Distributed Reed-Muller Coded-Cooperative Spatial Modulation

This paper proposes distributed Reed-Muller coded spatial modulation (DRMC-SM) scheme based on Kronecker product (KP) construction. This special construction enabled an effective distribution of classical Reed-Muller (RM) code along source and relay nodes. The proposed DRMC-SM scheme not only offers resilience in bit error rate (BER) performance but also enhances the spectral efficiency due to additional antenna index transmission inculcated by spatial modulation (SM). The usefulness of KP construction over classical Plotkin (CP) construction in coded-cooperation is analysed with and without incorporating SM. An efficient criteria for selecting the optimum bits is adopted at relay node which eventually results in better weight distribution of the mutually constructed (source and relay) RM code. The numerical results show that proposed KP construction outperforms CP construction by 1 dB in signal to noise ratio (SNR) at bit error rate (BER) of 7×10-7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,7\times 10^{-7}$$\end{document}. Moreover, the proposed DRMC-SM scheme outperforms its non-cooperative Reed-Muller coded spatial modulation scheme as well as distributed turbo coded spatial modulation scheme. This prominent gain in SNR is evident due to the path diversity, efficient selection of bits at the relay node and joint soft-in-soft-out (SISO) RM decoding employed at the destination node.


Introduction
Over the past decade, the multiple input multiple output (MIMO) techniques are widely deployed in wireless communication system to attain higher spectral efficiency [1,2] and ameliorate link reliability [3,4].These techniques can be broadly categorized in two categories namely, spatial multiplexing and space time coding.Space time coding schemes [3,5] provide the low decoding complexity by using linear decoding and enhances the link reliability by transmitting same information through independent channels to the receiver.However, the symbol rate of such schemes is no more than 1 symbol per symbol duration regardless of number of transmit antennas.While the spatial multiplexing schemes use layered space-time approach like vertical bell lab layered space time (V-BLAST) scheme which provides higher data-rates [6].The V-BLAST scheme allows all the transmit antennas to transmit their own stream of data simultaneously, which requires antenna synchronization and it also produces high inter-antenna interference (IAI) at the receiver.As an effective remedy to this problem was suggested by Mesleh et al. who devised new type of MIMO scheme named as spatial modulation (SM) [7].The SM exploited the unique idea of transmission by transmitting the information sequence not only via modulated symbols but also via antenna indices simultaneously.The modulated symbol is transmitted by using a single antenna known as a antenna index.Thus, the limitations of conventional MIMO schemes like antenna synchronization and IAI are efficiently averted by employing SM technique that provokes an active transmit antenna during transmission [8].The practical implementation of SM is discussed in [9].In [10], the SM is futher employed into relay aided systems.Moreover, different detection methods of SM along with their complexity analysis are discussed and investigated in literature [11].
Over the last three decades, channel coding has been proved as an effective means to improve the reliability of any communication channel.Therefore, the utilization of channel coding with cooperative schemes is an efficient method for enhancing the error performance of cooperative systems.These kind of cooperative schemes are named as codedcooperative diversity which was pioneered by Hunter in 2002 [12].In these schemes the channel codes at source and relay nodes mutually construct a powerful forward errorcorrection (FEC) code at the destination node.The source transmitted information is then decoded at the destination using joint decoding.During the last decade, various codedcooperative schemes based on channel codes like polar codes [13,14], turbo codes [16], low density parity check (LDPC) codes [15], convolutional codes [17] have been efficiently developed.However, the performance gain provided by these schemes was based on lengthy information sequences.Furthermore, the complex encoding and decoding is required for such coded-cooperative schemes.As there are myriad of applications which require short length message sequences particulary for extra low latency communications.Thus, a cooperation scheme based on RM code is proposed in [18], which offers better BER performance for small length information sequences with low encoding and decoding complexities.Furthermore, the encoding of partial information sequence at the relay node reduces the latency of cooperative scheme.Another RM coded-cooperative scheme based on rotated 4-QAM is reported in [19].An application of SM with space-time block code and its low complexity decoder is explained in [25].Moreover, the bit-interleaved coded SM with joint iterative demodulation and soft decoding is detailed in findings of [26].Furthermore, the performance analysis of trellis coded SM is carried out in [27].Recently, a network coding scheme using SM for two way networks is reported in [28].
The construction of RM code in both the aforementioned schemes is based on CP construction where shorter block-length codes recursively construct a larger block-length code.As in the literature it is suggested that RM codes can be recursively constructed by various construction methods [20][21][22].Another construction for generating the best RM sub-code using Greedy algorithm has been proposed in literature [23].Therefore, we have employed KP construction [20] for proposed coded-cooperative design.This design is also based on Plotkin's construction but it is different from the CP construction.Similar kind of Plotkin's construction is used by an author in polar coded-cooperation [24].In this construction, two short length codes can be extracted from a large length code which can be further employed in coded-cooperation like CP construction.Using this construction, the effective selection of the information bits at the relay yields a powerful code at the destination in comparison to CP construction.The intelligent selection of the information bits is performed using the efficient selection algorithm (ESA) proposed in [18].Further, we utilize this RM code in conjunction with spectrally efficient SM technique.Hence, distributed Reed-Muller coded spatial modulation (DRMC-SM) scheme has been proposed in this manuscript.The effective utilization of SM makes DRMC-SM scheme spectrally efficient because information sequence is transmitted from both via 4-QAM symbols and antenna index simultaneously.Moreover, the deployment of soft SM demodulator along with soft RM decoder resulted in increased BER performance of the proposed coded cooperative scheme.Hence, the novel aspect of this research exploration is outlined as follows • The KP construction is effectively utilized for the construction of distributed Reed-Muller (DRM) code and its efficient employment in coded-cooperation.• The DRMC-SM scheme for coded-cooperative and RMC-SM scheme for non-cooperative communication system have been devised.• The powerful joint soft RM decoder (at the destination node) has been proposed for DRMC-SM scheme.• The BER performance of the proposed scheme has also been presented for the imperfect channel case scenarios.
The remaining of the manuscript is structured as the following.Section 2 describes the preliminaries of the distributed RM code.Section 3 discusses the efficient sub-code formation, offered by the proposed construction.The spatial modulation and its correspoding soft spatial demodulation are briefed in Sect. 4. The coded spatial modulation schemes named as DRMC-SM and RMC-SM are detailed in Sect. 5.The joint SISO RM decoder for DRMC-SM scheme is presented in Sect.6.The complexity analysis of the proposed scheme is investigated and discussed in Sect.7. The simulation results of DRMC, DRMC-SM and RMC-SM schemes for different coded-cooperative scenarios are presented in Sect.8. Furthermore, the comparison of DRMC-SM scheme with DTC-SM scheme has also been performed in this section.Eventually, the manuscript is concluded in Sect.9.

Construction of DRM Codes
Classical Reed Muller code belongs to the class of linear block codes whose rich structural properties and simple construction make it distinct over other block codes.Mathematically a binary Reed Muller code R(r, n) of order r(0 ≤ r ≤ n) and code length N = 2 n [21], where r and n correspond to positive integer values, is a set of all vectors , where ( 1 , 2 , ⋯ , n ) is a binary function which is an at most r degree polynomial [21].The dimension and minimum hamming distance of RM code R(r, n) is defined by u = ∑ r a=0 � n a � and d min = 2 n−r [20], respectively.The RM codes are rich in the structural properties allowing them to split apart in its component RM codes.Such splitting of the codes are often termed as Distributed Reed-Muller (DRM) codes and that can be deployed in coded-cooperation scenrios.The RM codes can be constructed by various construction techniques such as CP construction, KP construction or by algebraic construction [20,21].In this paper, we have utilized KP construction for RM code construction.This construction has a built-in Plotkin's construction but there is subtle difference as compared to CP construction.In KP construction, a large length RM code A 3 (N 3 , u 3 , d 3 ) is decomposed into two short length RM codes A 1 (N 1 , u 1 , d 1 ) and A 2 (N 2 , u 2 , d 2 ) , where N k ,u k and d k (k = 1, 2, 3) that defines code length, information sequence length and code's minimum hamming distance, unlike CP construction.The obtained generator matrix through KP construction is given as where A 1 , A 2 and A 3 are the generator matrices of Reed Muller codes i.e., A 1 , A 2 and A 3 , respectively.The generator matrix A 3 has natural Plotkin's construction.Therefore, the large length RM code A 3 can be represented by short length RM codes A 1 and A 2 in Plotkin's form such as where addition operation is define over GF (2).The dimension and minimum hamming distance [20] of RM code A 3 are given by u 3 = u 1 + u 2 and d 3 = min{2d 1 , d 2 } , respectively.
The proposed KP construction of RM codes is opposite to CP construction of RM codes.The construction steps of RM code A 3 are detailed as follows where ≜ 1 1 0 1 and ⊗ defines the Kronecker product.• Expurgate all those rows of matrix ⊗n with hamming weights less than 2 n−r to obtained the generator matrix A 3 of RM code R(r, n).It should be noted that the generator matrix A k , (k = 1, 2, 3) obtained from KP construc- tion is the permuted form of the generator matrix obtained from CP construction.Therefore, better codes (in terms of BER) can be constructed by exploiting its construction in coded cooperative techniques.

Efficient Sub-code Formation by Proposed Construction
The distributed Reed-Muller coded (DRMC) scheme for coded-cooperation is comprised of two RM codes R(r, n − 1) and R(r − 1, n − 1) .The RM code R(r, n − 1) is placed at the source node which encodes u 1 information bits 1 = [p (1)  1 , p (1)  2 , ⋯ , p (1)   u 1 ] into N/2 coded bits 1 , q (1) 2 , ⋯ , q (1)  N∕2 ] .The encoded sequence 1 ∈ R(r, n − 1) is broadcasted to relay and destination nodes during first time-slot T 1 .The relay node correctly decodes the infor- mation sequence ̂ 1 , i.e. ̂ 1 = 1 , whereas, the source to relay channel is considered as an ideal channel.The dimension of RM code R(r − 1, n − 1) which is placed at relay node is less than source RM code R(r, n − 1) .Therefore, only u 2 information bits 2 = [p (2)  1 , p (2)  2 , ⋯ , p (2)   u 2 ] are encoded by relay into N/2 coded bits 2 = [q (2) (1) In this coded-cooperative scheme, only source node is responsible for the generation of all the message bits while the relay node is not producing any extra information bits.Therefore, the u 2 information bits 2 for RM code R(r − 1, n − 1) must be selected wisely from u 1 information bits 1 i.e. 2 is a subset of 1 .The coded sequences ̂ 1 and 2 are XOR together by relay node and transmit to destination node in second time-slot T 2 .In the desti- nation, the | 1 | ̂ 1 + 2 | construction forms Ã3 code with minimum hamming distance d min ( Ã3 ) and it can be represented as follows [18], where Ã3 is a subcode of Reed Muller code R(r, n) , i.e.Ã3 ⊂ R(r, n) or Ã3 ⊂ A 3 .Further- more, d min ( Ã3 ) ≥ 2 n−r and the optimum sub-code is represented as A opt .The optimum sub- code is determined by the algorithm proposed in [18].The difference between the two constructions is demonstrated by the following example. Example: ) is considered as Reed Muller codes employed by DRMC scheme at source and relay nodes, respectively.The generator matrices of A 1 and A 2 RM codes are given as follows Then the resultant RM code A 3 = R(2, 4) is constructed using CP construction at the destination node.The generator matrix representation of RM code A 3 using (1) is defined as In case of proposed KP construction for coded-cooperation, the generator matrix of jointly constructed RM code A 3 = R(2, 4) at the destination is determined by using the steps given in preceding section.The obtained generator matrix of code A 3 by the KP construction is given as follows The generator matrices of source and relay RM codes A 1 and A 2 are extracted from the gen- erator matrix A 3 such as It is evident that the generator matrices ) are the row permutation of each other.For the CP construction, the optimum sub-code A � opt ⊂ R(2, 4) is selected with the help of efficient selection algorithm (ESA) [18].On the basis of better weight distribution, this algorithm selects the best combination of u 2 = 4 information bits 2 = [p (2)  1 , p (2)  2 , p (2)  3 , p (2)  4 ] at relay node out of u 1 = 7 information bits 1 = [p (1)  1 , p (1)  2 , ⋯ , p (1)  7 ] which is transmitted by source node.For CP construction the optimum selected bits at relay node are given as follows The optimum sub-code A � opt ⊂ R(2, 4) has the following weight distribution Similarly, ESA is applied to the RM code constructed via KP construction yield the following optimum selected bits at relay node The optimum sub-code A opt ⊂ R(2, 4) using KP construction has the following weight distribution It can be observed from ( 5) and ( 7) that error coefficient or the number of minimum hamming weight codewords is less for optimum sub-code A opt ⊂ R(2, 4) which is caused by the KP construction.
In this paper, we have considered DRM code R(2, 5) for coded-cooperation.On the basis of minimum value of error coefficients K 1 and K 2 [18], the unique combinations of bit positions for relay node are obtained by ESA for each aforementioned RM construction.For sub-code R(2, 5) with proposed KP construction, the obtained unique combinations of bit positions by ESA are tabulated in Table 1.For the case of CP construction, the unique combinations of bit positions for sub-code R(2, 5) are given in [18].
For DRM code R(2, 5) , the relay node selects u 2 = 5 information bits 2 , ⋯ , p (2)  5 ] out of u 1 = 11 information bits 1 = [p (1)  1 , p (1)  2 , ⋯ , p (1)  11 ] transmitted by source node.On the basis of minimum value of error coefficients K 1 and K 2 , the opti- mum combination of bit positions is selected for the CP construction which is given as [18] and the obtained optimum sub-code A � opt ⊂ R(2, 5) using the sequence 2 mentioned in (8)  has the following weight distribution Similarly for proposed KP construction, the optimum combination of bit positions is selected from Table 1 based on the minimum value of error coefficients K 1 and K 2 , which can be represented as and the obtained optimum sub-code A opt ⊂ R(2, 5) using the selected combination of bits given in (10)

consist of the following weight-distributions
The weight distribution of the RM sub-codes A opt and A ′ opt are determined by exhaustive computer search.The weight distribution W A opt (X) of the optimum subcode A opt ⊂ R(2, 5) constructed from the proposed KP construction is better than the weight distribution W A � opt (X) of the optimum sub-code A � opt ⊂ R(2, 5) constructed from CP construction.This weight distribution of optimum sub-code A opt ⊂ R(2, 5) is the main cause of the enhanced BER performance.As, the KP construction has reduced the minimum hamming weight codewords from 20 to 12 that considerably enhances the bit error rate performance of the coded-cooperative DRMC scheme.

Preliminaries of SM
SM is a famous MIMO technique [7], which provides improve spectral efficiency due to the transmission of message bit sequence by both M-QAM symbols and antenna indices unlike tradition modulation schemes.The simple N t × N r SM technique with M-QAM modulation is illustrated in Fig. 1, where N r and N t represent no. of receive and transmit antennas, respectively.Initially, a train of = log 2 (MN t ) bits is given to SM mapper that assigns g = log 2 (M) bits ( map sequence) to M-QAM symbols z m and f = log 2 (N t ) bits ( ant sequence) to index It should be noted that also defines the spectral efficiency of SM technique.The mapping of bits to z m and z v is depend on a mapping table which is perfectly known to both source and destination nodes.An example of such a mapping table is given in [7].The output of SM mapper can be represented as [11] where mv is a complex sequence with N t elements, z m defines a complex symbol from M- and T defines the trans- pose.Since a transmit antenna z v only takes part in the transmission of z m symbol in SM.Thus, the sequence mv has a non-zero element z m at v-th position defining active transmit antenna while the rest of N t − 1 zero elements define the dormant transmit antennas.

Soft SM Demodulator
The sequence mv after mapping is transmitted over channel .The received sequence is represented as where shows N r × 1 received sequence, demonstrates fading channel matrix of dimen- sion N r × N t , and is N r × 1 AWGN sequence.Each element of and is identically inde- pendent distribution comprised of complex Gaussian distributed CU(0, 1) and CV(0, 2 ) , respectively.Moreover, v represents channel of active transmit antenna which is the v-th column of .After that the received sequence undergoes soft SM demodulation.The process of soft SM demodulation is as follows.Let Λ l,0 and Λ l,1 represent the antenna = mv + , = v m + , Fig. 1 Uncoded SM technique index subsets whose every element is mapped by bit sequences having 0 and 1 at l-th bit, respectively.Likewise, l,0 and l,1 represent the M-QAM symbol subsets whose every element is mapped by bit sequences having 0 and 1 at l-th bit, respectively.Hence, the loglikelihood ratio's (LLR's) can be mathematically expressed as [26] Finally, the bit-combiner concatenate the LLRs calculated from ( 14) and ( 16) to make the sequence D ( ) .The slicer recovers the information sequence ̂ .For the coded system, the soft sequence D ( ) is useful for decoder which will be discussed in subsequent section.

Coded Spatial Modulation Schemes
This section discusses the DRMC-SM scheme for coded-cooperative communication.Furthermore, it also explains the RMC-SM scheme for non cooperative communication.

Model of the Proposed Distributed Reed-Muller Coded Spatial Modulation (DRMC-SM) Scheme for Coded Cooperative Communication
The rich recursive structure of RM code allows to effectively distribute it over the source and relay nodes.Hence, an efficient DRMC-SM scheme (shown in Fig. 2) is proposed for single relay coded cooperative communication system.
In DRMC-SM scheme, the message sequence 1 of source node (S) takes two consecutive time-slots for an end-to-end transmission.During time-slot T 1 , the message sequence 1 of length u 1 is encoded by source node using RM code R(r, n − 1) .The encoded sequence 1 (a.k.a.S ) of length N 1 = 2 n−1 is then undergo SM as described in Sect. 4. The output sequence S mv 1 (i 1 ) = [0, ⋯ , z S m (i 1 ), ⋯ , 0] T of SM mapper is send to relay (R) and des- tination (D) nodes, where z S m (i 1 ) defines a complex symbol from M-QAM constellation having normalized is the number of SM symbols in a codeword at source node.
At relay node, the received sequence SR (i 1 ) during time-slot T 1 can be modelled as SR ] is a source-to-relay (SR) slow fading N r × N t channel matrix.Each element of channel matrix SR defines a column vector like The channel matrix SR follows the Rayleigh fading channel model with each element distributed according to Gaussian distribution CU(0, 1) .Moreo- ver, SR (i ] is an AWGN vector with every element n w SR (i 1 ) ,w ∈ [1 ∶ N r ] is distributed according to Gaussian distribution CV(0, 2 ) .Similarly at des- tination node, the received sequence SD (i 1 ) during time-slot T 1 can be mathematically rep- resented as where SD , v 1 SD and SD (i 1 ) represent source-to-destination (SD) fading channel matrix of order N r × N t , v 1 -th fading channel vector and AWGN vector, respectively, defined in a similarly as SR , v 1 SR and SR (i 1 ) in (15).All the I 1 RM coded SM symbols are send to relay and destination nodes in time-slot T 1 .
During time-slot T 2 , all I 1 received symbols at relay node are demodulated using soft SM demodulator and bit combiner.The soft maximum likelihood (S-ML) decoder of RM code R(r, n − 1) is used to decode the soft coded sequence SR ( S ) which is provided by soft SM demodulator and bit combiner block.The decoded (detected) message sequence ̂ 1 is partially encoded at relay node using Reed Muller code R(r − 1, n − 1) i.e.only u 2 (17) Fig. 2 Single relay DRMC-SM scheme for coded-cooperative communication systems message bits are chosen for encoding at relay node out of u 1 detected message bits.Since u 2 < u 1 , therefore the selection of u 2 message bits depends upon the weight distribution criteria is described in Sect.3. The effective selection of the information bits at relay node significantly enhances the bit error rate performance of the coded cooperative scheme [18].The selected message sequence 2 of length u 2 is encoded using RM code R(r − 1, n − 1) to obtain the coded sequence 2 of length N 2 = 2 n−1 .The coded sequence 2 and detected coded sequence ̂ 1 are then GF(2) sum to get the coded sequence R = ̂ 1 + 2 .The SM is performed on the coded sequence R and the output sequence . I 2 is the number of SM symbols in a codeword at relay node.During time-slot T 2 , the received sequence RD (i 2 ) at destination node is modelled as follows where RD , v 2 RD and RD (i 2 ) define relay-to-destination (RD) fading channel matrix of order N r × N , v 2 -th fading channel vector and AWGN vector, respectively, which are rep- resented in a same way as SR , v 1 SR and SR (i 1 ) in (15).All I 1 and I 2 RM coded SM symbols are received by destination nodes during their respective time-slots which undergo joint soft RM decoding to obtain the detected message sequence ̂ 1 .The joint soft RM decoding is discussed in Sect.6.

Model of the Reed-Muller Coded Spatial Modulation (RMC-SM) Scheme for Non-cooperative Communication System
A non-cooperative RMC-SM scheme also exploits the recursive structure of RM code.In RMC-SM scheme instead of placing the codes at relay and source nodes (as in DRMC-SM), both the codes are placed at source node as shown in Fig. 3.This scheme is also used a resonable benchmark for DRMC-SM scheme.In this scheme, the message sequences 1 and 2 of length u 1 and u 2 are encoded by RM codes R(r, n − 1) and R(r − 1, n − 1) , respectively at source node.Since 2 is dependent on 1 , so it is selected according to the same criteria used in DRMC-SM scheme which is explained in Sect.3. The coded sequences 1 and 2 of each RM code are then GF(2) sum to get the coded sequence 1 + 2 .This sequence is further concatenated with coded sequence 1 of RM code R(r, n − 1) to construct the coded sequence . This sequence is undergone SM and transmitted to the receiver.The soft SM demodulation and S-ML decoding at the destination node finally estimates information bit sequence ̂ 1 .

Joint Soft RM Decoding
The joint soft Reed Muller decoding of DRMC-SM scheme is depicted in Fig. 4. The decoding is based on component like soft SM demodulator, bit combiner and S-ML decoder.In case of hard decoding, the soft metric components are replaced with the hard metric components.( 19) At the destination node, soft SM demodulator provides LLRs SD (z m (i 1 )) and SD (z v 1 (i 1 )) of M-QAM symbol and antenna index, respectively for each received sequence SD (i 1 ) during time-slot T 1 .The LLRs of all I 1 RM coded SM symbols are combined by bit combiner block to construct the LLR sequence SD ( S ) .In a similar way, the LLR sequence RD ( R ) is con- structed by bit combiner block by combining all I 2 RM coded SM symbols during time-slot T 2 .
The DRM code R(r, n) constructed from RM codes R(r, n − 1) and R(r − 1, n − 1) can be decoded using an S-ML/MJL decoder in a single step [18,19].Therefore, a single joint coded sequence is formed by concatenating the output sequence of bit combiner blocks such The LLR sequence o is acted like a single codeword for RM decoder instead of two separate codewords SD ( S ) and RD ( R ) .Different decoders of RM code are suggested in literature, MJL decoder [20] was the first RM decoder.A low complexity soft MJL decoder and a simple soft decoder for RM codes are proposed in [29,30].In this paper, MJL and S-ML decoders are used for decoding.The decision metric of S-ML decoder can be defined as (20) where l ∈ {−1, +1} is the l th modulated/soft codeword, l ∈ [0 ∶ 2 u 3 ] and Ω is set of all possible modulated/soft codewords R ∈ {−1, +1} are the modulated/soft bit code- words of source and relay nodes, respectively.The well know Frobenius norm is denoted by ||.|| F .The S-ML decoder generates the detected message sequence by using the joint LLR sequence o .The source transmitted message sequence ̂ 1 is recovered by selecting only the initial u 1 bits of detected message sequence ̂ .In order to use the MJL decoder for the decoding of RM code, the whole process remains same except soft SM demodulator is replaced with hard SM demodulator [11].For the details of MJL decoder are given in [20].

Complexity Analysis
The Max-Log-LLR algorithm is used in the soft SM demodulator to avoid the logarithmic and complex exponential operations in the log-MAP algorithm.The Eqs. ( 14) and ( 16) define the soft SM demodulator that employs the Max-Log-LLR algorithm.The computational complexity of the soft SM demodulator is evaluated in terms of number of real additions and real multiplications.The required number of additions and multiplications for the term || − v z m || 2 is 6N r − 1 and 6N r , respectively.This term needs to be computed N t M times for the Max-Log-LLR algorithm.Moreover, LLR calculation for each bit one real addition and one real multiplication are required, respectively.The spatial modulation symbol is consists of log 2 (N t M) bits, therefore, the total number of real additions and real multiplications are given as The computational complexity of the hard SM demodulator is given in [31] such as The computational complexity of the MJL decoder and the S-ML decoder of RM codes are given in [32] and [33], respectively.The equations ( 23) and ( 24) define the computational complexity of the MJL decoder and the S-ML decoder, respectively which are given as follows The total computational complexity at the receiver for MLJ and S-ML decoders is given as where N log 2 (N t M) is the number of SM symbols in an RM codeword.The Table 2 tabulates the complexity analysis of the proposed schemes for different number of transmit and receive antennas under and 16QAM modulations.The Table 2 confirms that MJL decoder has much less complexity as compared to S-ML decoder.However, BER performance of S-ML decoder is better than MJL decoder as confirmed by simulation results.

Numerical Results
This section simulates and compares the bit error rate performance of DRMC, DRMC-SM and RMC-SM schemes.The optimally designed DRM code R(2, 5) is employed in all sim- ulations which is constructed using RM codes R(1, 4) and R(2, 4) .The DRMC scheme is simulated over AWGN channel using B-PSK modulation while the DRMC-SM and RMC-SM schemes with 4-QAM modulation are simulated over slow Rayleigh fading channels.The term 'slow' defines that the channel remains static for an entire RM codeword and it known to the receiving end.Furthermore, BER of DRMC-SM scheme is also evaluated for different transmit antennas.Hence, the spectral efficiency of DRMC-SM scheme for N t = 4 and N t = 8 with 4-QAM modulation is 4 b/s/Hz and 5 b/s/Hz, respectively.The SNR per bit ( Γ SD ) of SD channel is used in all BER simulations and the code rate R D is defined according to the code constructed at destination node i.e.R D = 11∕32 [18].
The coded-cooperative DRMC scheme is simulated using CP and KP constructions over AWGN channel with optimum bit selection at relay node, ideal SR channel ( Γ SR = ∞ ) and the relay node does not enjoy any additional gain in SNR over source node ( Γ RD = Γ SD ).The optimally selected bits at the relay node for both constructions are given as (8) and (10).The simulated results in Fig. 5 clearly indicate the better BER performance of DRMC scheme with KP construction over the simulated SNR region.At low SNR region, the bit error rate performance of DRMC scheme with KP construction is slightly better than the DRMC scheme with CP construction.However, at high SNR region the difference between two schemes becomes more prominent.The DRMC scheme with KP construction produces a gain of about 1 dB in SNR over CP constructed DRMC scheme at BER=7 × 10 −7 .This enhanced BER performance of DRMC scheme is only made possible due to better weight distribution offered by KP construction over existing CP construction.The spectrally efficient DRMC-SM scheme is analysed over slow Rayleigh fading channel with ideal SR channel ( Γ SR = ∞ ), N t = N r = 4 , Γ RD = SD and joint soft RM decoder at the destination.The comparison between the DRMC-SM schemes with KP and CP construction under optimum bit selection at relay is shown in Fig. 6.The KP construction performs better than CP construction even for DRMC-SM scheme as well.At SNR=10 dB, the DRMC-SM scheme with KP construction and DRMC-SM scheme with CP construction attain BER=9 × 10 −7 and BER=2 × 10 −6 , respectively.Furthermore, we have com- pared the proposed DRMC-SM schemes under the optimum and random bit selection scenarios at relay node.The scheme with random bit selection at relay is named as Distributed Random Reed-Muller Coded Spatial Modulation (DRRMC-SM) scheme.The DRMC-SM scheme obtains a bit error rate performance gain of approx.2.2 dB over DRRMC-SM scheme at bit error rate of 3.5 × 10 −5 .It shows the efficacy of DRMC-SM scheme over DRRMC-SM scheme.Thus, the proposed DRMC-SM scheme with KP construction is opted for further analysis.
For four transmit antennas ( N t = 4 ) BER performance of DRMC-SM and RMC-SM schemes under slow Rayleigh fading channel is depicted in Fig. 7.In this simulation soft SM demodulator along with S-ML decoder is used and ideal SR channel ( Γ SR = ∞ ) is considered.It can be observed from Fig. 7 that the coded cooperative DRMC-SM scheme performs better than the non cooperative RMC-SM scheme.In case relay enjoys no gain in SNR over source node, i.e.Γ RD = Γ SD , then the coded-cooperative DRMC-SM scheme provides a 1.2 dB better BER performance over non cooperative RMC-SM scheme at BER=4.5 × 10 −6 .If relay is given an extra gain of 2 dB in SNR over source node, i.e.Γ RD = Γ SD + 2 dB, then the bit error rate performance of coded cooperative DRMC-SM scheme is further enhanced as it yields 2.5 dB performance gain over non cooperative RMC-SM scheme at BER=4.5 × 10 −6 .Moreover, the coded cooperative DRMC-SM scheme is also compared with the equivalent coded cooperative DTC-SM scheme under similar conditions such as R D = 1∕3 , N = 33 and N t = N r = 4 etc.In DTC-SM scheme, source node uses a recursive systematic convolutional (RSC) encoder having the code rate 1/2 along with SM mapper while relay node employs the same RSC encoder with an interleaver and SM mapper but with punctured systematic bit.The symmetric RSC encoders having generator matrix = [1, 5∕7] 8 are employed at source and relay nodes.The turbo decoder (MAP) utilizes eight decoding iterations to detect the message sequence.Under similar conditions DRMC-SM scheme performs better than the DTC-SM scheme and at BER=2.5 × 10 −6 it attains a BER performance gain of 2.2 dB.The evident gain in per- formance of DRMC-SM scheme is due to the optimally design of relay node (effective bit selection) and the joint soft RM decoding performed at destination node.
In order to show the effectiveness of the joint soft RM decoding, we compare the bit error rate performance of DRMC-SM scheme under joint soft RM decoding and joint hard RM decoding.The simulated result shown in Fig. 8 that DRMC-SM scheme with joint soft RM decoding provides a gain of 4.3 dB at BER=4.5 × 10 −4 as compared to DRMC-SM scheme with joint hard RM decoding.The gain in bit error rate performance of DRMC-SM scheme with joint soft RM decoding is obvious due to deployment of soft SM demodulator and S-ML decoder.Since in joint hard RM decoding, the hard SM demodulator and MJL decoder is used which degrades the overall bit error rate performance of DRMC-SM scheme.
The DRMC-SM scheme is further analysed for the practical scenario.For the case of practical scenario the SR channel is assumed to be the non-ideal Γ SR ≠ ∞ and Γ RD = Γ SD + 2 dB.It is a known fact that the error performance of coded-cooperative scheme degrades due to lack of cooperation between source and relay node [12].This degradation may be due to bad or noisy SR channel.Therefore, we analyse the DRMC-SM scheme for Γ SR = 4 dB and Γ SR = 6 dB.It can be observed from Fig. 9 that the relay is in outage for both non-ideal SR channels Γ SR = 4 dB and Γ SR = 6 dB, which signifi- cantly degrades the BER performance of DRMC-SM scheme.In such cases relay propagates the error to the destination node due to the incorrect detection of source information.This problem can be control by the employing any error propagation control mechanism at relay such as cyclic redundancy check (CRC) proposed in [12].The interested reader may refer to [34] for the details of error propagation control mechanism.Further, we analyse the DRMC-SM scheme for Γ SR = 10 dB and Γ SR = 12 dB.It can be seen that for the case of Γ SR = 10 dB, the BER performance of DRMC-SM scheme is improved.It achieves the BER=6 × 10 −6 at SNR=7 dB which is 1 dB away from the bit error rate performance of ideal DRMC-SM scheme (Γ SR = ∞) .For the case of Γ SR = 12 dB, the bit error rate perfor- mance of DRMC-SM scheme is enhanced significantly and approximately reached the bit error rate performance of ideal DRMC-SM scheme (Γ SR = ∞).
In Fig. 10, the effect of no. of receive antennas in DRMC-SM scheme is examined.The bit error rate performance of DRMC-SM scheme is further improved by the addition of some receive antennas.In case receive antennas are five or six, a gain ranges from 1.5 dB to 2.5 dB is attained over four receive antennas by DRMC-SM scheme at BER= 10 −5 .However, if the no. of receive antennas decreases from four to three, the BER per- formance degradation of about 2.3 dB is observed at BER=8 × 10 −5 .Thus, there is a trade off between the bit error rate performance of DRMC-SM scheme and number of receive antennas.
Finally, the DRMC-SM scheme is analysed for eight transmit antennas ( N t = 8 ) which has the spectral efficiency of 5 b/s/Hz.It is observed from Fig. 11 that the optimum bit selection at relay node is also effective for DRMC-SM scheme with eight transmit Fig. 9 BER performance of DRMC-SM scheme under non-ideal SR channel and N t = N r = 4 antennas.Under similar conditions, the DRMC-SM scheme beats the DRRMC-SM scheme with the performance gain of 3 dB at BER=6 × 10 −4 .Further, the DRMC-SM scheme is compared to RMC-SM scheme under the assumption of Γ RD = Γ SD and optimum bit selec- tion.DRMC-SM scheme performs better than RMC-SM scheme.At SNR=10 dB, the DRMC-SM and RMC-SM schemes attain BER=9 × 10 −6 and BER=8 × 10 −5 , respectively.This shows the supremacy of DRMC-SM scheme over non-cooperative RMC-SM scheme irrespective of number of transmit antennas.

Conclusion
This manuscript presented KP construction of RM codes for coded cooperative communication system.This construction has offered better weight distribution for jointly constructed RM code at the destination node in comparison to CP construction.Therefore, the DRMC scheme with KP construction achieves 1 dB (interms of BER performance) gain over traditional DRMC scheme having CP construction over AWGN channel.However, both of these schemes were spectrally inefficient.Thus, we extended the proposed scheme to spectrally efficient DRMC-SM scheme that not only transmit the coded information via modulated symbols but also via transmit antenna indices.The numerical results confirm that the DRMC-SM scheme performs better than the non-cooperative RMC-SM scheme due to the joint RM code construction, efficient bit selection (at relay), joint soft RM decoding and path diversity.Moreover, we also compared DRMC-SM scheme with DTC-SM scheme under similar conditions.The proposed DRMC-SM scheme has also shown the superior BER performance in comparison to DTC-SM scheme.The dominance of joint soft RM decoder is evaluated by comparing Fig. 10 The effect of receive antennas on the BER performance of DRMC-SM scheme with four transmit antennas it with joint hard RM decoder.The DRMC-SM scheme is further investigated for more practical coded-cooperative scenarios by considering non-ideal SR channel.Finally, the effect of transmit and receive antennas on the bit error rate performance of DRMC-SM scheme has also been analysed.

Fig. 5 Fig. 6 4 Fig. 7
Fig. 5 BER performance of DRMC scheme for KP and CP construction over AWGN channel with optimum bit selection at relay and ideal SR channel

Fig. 8
Fig. 8 Joint soft RM decoding vs joint hard RM decoding of DRMC-SM scheme under ideal SR channel and N t = N r = 4 •The obtained generated matrix A 3 of Reed Muller code R(r, n) is already in Plotkin's form therefore the matrices A 1 and A 2 is extracted from matrix A 3 , where A 1 is a u 1 × N∕2 generator matrix of Reed Muller code R(r, n − 1) and A 2 is a u 2 × N∕2 gen- erator matrix of Reed Muller code R(r − 1, n − 1) , besides u 1 > u 2 .

Table 1
Combinations of bit indices which result in minimum value of K 1 and K 2