In this study, the minimum number of antennas at each user is derived to obtain 2 M 2 degrees of freedom (DOF) in the M × 1 × M multiple‐input–multiple‐output (MIMO) multi‐user two‐way relay X channels. Based on the design of beamforming vectors for the conventional signal space alignment scheme, it is shown that each user needs at least ⌈ M 2 − ( M /2)⌉ antennas to obtain DOF 2 M 2 with the relay having M 2 antennas. As the number of users increases, the number of antennas also increases, which makes the implementation difficult. In an effort to reduce the number of antennas, the authors propose a new beamforming scheme of MIMO multi‐user two‐way relay X channels using the time extension. Through the numerical analysis, it is confirmed that the proposed scheme with time extension is a good alternative scheme to replace the conventional scheme.

In this paper, for an odd prime p such that p ≡ 3 mod 4, odd n, and d = (pn+ 1)/(pk+ 1) + (pn- l)/2 with k|n, the value distribution of the exponential sum S(a, b) when a and b run through Fpn is calculated. The weight distribution of the relevant cyclic code C over Fpwith the length L = pn- 1 and the dimension dimFpC = 2n In is also derived. Our result generalizes the case in [5].

In this paper, for an integer k, we evaluate an upper bound for the cross-correlation of a ternary m-sequence of period N = 34k+2- 1 and its decimated sequence with decimation d = 34k+2- 32k+1+2/4 + 32k+1. It is found that the cross-correlation is upper bounded by 4.5 · 32k+1+ 1.

In this letter, a new performance measure for compressed sensing matrices is proposed. This new measure is based on thek-set correlation vectors whose components consist of the correlation values between two columns in thek-column submatrices of a sensing matrix. This measure is highly related to the restricted isometry property (RIP). And the proposed measure has less computational complexity than the condition number approach which is a typical approach for performance prediction with RIP check. It is shown by simulation that the proposed scheme works well as a performance measure for the compressed sensing matrices.

In this paper, for an odd prime , we investigate into the cross-correlation of a p-ary m-sequence m(t) of period p;n;-1 and its d-decimated sequences m(dt+l), 0≤l;m;+1)/2, where d=(pm+1)2/2(p+l), n=2m, and m is an odd integer. There are (pm+1)/2 distinct decimated sequences m(dt+l) since gcd(d,pn-1)=(pm+1)/2. It is shown that the magnitude of the cross-correlation values is upper bounded by (p+1)/2 pn/2+1 . We also construct the sequence family F from these sequences, where the family size is pmand the correlation magnitude is upper bounded by (pm+1)/2 pn/2+1.

For a prime p ≡ 1 mod 4, an odd integer n, and d = pn+1/p+1 + pn-1/2, we investigate the cross-correlation values of p-ary m-sequence m(t) of period pn- 1 and its decimated m-sequence m(dt). It is shown that the cross-correlation function between m(t) and m(dt) takes the values in {-1, -1 ± pn/2, -1 ± 1+√p/2 pn/2, -1 ± p-1/2 pn/2}.

In this paper, a new family ofM-ary sequences of periodpn-1 is proposed. The proposed family is constructed by the addition of cyclic shifts of anM-ary Sidel'nikov sequence and its reverse sequence. The number of sequences contained in this family is about (M-1)2times of their period and the maximum magnitude of their correlation values is upper bounded by 4 √(pn)+5.

For an odd primepcongruent to 3 modulo 4 and an odd integern, a new family ofp-ary sequences of periodN=(pn-1)/2 with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and 2d,d=N-pn-1. The upper bound for the maximum magnitude of nontrivial correlations of this family is derived using well known Kloosterman sums. The upper bound is shown to be 2√(N+1/2) = √(2pn) , which is twice the Welch's lower bound and approximately 1.5 times the Sidelnikov's lower bound. The size of the family is 2(pn-1) , which is four times the period of sequences.

In this paper, we investigate into the cross-correlation of a ternary m-sequence m(t) of period 3n- 1 and its decimated sequence m(dt) by d = (3m+1)2/8, where n = 2m = 4k + 2. It is shown that the magnitude of the cross-correlation values is upper bounded by 2√(3n+ 1).

In this paper, we propose a method to construct p-ary sequence family S which has the period of N = (pn- 1)/2 when p is an odd prime and n is an odd integer. The family has the size of 2(pn- 1) and the magnitudes of correlation values between any two sequences from the family are upper bounded by 2√N + 1/2. The bound can be derived from the well-known Kloosterman sum.

In this paper, we propose the best relay selection scheme for the soft-decision-and-forward cooperative network with multiple relays. The term `best relay selection' implies that the relay having the largest end-to-end signal-to-noise ratio is selected to transmit in the second phase transmission. The approximate performances are analyzed in terms of pairwise error probability. Using the Fox's H-function, it is shown that the proposed scheme has full diversity order.

In this paper, the Alamouti code combined with partial response signaling (PRS) is proposed. The bit error rate of the proposed quadrature PRS (QPRS) Alamouti code is derived for the quasi-static Rayleigh fading and additive white Gaussian noise channel. Furthermore, its performance in a severely bandlimited channel is simulated by using chopping filter and compared with the conventional Alamouti code. The numerical results show that the proposed QPRS Alamouti code outperforms the conventional Alamouti code under the chopping environment.

In this letter, the Alamouti code combined with partial response signaling (PRS) is proposed. The bit error rate of the proposed quadrature PRS (QPRS) Alamouti code is derived for the quasi-static Rayleigh fading and additive white Gaussian noise channel. Furthermore, its performance in a severely bandlimited channel is simulated by using chopping filter and compared with the conventional Alamouti code. The numerical results show that the proposed QPRS Alamouti code outperforms the conventional Alamouti code under the chopping environment.

In this paper, a new extending method of q-ary low correlation zone(LCZ) sequence sets is proposed, which is a generalization of binary LCZ sequence set by Kim, Jang, No, and Chung. Using this method, q-ary LCZ sequence set with parameters (N,M,L, isin) is extended as a q-ary LCZ sequence set with parameters (pN, pM, plfloor(L + 1)/plfloor - 1, pisin), where p is prime and p|q.

In this paper, for a positive integer M and a prime p such that M|pn- 1, a family of M-ary sequences using the M-ary Sidel'nikov sequences with period pn-1 is constructed. This family has its maximum magnitude of correlation values upper bounded by 3radic(pn) + 6 and the family size is (M-I)2(2n-1-1) + M - 1 for p = 2 or (M-1)2(pn-3)/2+ M (M-1)/2 for an odd prime p.

In this correspondence, for a positive integerMand a prime p such that M|pn- 1, three families of M-ary sequences using theM-ary Sidel'nikov sequences with period pn- 1 are constructed. Two small families contain [(pn- 1)/2]+M-2 or pn+M-3M-ary sequences, and both of their maximum magnitudes of correlation values are upper bounded by 2 radic{pn} + 6. The largest family has its maximum magnitude of correlation values upper bounded by 3 radic{pn}+5 and the family size is (M-1)2(2n-1-1) +M-1 for p = 2 or (M-1)2(pn- 3)/2 +M(M-1)/2 for an odd prime p.

A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices. In this paper, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i,i= 6,7,8,9,10, i.e., the cycles existing regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g,gges 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14.

In this correspondence, three methods of constructing low-correlation zone (LCZ) sequences are proposed. In the first method, we constructed binary LCZ sequence sets of period 2n-1 using the Legendre sequences of period 2m-1 as a column sequence when m|n. In the second method, we devise a column sequence set of length 2m+1-1 from a binary sequence of period 2m-1 having ideal autocorrelation property and this column sequence set is used to construct binary LCZ sequence sets of period 2n-1 when (m+1)|n. In the third method, p-ary LCZ sequence sets are constructed by adopting p-ary sequence of period pm-1 with ideal autocorrelation for integers n and m such that m|n as a column sequence. The second and third methods give us the optimal sets with respect to the bound by Tang, Fan, and Matsufuji. Finally, a construction method of pntimespnp-ary Hadamard matrices from optimal LCZ sequence sets is proposed

A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices (or circulants). In this correspondence, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i, i = 6, 7, 8, 9,10, i.e., the cycles that always exist regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g, g > 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14 or 18. We also propose a couple of shift value assigning rules for circulants of a QC LDPC code guaranteeing the girth 14.

In this paper, we propose the new construction methods of constructing optimal low correlation zone (LCZ) sequences. We construct the new p-ary LCZ sequence sets by adopting p-ary sequence of period pm- 1 with ideal autocorrelation for integers n and m such that m n as a column sequence. The new construction methods give us the optimal sets with respect to the bound by Tang, Fan, and Matsufuji

In this paper, given an integer e and n such that e|n, and a prime p, we propose a method of constructing optimal p2-ary low correlation zone (LCZ) sequence set with parameters (pn- 1,pe- 1, (pn- 1)/(p - 1), 1) from a p-ary sequence of the same length with ideal autocorrelation. The resulting p2-ary LCZ sequence set can be viewed as the generalization of the optimal quaternary LCZ sequence set by Kim, Jang, No, and Chung in respect of the alphabet size. But the method used in the proof is quite different from that used in the quaternary LCZ sequence. The proof used in this paper can be used for the proof of quaternary LCZ sequence

In this paper, we define quasi-cyclic (QC) generalized Hadamard matrices and balanced QC generalized Hadamard matrices. Then we propose a new construction method for QC generalized Hadamard matrices. The proposed matrices are constructed from the balanced optimal low correlation zone (LCZ) sequence set which has correlation value -1 within low correlation zone

Let p be an odd prime, q be a divisor of p - 1 and mu be a primitive root mod p. A g-ary PRS (power residue sequence) of period p is defined as s(n) = k if n isin Ckwhere Ck= {muqt+k|t = 0,1,2,...,T - 1} where T = (p - 1)/q. In this paper, we prove that the maximum absolute value of the periodic crosscorrelation of two distinct q-ary PRS's of period p is upper bounded by √ p + 2

In this paper, we present several construction methods for low-correlation zone (LCZ) sequence sets. First, we propose a design scheme for binary LCZ sequence sets with parameters (2n+1-2,M,L,2). In this scheme, we can freely set the LCZ length L and the resulting LCZ sequence sets have the size M, which is almost optimal with respect to Tang, Fan, and Matsufuji bound. Second, given a q-ary LCZ sequence set with parameters (N,M,L,epsi) and even q, we construct another q-ary LCZ sequence set with parameters (2N,2M,L,2epsi) or (2N,2M,L-1,2epsi). Especially, the new set with parameters (2N,2M,L,2) can be optimal in terms of the set size if a q-ary optimal LCZ sequence set with parameters (N,M,L,1) is used

In this correspondence, the cycles of Tanner (3,5) quasi-cyclic (QC) low-density parity-check (LDPC) codes are analyzed and their girth values are derived. The conditions for the existence of cycles of lengths 4,6,8, and 10 in Tanner (3,5) QC LDPC codes of length 5p are expressed in terms of polynomial equations in a 15th root of unity of the prime field F/sub p/. By checking the existence of solutions for these equations over F/sub p/, the girths of Tanner (3,5) QC LDPC codes are derived.

In this paper, we derive the cyclotomic numbers of order 5 over an extension field Fpnusing the well-known results of quintic Jacobi sums over Fp(B. C. Berndt, et al., 1998). For p ne 1 mod 5, we have obtained the simple closed-form expression of the cyclotomic numbers of order 5 over Fpn. For p equiv 1 mod 5, we express the cyclotomic number of order 5 over Fpnin terms of the solution of the diophantine system which is required to evaluate the cyclotomic number of order 5 over Fpn. Using the cyclotomic numbers of order 5 over Fpn, autocorrelation distributions of 5-ary Sidel'nikov sequences of period pn- 1 are also derived

In this paper, for integers e and n such that e|n and 2e- 1 is a prime, we propose a method of constructing binary low correlation zone (LCZ) sequences of period 2n- 1 by using the extended form sequence with the same period. These new LCZ sequences use Legendre sequences as their column sequences

In this paper, we derived the autocorrelation distributions, i.e., the values and the number of occurrences of each value of the autocorrelation function of Sidel'nikov sequences. The frequency of each autocorrelation value of an M-ary Sidel'nikov sequence is expressed in terms of the cyclotomic numbers of order M. It is also pointed out that the total number of distinct autocorrelation values is dependent not only on M but also on the period of the sequence, but always less than or equal to (2M) + 1

In this paper, we derive linear complexity over Fpof the M-ary Sidel'nikov sequences using discrete Fourier transform. As an example, we represent the linear complexity of the ternary Sidel'nikov sequences. It turned out that the ternary Sidel'nikov sequences have the linear complexity nearly close to their periods

In this paper, given a composite integer n, we propose a method of constructing quaternary low correlation zone (LCZ) sequences of period 2n$1 from binary sequences of the same length with ideal autocorrelation. These new sequences are optimal with respect to the bound by Tang, Fan, and Matsufuji. The correlation distributions of these new quaternary LCZ sequences constructed from m-sequences and GMW sequences are derived

In this paper, the cycles of Tanner's (3,5) quasicyclic (QC) low-density parity-check (LDPC) codes are analyzed and their girth values are derived. The conditions for the existence of cycles of lengths 4, 6, 8, and 10 in Tanner's (3,5) QC LDPC codes of length 5p are expressed in terms of polynomial equations in a 15-th root of unity of the prime field Fp. By checking the existence of solutions for these equations over Fp, the girths of Tanner's (3,5) QC LDPC codes are derived

In this paper, we introduce a new partial transmit sequence (PTS) orthogonal frequency division multiplexing (OFDM) scheme with low computational complexity. In the proposed scheme, 2/sup n/ - point inverse fast Fourier transform (IFFT) is divided into two parts. An input symbol sequence is partially transformed using the first l stages of IFFT to generate an intermediate signal sequence and the intermediate signal sequence is partitioned into a number of intermediate signal subsequences. Then, the remaining n - l stages of IFFT are applied to each of the intermediate signal subsequences and the resulting signal subsequences are summed after being multiplied by each member of a set of W rotating vectors to yield W distinct OFDM signal sequences. The one with the lowest peak to average power ratio (PAPR) among these OFDM signal sequences is selected for transmission. The new PTS OFDM scheme reduces the computational complexity while it shows almost the same performance of PAPR reduction as that of the conventional PTS OFDM scheme.

The authors introduce a new selected mapping (SLM) orthogonal frequency division multiplexing (OFDM) scheme with low computational complexity. The proposed SLM scheme transforms an input symbol sequence into a set of OFDM signals by multiplying the phase sequences to the signal after a certain intermediate stage of inverse fast Fourier transform (IFFT). Then, the OFDM signal with the lowest peak-to-average power ratio (PAPR) is selected for transmission. The new SLM OFDM scheme reduces the computational complexity, while it shows almost the same performance of PAPR reduction as that of the conventional SLM OFDM scheme.

For a prime p and positive integers M and n such that M|p/sup n/-1, Sidel'nikov introduced M-ary sequences (called Sidel'nikov sequences) of period p/sup n/-1, the out-of-phase autocorrelation magnitude of which is upper bounded by 4. In this correspondence, we derived the autocorrelation distributions, i.e., the values and the number of occurrences of each value of the autocorrelation function of Sidel'nikov sequences. The frequency of each autocorrelation value of an M-ary Sidel'nikov sequence is expressed in terms of the cyclotomic numbers of order M. It is also pointed out that the total number of distinct autocorrelation values is dependent not only on M but also on the period of the sequence, but always less than or equal to (M/2)+1.

In this paper, given a composite integer n, we propose a method of constructing quaternary low correlation zone (LCZ) sequences of period 2/sup n/-1 from binary sequences of the same length with ideal autocorrelation. These new sequences are optimal with respect to the bound by Tang, Fan, and Matsufuji. The correlation distributions of these new quaternary LCZ sequences constructed from m-sequences and Gordon-Mills-Welch (GMW) sequences are derived.

For a prime power q, we show that a cyclic relative difference set with parameters (q/sup n/-1/q-1,q-1,q/sup n-1/,q/sup n-2/) can be constructed from a d-homogeneous function from F/sub q//sup n//spl bsol/{0} onto F/sub q/ with difference-balanced property, where F/sub q//sup n/ is the finite field with q/sup n/ elements. This construction method enables us to construct several new cyclic relative difference sets with parameters (p/sup n/-1/p/sup l/-1,p/sup l/-1,p/sup n-l/,p/sup n-2l/) from p-ary sequences of period p/sup n/-1 with ideal autocorrelation property introduced by Helleseth and Gong. Using a lifting idea, other new cyclic relative difference sets can be constructed from the Helleseth-Gong (HG) sequences. Also, the 3-ranks and the trace representation of the characteristic sequences of cyclic relative difference sets from a specific class of ternary HG sequences and ternary Lin sequences are derived.

We present a construction for binary sequences {s(t)} of period N=p/sup m/-1 for an odd prime p based on the polynomial (z+1)/sup d/+az/sup d/+b, and discuss them in some cases of parameters p, m, d, a, and b. We show that new sequences from our construction are balanced or almost balanced and have optimal three-level autocorrelation for the case when the polynomial (z+1)/sup d/+z/sup d/+a can be transformed into the form z/sup 2/-c. We also derive the distribution of autocorrelation values they take on. The sequences satisfy constant-on-the-coset property, and we show that there are more than one characteristic phases with constant-on-the-coset property. Some other interesting properties of those sequences are presented. For the cases when the polynomial (z+1)/sup d/+z/sup d/+a cannot be transformed into the form z/sup 2/-c, we performed extensive computer search, and results are summarized. Based on these results, some open problems are formulated.

In this correspondence, the linear span of extended sequences of period 2/sup em/-1 which are constructed from the ideally correlated sequences of period of 2/sup m/-1 is derived. It is also shown that the linear span of cascaded Gordon-Mills-Welch (GMW) sequences can be derived in the same context. As examples, the linear span of extended Legendre sequences and cascaded GMW sequences with triple trace are computed.

We present a construction for binary pseudorandom sequences of period 2/sup m/-1 with ideal autocorrelation property using the polynomial z/sup d/+(z+1)/sup d/. We show that the sequence obtained from the polynomial becomes an m-sequence for certain values of d. We also find a few values of d which yield new binary sequences with ideal autocorrelation property when m is 3k/spl plusmn/1, where k is a positive integer. These new sequences are represented using a trace function and the results are tabulated.

We present a construction, in a closed form, for an optimal family of 2/sup m/ binary sequences of period 2/sup 2m/-1 with respect to Welch's (1974) bound, whenever there exists a balanced binary sequence of period 2/sup m/-1 with ideal autocorrelation property using the trace function. This construction enables us to reinterpret a small set of Kasami and No (1988) sequences as a family constructed from m-sequences. New optimal families of binary sequences are constructed from the Legendre sequences of Mersenne prime period, Hall's sextic residue sequences, and miscellaneous sequences of unknown type. In addition, we enumerate the number of distinct families of binary sequences, which are constructed from a given binary sequence by this method.

In this correspondence, it is shown that Legendre sequences of period p can be explicitly represented using the trace function defined on the finite field with 2/sup n/ elements, whenever p=2/sup n/-1 is prime for some n/spl ges/3.

The generalized Hamming weight of a linear code is a new notion of higher dimensional Hamming weights. Let C be an [n,k] linear code and D be a subcode. The support of D is the cardinality of the set of not-always-zero bit positions of D. The rth generalized Hamming weight of C, denoted by d/sub r/(C), is defined as the minimum support of an r-dimensional subcode of C. It was shown by Wei (1991) that the generalized Hamming weight hierarchy of a linear code completely characterizes the performance of the code on the type II wire-tap channel defined by Ozarow and Wyner (1984). In the present paper the second generalized Hamming weight of the dual code of a double-error-correcting BCH code is derived and the authors prove that except for m=4, the second generalized Hamming weight of [2/sup m/-1, 2m]-dual BCH codes achieves the Griesmer bound.>

A technique for constructing optimal OOCs (optical orthogonal codes) is presented. It provides the only known family of optimal (with respect to family size) OOCs having lambda =2. The parameters (n, omega , lambda ) are respectively (p/sup 2m/-1, p/sup m/+1,2), where p is any prime and the family size is p/sup m/-2. Three distinct upper bounds on the size of an OOC are presented that, for many values of the parameter set (n, omega , lambda ), improve upon the tightest previously known bound.>

A simple, yet general method of constructing bent functions is presented. For certain applications it is of interest to minimize the size of the image of a bent function. In this study a tight lower bound on the number of elements in the image is established under certain conditions.>