Research Article  Open Access
A Deterministic Construction for Jointly Designed Quasicyclic LDPC CodedRelay Cooperation
Abstract
This correspondence presents a jointly designed quasicyclic (QC) lowdensity paritycheck (LDPC) codedrelay cooperation with jointiterative decoding in the destination node. Firstly, a designtheoretic construction of QCLDPC codes based on a combinatoric design approach known as optical orthogonal codes (OOC) is presented. Proposed OOCbased construction gives three classes of binary QCLDPC codes with no length4 cycles by utilizing some known ingredients including binary matrix dispersion of elements of finite field, incidence matrices, and circulant decomposition. Secondly, the proposed OOCbased construction gives an effective method to jointly design length4 cycles free QCLDPC codes for codedrelay cooperation, where sumproduct algorithm (SPA) based jointiterative decoding is used to decode the corrupted sequences coming from the source or relay nodes in different time frames over constituent Rayleigh fading channels. Based on the theoretical analysis and simulation results, proposed QCLDPC codedrelay cooperations outperform their competitors under same conditions over the Rayleigh fading channel with additive white Gaussian noise.
1. Introduction
Multipleinput multipleoutput (MIMO) has been recognized as an effective approach to combat the effect of fading by offering diversity [1, 2]. However, for some practical scenarios in wireless communication (e.g., wireless sensor networks), it is not feasible to install multiple antennas due to terminal size, power consumption, and hardware limitation. To solve this crucial problem, cooperative communication is determined as a virtual MIMO, where the devices with single antenna terminals can share their antennas to acquire multiplexing gain and diversity [3–5]. The basic idea of coded cooperation is that each relay node, instead of transmitting the whole codeword block, only transmits the redundant parity data. In the literature [6–8], three basic user cooperation protocols, that is, amplifyandforward (AF), estimateandforward (EF), and decodeandforward (DF), have been presented. In an AF cooperation, the relay nodes send only the amplified version of the signal received from source to the destination node, where the strength of transmitted signals is controlled by the amplification or scaling coefficients at the relay nodes. In an EF approach, the signals received from the source node are first estimated by the relay nodes based on some harddecision statistics, and then these estimated signals are transmitted to the destination node. Generally, an AF cooperation protocol seems to be more attractive as compared to an EF approach since it does not need extra computational complexity for harddecision detection in the relay node. However, a serious flaw of an AF cooperation is that it also amplifies the noise received from source to relay (SR) broadcast channel and sends to the destination node. On the contrary, both AF and EF cooperation protocols are not feasible for low bit error rate (BER) applications.
For higherror performance applications, channel coding coupled with the conventional relay cooperation is called coded cooperation. The conventional AF and EF user cooperation protocols have been replaced by coded cooperation employing forward error correction in the relay and destination node. In coded relay cooperative communication, each relay node instead of transmitting the whole data frame only sends the redundant parity bits to the destination node. The performance of coded cooperation has been investigated based on turbo and LDPC codes [9–14]. However, LDPCcoded cooperation provides more advantages over turbocoded cooperation in terms of lowcost decoding and delay for hardware implementation of the decoder. To the best our knowledge, most of the previous studies have investigated LDPCcoded cooperation by utilizing random LDPC codes. However, the investigation on QCLDPCcoded cooperation is rarely discussed [14]. As compared to QCLDPC codedrelay cooperation, random LDPCcoded cooperation provides a limited spectrum for code length and rate selection and encoding complexity is quadratic in nature. However, QCLDPCcoded cooperation provides more flexibility for code length and rate selection, and encoding complexity is linear in nature.
In this correspondence, we propose a jointly designed QCLDPC codedrelay cooperation with jointiterative decoding in the destination node over a Rayleigh fading channel. Firstly, a design theoretic construction based on optical orthogonal codes [15–21] gives three classes of binary QCLDPC codes with no length4 cycles. Secondly, the proposed OOCbased construction is used to jointly design length4 cycles free QCLDPC codes for codedrelay cooperation, where sumproduct algorithm SPAbased jointiterative decoding is used to decode the corrupted sequences coming from source or relay nodes in their respective time slots. Based on the performance analysis, the proposed QCLDPCcoded cooperations outperform their competitors under the same conditions over a Rayleighfading channel in the presence of additive white Gaussian noise.
Lowdensity paritycheck codes [22] have been included in the list of capacityapproaching codes because of their excellent error correcting capability and lowcost iterative decoding over various communication channels. Besides an efficient error correction performance over noisy information channels, LDPC codes provide a flexible spectrum in terms of code length and rate selection. Therefore, many communication systems prefer LDPC codes over traditional errorcorrecting codes such as Wifi, WiMAX, satellite communication, and 10 gigabit ethernet. Furthermore, LDPC codes are gaining attention as one of the contenders for 5G wireless communications. These significant efforts provide the fundamental reasons for many digital communication and storage systems to adopt LDPC codes as a primary choice.
The null space of a paritycheck matrix gives a regular LDPC code if it has constant columnweight and constant rowweight . If has variable column and/or row weight, then its null space gives an irregular LDPC code. If a paritycheck matrix composed of an array of circulant permutation matrices, then its null space gives a QCLDPC code over a finite field [23–25]. If a paritycheck matrix satisfies a constraint that no two rows or columns can overlap, in terms of a nonzero element, at more than one positions, then this constraint is known as rowcolumn (RC) constraint which ensures that the Tanner graph representation of has a girth of at least 6. The construction spectrum of LDPC codes is divided into two categories: (A) computerbased LDPC codes are constructed based on random construction methods, PEGLDPC [26] codes are considered as one of the most promising randomLDPC codes, and protographbased [27] techniques; (B) structuredLDPC (e.g., QCLDPC) codes are obtained from deterministic or algebraic techniques such as finite fields [23, 24, 28–30], finite geometries [31, 32], and combinatorial designs [33–40]. Quasicyclic or architectureaware LDPC codes have been adopted by many communication standards because of their efficient architecture which reduces computational cost of an encoder and decoder.
Remainder of this manuscript is arranged as follows: In Section 2, the fundamental concepts about the general QCLDPCcoded cooperative communication are given. Basic concepts about the existence and construction of optical orthogonal codes (OOC) are presented in Section 3. Section 4 presents an OOCbased construction of QCLDPC codes. A construction method for jointly designed QCLDPC codes for coded cooperative communication is given in Section 5. Section 6 presents the performance analysis based on the numerical results. Finally, the conclusion of this manuscript is given in Section 7.
2. Fundamental Model for CodedRelay Cooperation
For the onerelay coded cooperative communication system, a fundamental model with three nodes such as source , relay , and destination is depicted in Figure 1. All these nodes are supposed to have only one antenna, and they communicate with each other over a half duplex Rayleigh fading channel.
The source and relay nodes transmit their signals to the destination node in two consecutive time frames, time frame 1 and time frame 2, respectively. In the time frame 1, the information data are encoded by the first encoder in the source node, denoted as Encod1, and transmitted to the relay and destination nodes simultaneously over the constituent Rayleigh fading channels and , respectively. Suppose that the broadcast channel is Rayleigh fading channel with additive white Gaussian noise, and then the received signal at the relay node is given as follows:where denotes the codeword symbols sent by the source to the relay node over broadcast channel and is a zeromean complex Gaussian random variable with variance . For the Rayleigh fading channel, is also a zeromean and unitvariance complex Gaussian random variable. Similarly, the received signal at the destination sent from source over the Rayleigh fading channel is given aswhere and represent the corresponding zeromean complex Gaussian random variables with variance and 1, respectively.
In time frame 2, the decoder in the relay node, denoted as Decod2, first decodes the data received from the source node over a noisy broadcast channel . Then, the decoded data are encoded by the second encoder in the relay node, denoted as Encod2, and whole or a part of the coded symbols is transmitted to the destination node over a broadcast channel . For ideal coded cooperation, it is assumed that Decod2 can successfully decode the data received from the source node. In the destination, the received signal over a broadcast channel is described aswhere and are the corresponding additive white Gaussian noise and fading, respectively, denotes the coded symbols sent from relay to the destination over a Rayleigh fading channel , and q represents the power gain for the signals transmitted by the relay node to that transmitted by the source node.
In the onerelay coded cooperative communication, two distinct QCLDPC codes and defined by the null space of two paritycheck matrices and were used to define the source and relay nodes, respectively. The code in the systematic form is , where denotes the redundant parity data with length . The relay node first decodes the information sent from , then Encod2 encodes the estimated data by adding new redundant parity bits with length using paritycheck matrix . In the destination node, a matched filter alternately receives the coded symbols transmitted by the source node and relay node in two different time frames which are then multiplexed for further processing. Finally, joint iterative decoder uses the paritycheck matrix , comprised of and , to jointly decode the information sent from the source node. Note that the relay node only sends the redundant parity data to the destination node as it has already received the information bits from the source.
3. Optical Orthogonal Codes
3.1. Fundamental Concepts
Optical orthogonal codes are a special type of balanced incomplete block design (BIBD), so we begin with the definition of BIBD.
Definition 1 (see [41]). A pair is called a design, where denotes a set of elements or varieties and denotes the nonempty subsets of called blocks. Suppose μ, k, and η be positive integers such that . A design is called BIBD if all of the following properties are satisfied:(1)(2)Each nonempty subset (block) of have k varieties(3)Each pair of elements exists in exactly η subsets of
Definition 2 (see [41]). A family of binary codewords is known as an optical orthogonal code or briefly OOC if the following two correlation properties are satisfied:(1)The autocorrelation property:(2)The crosscorrelation property:where the subscripts are treated over . An OOC is said to be optimal if it has codewords:
Example 1. Two optical orthogonal codes, OOC and OOC, are given as follows:(1)An OOC has two binary codewords:(2)An OOC has three binary codewords:Clearly, the above two optical orthogonal codes are optimal. Besides a binary notation, an OOC can be alternatively represented as a collection of k subsets of μ, called blocks, where each k subset denotes a binary codeword and the elements of each block denote the position of nonzero elements.
Example 2. The set notations of two optical orthogonal codes, OOC and OOC, given in Example 1 are as follows:(1)The set notation of OOC is(2)The set notation of OOC isNext, the existence of OOC in terms of the necessary conditions is given as follows:
Lemma 1 (see [42]). An optimal OOC exists if and only if , where or .
Lemma 2. An optimal OOC exists for any positive integer μ if and only if(i) [43](ii) [43](iii) [20](iv), where and denotes a positive integer with prime factors congruent to 7 or [44](v), where and denotes a positive integer such that or 25 [43](vi), where and denotes a positive integer with prime factors congruent to [43]
Lemma 3. An optimal OOC exists for any positive integer μ if and only if(i), where f denotes a prime number with [45](ii), where f denotes a prime number with [46](iii), where f is the product of prime factors congruent to and greater than 5 [47](iv), where f denotes a prime number with [48](v), where f denotes a prime number with and [48](vi), where represents any nonnegative integer and f is the product of prime factors congruent to [16]
Lemma 4. An optimal OOC exists for any positive integer μ if and only if(i), where f denotes a prime number with [17](ii), where f denotes a prime number with and [49]
3.2. Construction of Optical Orthogonal Codes
In the literature [50], it has already been shown that an optimal OOC is equivalent to  cyclic difference packing or briefly βCDP.
Definition 3 (see [41]). Let be the positive integers with . A pair is called a β packing design, where and denotes the nonempty k subsets of called blocks such that every β subset of distinct elements from appears in at most η blocks.
Let be a β packing design. Suppose α is a permutation on such that for a block B, , then α is called an automorphism of the β packing design . A β packing design having cyclic automorphism is known as cyclic β packing design, where cyclic automorphism is a bijection α: . Suppose be a block of cyclic β packing design , then the block orbit consists of the following distinct blocks:where . A block orbit containing μ distinct blocks is called full orbit; otherwise short orbit. Any fixed block from each block orbit is called a base block.
Let denote the set of all base blocks of a cyclic β packing design. Then, the pair is called a cyclic β difference packing or briefly a βCDP. A βCDP with base blocks is called maximum βCDP. A fundamental equivalence relation between an optimal OOC and a maximum βCDP is given by [50]
Theorem 1 (see [50]). An optimal OOC is equivalent to a maximum βCDP if and only if .
Therefore, the construction of a maximum βCDP gives its corresponding optimal OOC. Some known results about the existence of a maximum βCDP are given as follows:
Lemma 5. Under any of the following conditions, a maximum βCDP exists:(i)A 36regular 2CDP exists if is a prime and [15](ii)A 48regular 2CDP exists if is a prime and [15](iii)A 60regular 2CDP exists for any positive integer such that or 25 [15](iv)A fregular 2CDP exists for , and denotes a positive integer such that or 25 [15](v)There exists an optimal CDP for [16](vi)There exists an optimal CDP for any nonnegative integer r, where f denotes the product of prime factors congruent to [16](vii)There exists an optimal 15regular CDP, where and [17](viii)There exists an optimal 20regular CDP, where and [17]Next, we construct QCLDPC codes by utilizing the maximum CDPbased construction of optimal OOC’s with .
4. OOCBased Construction of QCLDPC Codes
We provide a construction of QCLDPC codes based on the optimal OOC’s with given in Sections 3.1 and 3.2. Consider a matrix given as follows:where each , , represents a circulant matrix. The rightcyclic shift of each row of , , returns the next row. However, the first row of is obtained from one of the ω kelement base blocks of the OOC’s with . The null space of base matrix gives a length4 cycles free QCLDPC code with rate and a minimum distance lower bounded by .
Example 3. Consider an optimal OOC with base blocks , , and for . The base blocks for a OOC areBased on equation (13), consider a base matrix given as follows:To make the idea more clear, consider a submatrix Q given aswhere and , . Submatrix Q has length4 cycles if and only if [30]. Clearly, the above base matrix does not satisfy the condition . Therefore, the base matrix has no length4 cycles.
4.1. OOCBased QCLDPC Codes: ClassI
Let be a finite field with ρ elements. For each nonzero element in , , form a tuple over , , with all the components of zero except the ith component [25]. The subscript “b” stands for the binary. This tuple is referred as the binary locationvector of . The binary locationvector of 0element is .
Let λ be an element of . If denotes the binary locationvector of λ, then the right cyclic shift of gives the binary location vector of element . If θ denotes a primitive element of , then the tuples of give a circular permutation matrix . Matrix is referred as fold binary dispersion [25] of λ over . The fold binary dispersion of 0element is a allzero matrix over .
Next, all entries of base matrix given by are replaced by their fold matrix dispersions over . We obtain a array given as follows:where is an circular permutation matrix over , for and . Array gives a matrix over . Since, the matrix satisfies the RC constraint. The null space of gives a QCLDPC code whose Tanner graph has no length4 cycles.
For a pair of integers and , and . Let be a subarray of . Subarray gives a matrix over . The null space of matrix gives a QCLDPC code of length with rate at least and minimum distance lower bounded by .
4.2. OOCBased QCLDPC Codes: ClassII
A class of length4 cycles free QCLDPC codes is constructed based on the incidence matrices obtained from OOC with . A design with and n blocks, , satisfying the following properties is called a BIBD: (A) each entry in participates in exactly r blocks; (B) each pair of the elements in participates in exactly η blocks of ; and (C) the size of each block k is small compared to the cardinality of . A BIBD can also be represented by a matrix over :where matrix is known as the incidence matrix [25]. The incidence matrix satisfies the following properties: (A) the incidence matrix has columnweight equal to k; (B) the incidence matrix has rowweight equal to r; and (C) any two rows or columns of have 1element common at most η points.
Example 4. Let be the following BIBD:This BIBD has seven blocks, so it will give a incidence matrix over . Each row of corresponds to elements of , and each column of corresponds to the blocks of . The first element 0 of participates in 1st, 2nd, and 6th block of . Therefore, the first row of incidence matrix has nonzero elements at 1st, 2nd, and 6th position. The element 1 of participates in 2nd, 3rd, and 7th block of . So, the second row of has nonzero elements at 2nd, 3rd, and 7th position. Similarly, the last element 6 of participates in 1st, 5th, and 7th block of . Therefore, the last row of has nonzero elements at 1st, 5th, and 7th position. The incidence matrix of above BIBD is given as follows:The cyclic shift of each row returns a next row of , and the first row is obtained by the cyclic shift of last row. Also, the downward cyclic shift of each column of gives a column on its right. So, the matrix is a circulant permutation matrix over . Note that the circulant permutation matrix fulfills all the required properties of paritycheck matrix and satisfies the RC constraint. Therefore, the null space of gives an LDPC code with a girth of at least 6.
Based on a OOC with , consider a incidence matrix obtained from ω base blocks. The incidence matrix can be arranged in a cyclic manner consisting of a array of circulant submatrices given as follows:where each , , represents a circulant submatrix over . Note that the matrix satisfies all the required properties of a paritycheck matrix. Therefore, the null space of matrix gives a QCLDPC code with a rate lower bounded by , and a girth of at least 6.
4.3. OOCBased QCLDPC Codes: ClassIII
In Section 4.2, QCLDPC codes have been constructed based on incidence matrices obtained from OOC. The QCLDPC codes of this class have a paritycheck matrix consisting of an array of circulant matrices. A new class of QCLDPC codes can be constructed based on the circulant decomposition [25] of each circulant matrix in the array. For illustration, consider the paritycheck matrix of QCLDPC codes given by equation (21):where each is a circulant matrix with both row and column weight 5, for . Let be the first row of circulant obtained from OOC with . Decompose into five rows of length μ, , , , , and , by distributing the five 1components of among the five new rows. The first 1component of is placed in , the second 1component of is placed in , …, and the fifth 1component of is placed in . Form a circulant matrix for each new row by using and its right cyclic shifts. Both the row and column weights of are equal to 1. The above decomposition and cyclic shifting of each circulant , , give five circulant matrices, , , …, and , which are called descendants of .
Let . A array of circulant matrices is obtained as follows:where is a matrix over . is constructed based on the OOC with , so it satisfies all the required properties of a paritycheck matrix. Therefore, the null space of gives a QCLDPC code with rate lower bounded by , and a girth of least 6.
5. QCLDPC CodedRelay Cooperation
5.1. Jointly Designed QCLDPC Codes
In this section, a joint design of QCLDPC codes for onerelay coded cooperative communication system is presented. Jointly designed QCLDPC codes are constructed based on the proposed three classes of binary QCLDPC codes in Sections 4.1, 4.2, and 4.3, respectively. Jointly designed QCLDPC codes for coded cooperative communication over the Rayleigh fading channel are jointly decoded by a joint iterative decoder in the destination.
Suppose a onerelay coded cooperative communication system where source and relay nodes are realized by two distinct QCLDPC defined by the null space of paritycheck matrices and , respectively. and are designed based on the proposed OOCbased construction of QCLDPC codes given by , , and , respectively. Abovementioned QCLDPC codes defined by the null space of and are regular and denoted as and , where and denote the number of 1’s in each row and column of and , respectively.
Joint iterative decoding, defined by a joint Tanner graph, can be applied for double regular QCLDPC codes and for onerelay coded cooperation. A general joint triplelayer Tanner graph used in onerelay coded cooperative system realizing source and relay nodes by two distinct QCLDPC codes and is shown in Figure 2. In one relay cooperation, the overall code rate is [13], where and denote the code rates of and , respectively. In particular, the resultant paritycheck matrix for onerelay cooperation is given as follows:where and are the paritycheck matrices of and , respectively, where , , and . So, the null space of gives an irregular QCLDPC code with no length4 cycles. Note that the idea can easily be extended for multirelay coded cooperation communication system over the Rayleigh fading channel.
5.2. SPABased Joint Iterative Decoding
In the destination node, the output sequence obtained from the multiplexer, as depicted in Figure 1, can be expressed as . Using binaryphaseshiftkeying (BPSK) transmission over a Rayleighfading channel, let that are treated as the inputs of the jointiterative decoder.
5.2.1. Initialization
The loglikelihood ratio (LLR) with known channelstate information (CSI) can be computed as
Let the a posteriori LLR for corresponding sequences be computed aswhere if all the check nodes , except the check node , connected to variable node are satisfied simultaneously.
5.2.2. CheckNode Update
Each check node in either first or third layer estimates the extrinsic information and sends to a variable node aswhere denotes the set of variable nodes connected to check node excluding the variable node .
5.2.3. VariableNode Update
The extrinsic information from a variable node to a check node in the first layer of the joint Tanner graph, as depicted in Figure 2, can be computed as
Similarly, extrinsic information from a variable node to a check node in the third layer of the joint Tanner graph can be computed as
5.2.4. Final A Posteriori LLR
Perform the processes given in Sections 5.2.2 and 5.2.3 until a maximum number of iterations are reached. Final a posteriori LLR can be computed as
Finally, if , the estimated decoded bit , otherwise .
6. Numerical Results
In this section, the error correcting performance of proposed jointly designed QCLDPC coded cooperation is compared with randomly constructed LDPCcoded cooperation and some related works under same conditions. Simulation results are obtained by SPAbased joint iterative decoding with BPSK transmission. Also, the constituent channels are all Rayleigh fading channels with additive white Gaussian noise.
6.1. Jointly Designed QCLDPC Coded Cooperation Based on ClassI QCLDPC Codes
Firstly, the BER performance of a proposed jointly designed QCLDPC code under various decoding iterations is shown in Figure 3. It is important to note that the BER reduces significantly by increasing the decoding iterations. For instance, about 1.5 dB gain is achieved for the 3rd iteration over the 2nd iteration at . Similarly, at , about 1.2 dB gain is achieved for the 4th decoding iteration over the 3rd decoding iteration, and it is about 1 dB for the 5th decoding iteration over the 4th decoding iteration at . Secondly, the BER performance of a proposed jointly designed QCLDPC code compared with a randomly constructed LDPCcoded cooperation [26] under same conditions is shown in Figure 4. The relevant parameters for component QCLDPC codes adopted for and randomly designed LDPCcoded cooperation are given in Table 1, where the overall code rate is from the destination. Simulation results show that the proposed jointly designed QCLDPC coded cooperation outperforms the randomly codedcooperation for the same code rate and decoding iterations over a Rayleigh fading channel in the presence of additive white Gaussian noise.

6.2. Jointly Designed QCLDPCCoded Cooperation Based on ClassII QCLDPC Codes
In this subsection, the BER performance of the proposed jointly designed QCLDPC code compared with a QCLDPC codedcooperation [38] under various decoding iterations is shown in Figure 5. Simulation results show that the proposed jointly designed QCLDPCcoded cooperation outperforms its competitor under the same conditions over a Rayleigh fading channel in the presence of additive white Gaussian noise. For instance, at BER , the proposed jointly designed QCLDPCcoded cooperation provides a gain of about 1 dB with 4 decoding iterations. The relevant parameters for component QCLDPC codes adopted for and QCLDPCcoded cooperation [38] are given in Table 2. Note that the overall code rate is from the destination.

6.3. Jointly Designed QCLDPCCoded Cooperation Based on ClassIII QCLDPC Codes
Figure 6 shows the BER performance of the proposed jointly designed QCLDPC code and the QCLDPCcoded cooperation [39] under different decoding iterations. Simulation results show that the error correcting performance of the QCLDPCcoded cooperation [39] is inferior than the proposed jointly designed QCLDPCcoded cooperation under the same conditions over a Rayleigh fading channel in the presence of additive white Gaussian noise. For instance, at , jointly designed codedrelay cooperation provides a gain of about 0.6 dB with 4 decoding iterations. The relevant parameters for component QCLDPC codes adopted for and QCLDPCcoded cooperation [39] are given in Table 3, where the overall code rate is from the destination.

7. Conclusion and Remarks
In this correspondence, a jointly designed QCLDPC codedrelay cooperation with SPAbased jointiterative decoding in the destination over a Rayleighfading channel has been presented. Three classes of binary length4 cycles free QCLDPC codes have been constructed based on optical orthogonal codes (OOC) and some known ingredients like binary matrix dispersion of elements of finite field, incidence matrices, and circulant decomposition. Firstly, a class of binary QCLDPC codes is constructed by binary matrix dispersion of elements of finite field based on the OOC with , where the resultant paritycheck matrices of this class have a minimum distance of at least 8. Secondly, a class of binary regular QCLDPC codes is constructed based on the incidence matrices obtained from OOC with , where the resultant paritycheck matrices of this class have a minimum distance lower bounded by 8. Thirdly, a class of binary regular QCLDPC codes is constructed using circulant decomposition of incidence matrices obtained from OOC with , where the resultant paritycheck matrices of this class have a minimum distance of at least 6. Furthermore, the proposed OOCbased construction of QCLDPC codes is utilized to jointly design length4 cycles free QCLDPC codes for codedrelay cooperation, where SPAbased joint iterative decoding is used to decode the corrupted sequences coming from source or relay nodes in their respective time slots over constituent Rayleighfading channels. Theoretical analysis and simulation results show that the proposed jointly designed QCLDPC codedrelay cooperations outperform their counterparts under the same conditions over a Rayleighfading channel in the presence of additive white Gaussian noise.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication this work.
Acknowledgments
Muhammad Asif acknowledges the support of the Chinese Academy of Sciences and TWAS for his PhD studies at the University of Science and Technology, China, as a 2016 CASTWAS President’s Fellowship Awardee (CASTWAS no. 201648). This research project is sponsored by the National Natural Science Foundation of China (grant nos. 61461136002 and 61631018) and Fundamental Research Funds for the Central Universities.
References
 A. H. Mehana and A. Nosratinia, “Diversity of MMSE MIMO receivers,” IEEE Transactions on Information Theory, vol. 58, no. 11, pp. 6788–6805, 2012. View at: Publisher Site  Google Scholar
 S. Karmakar and M. K. Varanasi, “The diversitymultiplexing tradeoff of the MIMO halfduplex relay channel,” IEEE Transactions on Information Theory, vol. 58, no. 12, pp. 7168–7187, 2012. View at: Publisher Site  Google Scholar
 E. C. Van der Meulen, “Threeterminal communication channels,” Advances in Applied Probability, vol. 3, no. 1, pp. 120–154, 1971. View at: Publisher Site  Google Scholar
 S. Ikki and M. H. Ahmed, “Performance analysis of incremental relaying cooperative diversity networks over Rayleigh fading channels,” in Proceedings of the 2008 IEEE Wireless Communications and Networking Conference, pp. 1311–1315, Las Vegas, NV, USA, April 2008. View at: Publisher Site  Google Scholar
 T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Transactions on Information Theory, vol. 25, no. 5, pp. 572–584, 1979. View at: Publisher Site  Google Scholar
 A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversityPart I: system description,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1927–1938, 2003. View at: Publisher Site  Google Scholar
 K. Azarian, H. ElGamal, and P. Schniter, “On the achievable diversitymultiplexing tradeoff in halfduplex cooperative channels,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4152–4172, 2005. View at: Publisher Site  Google Scholar
 J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, 2004. View at: Publisher Site  Google Scholar
 P. Razaghi and W. Yu, “Bilayer lowdensity paritycheck codes for decodeandforward in relay channels,” IEEE Transactions on Information Theory, vol. 53, no. 10, pp. 3723–3739, 2007. View at: Publisher Site  Google Scholar
 J. P. Canees and V. Meghdadi, “Optimized low density parity check codes designs for half duplex relay channels,” IEEE Transactions on Wireless Communications, vol. 8, no. 7, pp. 3390–3395, 2009. View at: Publisher Site  Google Scholar
 C. X. Li, G. S. Yue, M. A. Khojastepour, X. Wang, and M. Madihian, “LDPCcoded cooperative relay systems: performance analysis and code design,” IEEE Transactions on Communications, vol. 56, no. 3, pp. 485–496, 2008. View at: Google Scholar
 C. Li, G. Yue, X. Wang, and M. Khojastepour, “LDPC code design for halfduplex cooperative relay,” IEEE Transactions on Wireless Communications, vol. 7, no. 11, pp. 4558–4567, 2008. View at: Publisher Site  Google Scholar
 F. Yang, J. Chen, P. Zong, S. Zhang, and Q. Zhang, “Joint iterative decoding for pragmatic irregular LDPCcoded multirelay cooperations,” International Journal of Electronics, vol. 98, no. 10, pp. 1383–1397, 2011. View at: Publisher Site  Google Scholar
 Y. Zhang, F.F. Yang, and W. Song, “Performance analysis for cooperative communication system with QCLDPC codes constructed with integer sequences,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 649814, 7 pages, 2015. View at: Publisher Site  Google Scholar
 Y. Chang, R. FujiHara, and Y. Miao, “Combinatorial constructions of optimal optical orthogonal codes with weight 4,” IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1283–1292, 2003. View at: Publisher Site  Google Scholar
 M. Shikui and Y. Chang, “A new class of optimal optical orthogonal codes with weight five,” IEEE Transactions on Information Theory, vol. 50, no. 8, pp. 1848–1850, 2004. View at: Publisher Site  Google Scholar
 S. Wang, L. Wang, and J. Wang, “A new class of optimal optical orthogonal codes with weight six,” in Proceedings of the 2015 Seventh International Workshop on Signal Design and its Applications in Communications (IWSDA), pp. 66–69, Bengaluru, India, September 2015. View at: Publisher Site  Google Scholar
 C. Wensong and J. C. Charles, “Optimal OOC of small orders,” Discrete Mathematics, vol. 279, no. 1–3, pp. 163–172, 2004. View at: Publisher Site  Google Scholar
 C. Wensong and J. C. Charles, “Recursive constructions for optimal OOCs,” Journal of Combinatorial Designs, vol. 12, no. 5, pp. 333–345, 2004. View at: Publisher Site  Google Scholar
 Y. Chang and Y. Miao, “Constructions for optimal optical orthogonal codes,” Discrete Mathematics, vol. 261, no. 1–3, pp. 127–139, 2003. View at: Publisher Site  Google Scholar
 K. Chen, G. Ge, and L. Zhu, “Starters and related codes,” Journal of Statistical Planning and Inference, vol. 86, no. 2, pp. 379–395, 2000. View at: Publisher Site  Google Scholar
 R. Gallager, “Lowdensity paritycheck codes,” IEEE Transactions on Information Theory, vol. 8, no. 1, pp. 21–28, 1962. View at: Publisher Site  Google Scholar
 Y. Kou, S. Lin, and M. P. C. Fossorier, “Lowdensity paritycheck codes based on finite geometries: a rediscovery and new results,” IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 2711–2736, 2001. View at: Publisher Site  Google Scholar
 L. Chen, J. Xu, I. Djurdjevic, and S. Lin, “NearShannonlimit quasicyclic lowdensity paritycheck codes,” IEEE Transactions on Communications, vol. 52, no. 7, pp. 1038–1042, 2004. View at: Publisher Site  Google Scholar
 W. E. Ryan and S. Lin, Channel Codes: Classical and Modern, University of Cambridge, New York, NY, USA, 2009.
 X. Y. Hu, E. Eleftheriou, and D. M. Arnold, “Regular and irregular progressive edgegrowth Tanner graphs,” IEEE Transactions on Information Theory, vol. 51, no. 1, pp. 386–398, 2005. View at: Publisher Site  Google Scholar
 D. Divsalar, S. Dolinar, C. Jones, and K. Andrews, “Capacityapproaching protograph codes,” IEEE Journal on Selected Areas in Communications, vol. 27, no. 6, pp. 876–888, 2009. View at: Publisher Site  Google Scholar
 Q. Diao, Q. Huang, S. Lin, and K. AbdelGhaffar, “A matrixtheoretic approach for analyzing quasicyclic lowdensity paritycheck codes,” IEEE Transactions on Information Theory, vol. 58, no. 6, pp. 4030–4048, 2012. View at: Publisher Site  Google Scholar
 J. Li, K. Liu, S. Lin, and K. AbdelGhaffar, “Algebraic quasicyclic LDPC codes: construction, low errorfloor, large girth and a reducedcomplexity decoding scheme,” IEEE Transactions on Communications, vol. 62, no. 8, pp. 2626–2637, 2014. View at: Publisher Site  Google Scholar
 M. P. C. Fossorier, “Quasicyclic lowdensity paritycheck codes from circulant permutation matrices,” IEEE Transactions on Information Theory, vol. 50, no. 8, pp. 1788–1793, 2004. View at: Publisher Site  Google Scholar
 J. Xu, L. Chen, I. Djurdjevic, S. Lin, and K. AbdelGhaffar, “Construction of regular and irregular LDPC codes: geometry decomposition and masking,” IEEE Transactions on Information Theory, vol. 53, no. 1, pp. 121–134, 2007. View at: Publisher Site  Google Scholar
 Q. Diao, Y. Y. Tai, S. Lin, and K. AbdelGhaffar, “LDPC codes on partial geometries: construction, trapping set structure, and puncturing,” IEEE Transactions on Information Theory, vol. 59, no. 12, pp. 7898–7914, 2013. View at: Publisher Site  Google Scholar
 B. Vasic and O. Milenkovic, “Combinatorial constructions of lowdensity paritycheck codes for iterative decoding,” IEEE Transactions on Information Theory, vol. 50, no. 6, pp. 1156–1176, 2004. View at: Publisher Site  Google Scholar
 S. J. Johnson and S. R. Weller, “Resolvable 2designs for regular lowdensity paritycheck codes,” IEEE Transactions on Communications, vol. 51, no. 9, pp. 1413–1419, 2003. View at: Publisher Site  Google Scholar
 S. J. Johnson and S. R. Weller, “A family of irregular LDPC codes with low encoding complexity,” IEEE Communications Letters, vol. 7, no. 2, pp. 79–81, 2003. View at: Publisher Site  Google Scholar
 H. Park, S. Hong, J.S. No, and D.J. Shin, “Construction of highrate regular quasicyclic LDPC codes based on cyclic difference families,” IEEE Transactions on Communications, vol. 61, no. 8, pp. 3108–3113, 2013. View at: Publisher Site  Google Scholar
 M. Fujisawa and S. Sakata, “A construction of high rate quasicyclic regular LDPC codes from cyclic difference families with girth 8,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E90A, no. 5, pp. 1055–1061, 2007. View at: Publisher Site  Google Scholar
 H. Falsafain and M. Esmaeili, “Construction of structured regular LDPC codes: a designtheoretic approach,” IEEE Transactions on Communications, vol. 61, no. 5, pp. 1640–1647, 2013. View at: Publisher Site  Google Scholar
 M. Asif, W. Y. Zhou, M. Ajmal, A. Zain ul Abiden, and A. K. Nauman, “A construction of high performance quasicyclic LDPC codes: a combinatoric design approach,” Wireless Communications and Mobile Computing, vol. 2019, Article ID 7468792, 10 pages, 2019. View at: Publisher Site  Google Scholar
 V. Sina and N. R. Majid, “A new scheme of high performance quasicyclic LDPC codes with girth 6,” IEEE Communications Letters, vol. 19, no. 10, pp. 1666–1669, 2015. View at: Publisher Site  Google Scholar
 C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC, Boca Raton, FL, USA, 1996.
 F. R. K. Chung, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes: design, analysis, and applications,” IEEE Transactions on Information Theory, vol. 35, no. 3, pp. 595–604, 1998. View at: Publisher Site  Google Scholar
 G. Ge and J. Yin, “Constructions for optimal (, 4, 1) optical orthogonal codes,” IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 2998–3004, 2001. View at: Publisher Site  Google Scholar
 S. Bitan and T. Etzion, “Constructions for optimal constant weight cyclically permutable codes and difference families,” IEEE Transactions on Information Theory, vol. 41, no. 1, pp. 77–87, 1995. View at: Publisher Site  Google Scholar
 H. Hanani, “Balanced incomplete block designs and related designs,” Discrete Mathematics, vol. 11, no. 3, pp. 255–369, 1975. View at: Publisher Site  Google Scholar
 K. Chen and L. Zhu, “Existence of (q, k, 1) difference families with q a prime power and k = 4, 5,” Journal of Combinatorial Designs, vol. 7, no. 1, pp. 21–30, 1999. View at: Publisher Site  Google Scholar
 Y. Tang and J. Yin, “Combinatorial constructions for a class of optimal OOCs,” Science in China, vol. 45, pp. 1268–1275, 2002. View at: Google Scholar
 Y. Chang and L. Ji, “Optimal (4up, 5, 1) optical orthogonal codes,” Journal of Combinatorial Designs, vol. 12, no. 5, pp. 346–361, 2004. View at: Publisher Site  Google Scholar
 K. Chen and L. Zhu, “Existence of difference families with q a prime power,” Designs, Codes and Cryptography, vol. 15, no. 2, pp. 167–173, 1998. View at: Google Scholar
 Y. Miao and R. FujiHara, “Optical orthogonal codes: their bounds and new optimal constructions,” IEEE Transactions on Information Theory, vol. 46, no. 7, pp. 2396–2406, 2000. View at: Publisher Site  Google Scholar
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