − ۯ ିଵ ۰ܺ ෘ ଶ ൯ (17) Having a pr ecise look at (1 6) and (17) , we discover that it only needs to search a mong 2 ௄ି௅ vecto rs rather t han 2 ௄ vectors. It means a significant decr ease in the complexity of the d ecoder. Now notice that i f ۯ is a unit ary matrix, then ܰ ′ is a noise vector with the same prop erties as ܰ and thus ܺ ෘ = ܺ ෠ . T his means that by following ( 16) and (17) we can imple ment the SER ML decoder with a much lower computational complexity than the direct im plementatio n of the SER ML decoder . However, we call the decoder that is obtained by (16) and (17) Alm ost ML ( AML) decoder. It is worth mentioning t hat if the multiplication of th e matrix ۱ with ሼ − 1, +1 ሽ -vectors is not o ne-to-one, then there are always case s that ܺ ෠ and ܺ ෘ are not uniq ue. This leads to a n interference that cannot be re m oved and an intri nsic non-zero probabilit y of er ror. Consid ering this point, we will sho w through simulatio ns that despite the fac t that the codes with minimum T SC maximize the channel capacit y when t he inp ut data are real or complex, t hey are not neces sarily appro priate for a CDMA s ystem with bi nary inputs. In other words, we will show that the T SC of a c ode is not a proper criterion for comparing the performance o f the codes w hen the i nput alphabet is bi nary. Particularl y, we will see W BE co des with very high noiseless BER and two codes with the same TSC and d iff erent perfor m ances. These phe nomena bring t his idea to mind that when the input alphabet is binar y the main criterion that should be considered for designing a code is the extent to which it s hows injective prop erties. Simulation results for verif ying the discussed facts a re covered in the next section. V. S IMULATI ON R ESULTS To verify the previous discussions, we simulated four binary i nput CDMA systems with differe nt binary a ntipodal signature matrices and dif ferent d ecoding schemes. In t he following, ۱ ௅ × ௄ and ۱ ′ ௅ × ௄ denote different ܮ × ܭ BWBE codes which are c onstructed through the flo wch art that is depicted in [7]. In add ition, ۹ ௗ = ଵ ξ ௗ ۶ ௗ where ۶ ௗ denotes a ݀ × ݀ Hadamard matrix. Simulation 1 The first syste m has a spread ing factor of 56 with 6 4 users. Accord ing to Tab. 2 and equation (3 ), when ܮ = 7 a nd ܭ = 8 the KP bound equals the W elch bound. Thus, ac cording to T heorem 1, ۹ ଼ ⊗ ۱ ଻ × ଼ is a 56 × 64 BWBE code. The perfor m ance c urves for this cod e, using BER ML, SER AML and Iterative inter ference cancellati on with soft-li miter (IT) are depicted in Fig. 1 . Additionally, t he performance curve for ۱ ହ଺ × ଺ସ with IT deco der is derived. Fig. 1 . B ER versus ܧ ௕ ܰ ଴ Τ for a system wi th 56 chips a nd 64 users using different decoding methods. I n the figure, * indicates the Kronecker product. As it is clear fro m Fig. 1 there is a signi ficant amount o f error that cannot be removed even b y increasing ܧ ௕ ܰ ଴ Τ (notice that the op timum decoder is performed). Simulation 2 The second syste m has a spreading factor of 64 with 72 users. Accor ding to Tab . 2 and eq uation (3) , when ܮ = 8 and ܭ = 9 the KP bound is greater than the Welch bound. T hus, accord ing to (7), ۹ ଼ ⊗ ۱ ଼ × ଽ and ۹ ଼ ⊗ ۱ ′ ଼ × ଽ are not B WBE codes but hav e the same TSCs which is ver y n ear to the KP bound. We call such codes Almost BWBE (ABWBE). The advantage of t hese codes is that t heir optimum and suboptimum decode r can be implemented just through the method proposed in section I II and IV. T he p erf ormance curves of ۹ ଼ ⊗ ۱ ଼ × ଽ , ۹ ଼ ⊗ ۱ ′ ଼ × ଽ and ۱ ଺ସ × ଻ଶ using differ ent decoding methods ar e depicted in Fig. 2. Some intere sting pheno mena can be seen through this simulation. T he first intere sting p oint is th at, desp ite the f act 2 6 10 14 18 22 0 0.01 0.03 0.05 0.07 0.09 E b / N 0 ( dB) BER K 8 *C 7x8 A ML C 56x64 IT K 8 *C 7x8 IT K 8 *C 7x8 ML K 8 *C 7x8 MAP T h e c urv es are quit e c oinc ide that the T SC of ۹ ଼ ⊗ ۱ ଼ × ଽ and ۹ ଼ ⊗ ۱ ′ ଼ × ଽ are equal, they have co m pletel y different per formances. U sing ۹ ଼ ⊗ ۱ ଼ × ଽ the BER tends to 0 when ܧ ௕ ܰ ଴ Τ increases. Ho wev er, when we use ۹ ଼ ⊗ ۱ ′ ଼ × ଽ the BER cannot be lower t han a speci fic value. T he second p oint is t hat, although the B WBE matrix ۱ ଺ସ × ଻ଶ has better performance in low values of ܧ ௕ ܰ ଴ Τ , its BER curve sat urates and is above the BER curve of the ABWBE ۹ ଼ ⊗ ۱ ଼ × ଽ matrix. Although some a mount of the error o f ۱ ଺ସ × ଻ଶ m ay be introd uced through the non-optimum IT decoder, it has an intrinsic amount o f error beca use of t he non-invertibilit y o f its mapping on ሼ − 1, +1 ሽ ଻ଶ . Notice that t he SER AM L d ecoder of ۱ ଼ × ଽ is e quivalent to its SER ML decoder because its first 8 columns is a unitar y matrix. Fig. 2 . B ER versus ܧ ௕ ܰ ଴ Τ for a system wi th 64 chips a nd 72 users using different codes a nd decoding methods. In th e figure, * i ndicates the Kronecker product. Simulation 3 T he third si mulation is a system with 64 chips and 96 users. The si mulations a re d one for ۹ ଼ ⊗ ۱ ଼ × ଵଶ and ۱ ଺ସ × ଽ଺ w ith di fferent decoding schemes. Notice that a ccording to T ab. 2, eq uation (3) and Theore m 1, ۹ ଼ ⊗ ۱ ଼ × ଵଶ is a 64 × 96 B WBE code. The results are depicted in Fig. 3. It is surprising to see that in this ca se rando m code s perform better than WBE codes in high ܧ ௕ ܰ ଴ Τ . This again shows that being WBE and having low TSC is not a proper criterion when the input alp habet is binary. Simulation 4 The last simulatio n is a s ystem with 64 chips and 104 users. Again by referring to Tab. 2 a nd equation (3), we find that ۹ ଼ ⊗ ۱ ଼ × ଵଷ and ۹ ଼ ⊗ ۱ ′ ଼ × ଵଷ are not BWBE codes. The c ontradicting point i n this simulation is the d ifferent performance o f ۹ ଼ ⊗ ۱ ଼ × ଵଷ and ۹ ଼ ⊗ ۱ ′ ଼ × ଵଷ that h ave the same T SCs. H owever, in this case the B WBE codes perfor m better than the others. VI. C ONCLUSION AND F UTURE W ORKS In this pap er we introduced a ne w method for co nstructing large WBE co des from s maller ones. T he advantage o f th ese codes is that their deco ding can be reduced to the decoding of the smaller codes. This lea ds to a dra matic decrease in decoding complexit y in overloaded cases. Additionally, we show that this met hod can be used for enlargin g BWBE codes and arriving at o ther B WBE c odes or ABWBE co des. Using these enlarged codes, w e si mulated some CDMA systems with binary input a nd binary signatures that e m ploy lo w computational o ptimum or sub- optimum d ecoders. An important result o f these sim ulations i s that, T SC criter ion is not an approp riate figure o f merit when the input a lphabet is binary ( which is a case that is most encouraging in the practice). Hence despite the fact that WBE codes are opti m um when the input data vector is real or c omplex [1], the y ar e not appropriate for systems with binar y inputs. Fig. 3. BER versus ܧ ௕ ܰ ଴ Τ for a system with 64 chips and 96 users using diff erent co des and decoding methods. In the figure, * indicates the Kronecker product. Fig. 4. BER v ersus ܧ ௕ ܰ ଴ Τ for a system with 64 chips and 104 users using different codes a nd decoding methods. In the fi gure , * indicates the Kronecker product. R EFERENCES [1] M. Rupf and J.L. Massey , “ Optimum Sequences multisets for synchronous c ode-divis ion multiple-access channels, ” IEEE Trans. Inform. Theory , vo l. 40, pp. 1261 – 1266, Jul. 1 994. [2] R.L . Welch, “ Lower bounds on the maximum c ross correlation of signals, ” IEEE Trans. Inform. Theory , vol . IT-20, pp. 397 – 399, May 1974. 2 6 10 14 18 22 10 -7 10 -5 10 -3 10 -1 E b /N 0 (dB) BER K 8 *C' 8x9 ML K 8 *C 8x9 IT C 64x72 IT K 8 *C 8x9 AML K 8 *C 8x9 ML, MAP 2 6 10 14 18 22 0. 03 0. 05 0. 07 0. 09 0. 11 0. 13 0. 15 0. 17 E b /N 0 ( dB) BER Ra ndom IT K 8 *C 8x12 A ML C 64x96 IT K 8 *C 8x12 IT K 8 *C 8x12 ML K 8 *C 8x12 MAP 2 6 10 14 18 22 0.02 0.06 0.1 0.14 0.18 E b /N 0 (dB) BER K 8 *C' 8x13 ML K 8 *C 8x13 IT K 8 *C 8x13 A ML K 8 *C 8x13 ML K 8 *C 8x13 MAP C 64x10 4 IT [3] J.L . Massey an d T. Mittelholzer , “ Welch ’ s bound a nd sequence sets for code-division multiple-access sy stems, ” in Sequences II, Methods in Communication, Security, and C omputer Sciences , R. Capocell i, A. De Santis, and U. Vacc aro, Eds. Ne w York: Springe r-Ve rlag, 1993. [4] P. Cotae, “ A n algo rithm for obtaining Welch bound equality se quences for S-CDMA ch annels, ” AE Ü Int. J. Electron. Co mm un. , vol. 55, pp. 95 – 99, Mar. 20 01. [5] S. Ulukus and R. D. Yates, “ Iterative construction of optimum signature sequence sets in sy nchronous CD MA systems, ” IEEE Trans. Inform. Theory , vol. 47, p p. 1989 – 1998, J ul. 2001. [6] C. Rose, “ CDMA codeword o ptimization: interfe rence avoidance and converge nce via class warfare, ” IEEE Trans. Inform. Theory , vo l. 47, pp. 2368 – 2382, Se p. 2001. [7] G .N. K arystinos and D.A. Pad os , “ New bounds o n the total s quared correl ation and optimum design of DS- CDMA bin ary si gnature sets, ” IEEE Trans. Comm un. , vol. 51, no. 1, pp. 48 – 5 1, Jan. 2003. [8] V.P. I patov, “ On the Karystinos-Pado s Bounds and Optimal Binary DS- CDMA Signature En sembl es ” IEEE Commun. Letters , vol . 8, no. 2, pp. 81 – 83, Feb. 2004. [9] D. Sarw ate, “ Meeting the Welch bound w ith equality, ” in Sequences and Their Applications: Pro ceedings of SETA ’ 9 8 , C. Ding, T. Helle seth, and H. Niederre iter, Eds. Lo ndon, U.K.: Springer- Verlag, 1999. [10] H.D. Schotte n, “ Sequence families with optimum aperiod ic mean -square correl ation pa r ameters, ” in Sequences and Their Applicati ons: Proceedings of SETA ’ 98 , C. Ding, T. Hel leseth, and H. Niederre iter, Eds. Lo ndon, U.K.: Springer -Verlag, 1999. [11] S. Verdu, Multiuser Dete ction, Cambr idge Univer sity Press, New York, NY, USA , 1998. [12] A. Kapur and M.K. Varanasi, “ Multiuser detection for overloaded CDMA systems, ” IEEE Trans. Inform. Theor y , vol. 49, no. 7, pp. 1 728 – 1742, Jul. 2003. [13] S. Moshavi, “ Multi -user detection for DS -CDMA communications, ” IEEE Commun. M ag. , vol. 34, no. 10, pp. 124 – 136, Oct. 1996. [14] D. Guo, L.K. Rasmussen, S. Sun, and T.J. L im, “ A matrix -alg ebraic approach to linear p aralle l in terfe rence cancellation in CDMA , ” IEEE Trans. Commun. , vol . 48, no. 1, pp. 152 – 161, Ja n. 2000. [15] G. Xue, J. Weng, T. Le -Ngoc, and S. Tahar, “ Adaptive multistage parallel int erf erence cancellatio n for CDMA, ” IEEE JSAC , vol . 17, no. 10, pp. 1815 – 1827, Oct. 1999. [16] M. Kobayashi, J. Boutros, and G. Caire, “ Successive interference cancellation with S I SO decoding and EM channel estimation ” , IEEE JSAC , vol. 19, no. 8, pp. 1450 – 14 60, Aug. 2001. [17] X. Wang and H.V. Poor, “ I terative (turbo) soft interfe rence ca ncell ation and decoding for coded CDMA, ” IEEE Trans. Commun. , vol. 47, no. 7, pp. 1046 – 1061, Jul. 1999. [18] M.C. Reed, C.B. Schlegel, P.D. Alexander, and J.A. Asenstorfer, “ Iterative multiuse r detection for CDMA with FEC: near -single -user performance, ” IEEE Trans. Commun. , vol. 46, no. 12, pp. 1693 – 1699, Dec. 1998.