Optimized-Goppa Codes Based on the Effective Selection of Goppa Polynomials for Coded-Cooperative Generalized Spatial Modulation Network

. This paper proposes a novel optimized-Goppa-coded cooperative generalized spatial modulation (OGCC-GSM) scheme for short-to-medium information block transmission. In the proposed OGCC-GSM scheme, an efficient Goppa polynomial selection approach is designed to ensure that the selected Goppa codes applied in the source and relay nodes both have the largest minimum Hamming distance (MHD) and the optimal weight distribution. Compared to conventional coded cooperation (CC) with a single antenna, the proposed scheme employs the generalized spatial modulation (GSM) technique to achieve more diversity gains, where each node is equipped with multiple antennas and more than one transmit antenna (TA) is activated at each time-instant transmission. As a benchmark comparison, the OGCC spatial modulation (OGCC-SM) scheme is also investigated with a single TA active. Moreover, the reduced-complexity transmit antenna combination (RC-TAC) selection algorithm utilized in GSM is first developed with the aid of the channel state information (CSI) to reconcile computational complexity and system performance. In addition, joint decoding is conducted on the destination terminal to further enhance the performance of the proposed scheme. The simulated results indicate the performance of the proposed OGCC-GSM scheme is superior to that of its benchmark OGCC-SM scheme, with a substantial reduction in the number of TAs. Besides, Monte Carlo simulations demonstrate that the proposed OGCC-GSM scheme prevails over its counterparts by a margin of over 4.2 dB under identical conditions.


Introduction
In wireless communication systems, channel fading has been proven to deleteriously impair overall performance due to multipath propagation.The diversity technique is one of the most effective techniques to combat fading, which increases the reliability of the communication system by providing multiple remediable transmission paths that fade independently [1].Currently, a relatively new diversity technique called user-cooperative diversity has emerged, which intelligently leverages path diversity.The fundamental concept of user cooperation is to share the whole/part information with at least two mobile users (source and relay) and transmit it with a single antenna to the common terminal (destination).Among the existing cooperative diversity techniques, coded cooperation (CC), combining channel coding and path diversity, demonstrates a superior bit error rate (BER) performance that offers both coding gain and diversity gain [2].Many binary long channel codes including turbo-codes [3], polar codes [4], and low-density parity-check (LDPC) codes [5] have been deployed in CC.Recently, short-length non-binary codes have also been commonly utilized in CC to facilitate the transmission of small information blocks, such as for developing cellular and sensor networks [6].Guo et al. [7] proposed the non-binary Reed Solomon coded cooperative (RSCC) scheme where two selection approaches are developed in the relay node to generate a strong final code in the destination node.As a maximum distance separable (MDS) code, the RS code has a great capacity to rectify random burst errors.Unfortunately, its information dimension and code length are conditional, limiting the proposed RSCC system from being generalizable.The focus then turns to another short-to-medium length code, namely Goppa code, which allows for more flexibility in selecting the code length and dimension, and has a low encoding and decoding complexity [8].Clearly, the Goppa code is more appropriate to be implemented in practical engineering.Waweru et al. [9] and Feng et al. [10] both proposed the classic binary Goppacoded cooperation (GCC) scheme.They employed relay selection methods similar to that in [7] to acquire a resultant code with superior weight distribution.However, the complexity of the two selection approaches is extremely high.To avoid the complexity induced by the selection without affecting the overall performance, we have opted to transfer all the source's messages to the relay in this manuscript.In addition, two factors are required for the construction of a Goppa code: the code locator set and the Goppa polynomial.Under the premise of a given locator set, different Goppa polynomials may influence the performance of a Goppa code.In literature [9] and [10], the Goppa polynomial is selected randomly, ignoring this issue.Therefore, this paper proposes a simple but effective optimization selection approach for determining the optimum Goppa polynomial to achieve the Goppa code with the maximum free/minimum Hamming distance (FHD/MHD).And two optimized Goppa codes are employed in the CC to construct an optimized-Goppa-coded cooperative (OGCC) scheme.
Typically, coded cooperation is equipped with a single antenna at each node, resulting in low spectrum efficiency.Moreover, only binary-phase shift-keying (BPSK) modulation was considered in the previous GCC systems, which leads to poor bandwidth efficiency.Compared to singleantenna systems, multiple-input multiple-output (MIMO) systems are acknowledged to be a vital technology to enhance link reliability and spectrum efficiency [11].Among many MIMO systems, spatial modulation (SM) may effectively avoid inter-channel interference (ICI) and inter-antenna synchronization (IAS) by conveying information with amplitude/phase modulated (APM) symbol and antenna index due to only one antenna being activated during a one-time instant transmission [12].Hence, the total spectral efficiency increases by the base-two logarithms of the number of transmit antennas (TAs), implying that a large number of TAs is required to improve the spectral efficiency.To address this challenge, generalized spatial modulation (GSM) was proposed where more than one TA is activated concurrently to transmit a modulated symbol [13].Consequently, for a certain spectrum efficiency, the number of TAs of GSM is lowered by more than half compared to SM.In the latest available literature [17], [18], the authors proposed distributed Goppa-coded spatial modulation (DGC-SM) and distributed Goppa-coded generalized spatial modulation (DGC-GSM) schemes, respectively, only adopting the relay selection approaches similar as ones developed by [9].However, to our best knowledge, optimized Goppa codes by selecting Goppa polynomials have not been thoroughly investigated and implemented on CC.Inspired by this, the optimized-Goppacoded cooperative GSM (OGCC-GSM) scheme is proposed as a fusion of the OGCC system and GSM technique in this manuscript.Moreover, its benchmark OGCC-SM system is presented as a counterpoint to the OGCC-GSM scheme.Note that not only one transmit antenna is selected in the GSM technique.Therefore, we also develop an appropriate transmit antenna combination (TAC) selection algorithm to optimize the activated antenna selection.Since the message at the relay node is dependent on that at the source, a joint decoding approach is established to obtain the final estimated messages.
Hence, the primary contributions of this paper are as below: • An efficient selection approach of the Goppa polynomial is proposed to select an optimized Goppa code with the largest MHD and an optimal weight distribution.
• The novel OGCC-GSM scheme is proposed over the Rayleigh quasi-static fading channels where two optimized Goppa codes are utilized in the source and relay, respectively.And its benchmark OGCC-SM scheme is also developed as a comparison.
• Based on the optimum TAC algorithm [15], a reducedcomplexity TAC (RC-TAC) selection algorithm is presented to maximize the minimum Euclidean distance (MED) of the transmitted signals using the channel state information (CSI).
• A joint decoding approach is developed for both OGCC-GSM and its benchmark OGCC-SM schemes to enhance the overall performance.
• The performance of the OGCC-GSM scheme is evaluated under the ideal and imperfect mobile user channels (source-to-relay links).
The remaining sections of the manuscript are structured as follows: Section 2 provides a concise overview of the Goppa code generation process and its fundamental properties.In Sec. 3, an optimized selection algorithm of the Goppa polynomials is proposed along with an example of the selection procedure.The OGCC-GSM scheme and the benchmark OGCC-SM scheme are presented over the Rayleigh fading channel in Sec. 4. Section 5 proposes a novel RC-TAC algorithm to trade off the complexity of the TAC selection and performance of the GSM.In Sec. 6, a joint decoding approach is developed in the destination.The various Monte-Carlo simulations are presented to validate the proposed OGCC-GSM scheme in Sec. 7. Ultimately the manuscript concludes with Sec. 8.

Fundamentals of Goppa Codes
Given the Galois field GF(2 m ) ( is any positive integer), define the set is determined as a (, , ) linear code where all components of the codeword c = { 0 ,  1 , . . .,   −1 } are in GF (2) and satisfy the condition, It is noted that L is referred as the code locator set, and its cardinality determines the code length of Γ(L, G(x)), i.e., |L| = .Also, G() is recognized as the Goppa polynomial whose degree  also represents the number of rectifiable errors of this Goppa code.Generally, the MHD  satisfies  ≥  + 1 [9].
Furthermore, the parity-check matrix H of the Goppa code Γ(L, G(x)) for decoding is provided as The dimension of the Goppa code Each element in GF(2 m ) corresponds to an -bit vector.
Therefore, H may need to extend to the  ×  binary matrix H, and then the systematic generator matrix G can be acquired by simple row transformation of H for Goppa encoding.
(3) Table 1 illustrates the corresponding 4-bit vector for each non-binary element in GF(2 4 ), and H is then extended into a 4 × 10 binary matrix H as follows, Moreover, the systematic generator matrix G of the Goppa code for encoding is obtained by row operations and transformation of H.

Optimized Selection of the Goppa Polynomial
Unlike MDS codes such as RS code whose minimum distance can reach 2 + 1, the MHD  of Goppa code Γ(L, G(x)) has a low bound as  + 1.As the components of the code locator set L are selected independently and equitably from GF(2 m ), L = { 0 ,  1 , . . .  −1 } is predetermined to meet the objective of controlling variables.Thus, the different selections of Goppa polynomials may result in various values of .The unique Goppa polynomial is determined from 2  potential candidates.A random selection of G(x) may induce the error correction performance of a single Goppa code to be extremely poor.Hence, an efficient selection of G(x) for obtaining the maximum value max () and optimizing the weight distribution is essential.

Goppa Polynomial Selection Steps
Before presenting the design steps, the following terms are defined: 1) " →  " signifies that event  that causes event  ; 2) | | denotes the cardinality of set ; 3) the weight distribution of Γ(L, G(x)) is provided as where   ( = ,  + 1, ..., ) is the number of the codewords with the weight .The specific steps of the optimized selection are outlined below: Step 1 Determine the set Λ 0 to store the Goppa polynomials such that G( i ) ≠ 0 (∀  ∈ L, i = 0, 1, . . ., N − 1) from all 2  candidates.
Step 2 Through the calculation or the computer search, some Goppa polynomials that lead the dimension of Γ(L, G(x)) to be more than  are removed, and the rest polynomials are kept in the set Λ 1 .
Step  Step 6 The optimum Goppa polynomial G opt () is acquired and the selection approach is completed.
Note that if the selection procedure comes to acquiring Moreover, the weight distribution  () is also optimized by this selection approach.To make the proposed Goppa polynomial selection clearer, Figure 1 summarizes the effect of each step.
The selection is finalized.This optimized selection approach ensures that a single Goppa code has the maximum MHD and as few codewords as possible have small weights.In this example, the weight distribution is optimized as

The Proposed Model of the OGCC-GSM Scheme
Figure 2 displays the system structure of the proposed OGCC-GSM scheme with a single relay.Source and relay nodes in the OGCC-GSM scheme have the same numbers of antennas and activated TAs, designated as  t and  a ( a <  t ), respectively, while the number of receive antennas (RAs) is represented as  r at the destination node.Note that the relay node uses the same set of antennas to receive signals from the source and transmit signals to destination nodes, i.e., antenna multiplexing.However, the number of activated antennas is different in two phases after the inclusion of GSM technology.Furthermore, two Goppa codes from the same Galois field are optimized by the proposed selection approach and then distributed at the source and relay, respectively.The entire message transmission requires two phases to complete in a half-duplex manner.
During phase-1, the information bit sequence u is encoded by the optimized Goppa code Γ 1 (L 1 , G opt,1 (x)) with code length  1 , code dimension  and the maximum MHD  1 to obtain the encoded sequence c 1 .Through the buffer, c 1 is split into the short sequences c 1 ( 1 ) ( the number of dummy bits adding in c 1 to ensure the length ) is a multiple of ,⌊•⌋ represents the floor operation, and  stands for the modulation order.Then, all short sequences c 1 ( 1 ) enter the 'GSM' module in turn, where first denotes the index of the activated TA referred to the selected TAC index table I sel and the remaining bit vector c map 1 ( 1 ) of length  2 = log 2  is modulated as an -PSK/QAM symbol  1 ( 1 ).Combined with I 1 ( 1 ) and  1 ( 1 ), the  t × 1 GSM vector X S ( 1 ) for each  1 at the source is given as where the modulated symbol  1 ( 1 ) is only placed in the   -th TA (  ∈  1 ( 1 ),  = 1, 2, . . .,  a ) while other TAs keep dormant, and [•] T is the transpose operation.Shown in the 'GSM Mapper' block of Fig. 2, the TAC index table is pre-selected based on the CSI that will be elaborated on in the next section.Subsequently, X S ( 1 ) is broadcast where z S,R   represents the   -th column of Z S,R that is the  r × 1 channel vector from the   -th activated TA at the source to all receive antennas at the relay.z S,D   is also defined similarly as z S,R   .Moreover, N S,R and N S,D are the  r × 1 noise vectors from the source to the relay and destination nodes, yielding the complex Gaussian distribution as CN (0,  2 ) where  2 stands for the power spectral density (PSD) of noise.
During phase-2, supposing that the CSI is known at the relay, the optimal maximum likelihood (ML) detection is first performed on Y S,R ( 1 ), given as Ĩ1 ( 1 ), s1 ( 1 ) = arg min where S is the set of all signal constellation points and 2 )/), where  ′ 2 is also the number of dummy bits defined similarly as  ′ 1 .Likewise, c 2 ( 2 ) is sent to the 'GSM' module to attain the  t × 1 GSM vector X R ( 2 ).The detailed process can be referred to X S ( 1 ) at the source node.Next, X R ( 2 ) is transmitted to the destination terminal, and the received signal vector Y R,D ( 2 ) is mathematically modeled as where Z R,D and N R,D are the Rayleigh quasi-static fading MIMO channel and the noise vector from the relay to the destination node, defined like Z S,R and N S,R , respectively.Moreover, z R,D   represents the   -th column of Z R,D .Then, the received signals Y S,D ( 1 ) and Y R,D ( 2 ) are demodulated at the corresponding phase through 'GSM Demapper' block.Lastly, the 'Joint Decoding' block performs on the detected vectors c1 and c2 acquire to the estimated final information sequence û, which will be discussed in detail in Sec. 6.

Benchmark: OGCC-SM Scheme
The OGCC-SM scheme is introduced as a benchmark for the proposed OGCC-GSM scheme with the aid of Fig. 2. The primary distinction between SM and GSM is that only one TA is active in SM.Hence, the TAC selection is not required.More specifically, the modifications for the message transmission of the OGCC-SM scheme are discussed in detail below.
During the phase-1, the short sequence c 1 ( τ1 ) ( τ1 = 1, 2, . . ., l1 , l1 = ( 1 + N′ 1 )/ n) of length n = log 2  t + log 2  is also divided into two parts in SM: the first sequence c ant 1 ( τ1 ) of length n1 = log 2  t corresponds to one activated TA index  1 ( τ1 ) ∈ {1, 2, . . .,  t }, and the rest  2 -bits sequence c map 1 ( τ1 ) is modulated as an M-PSK/QAM symbol  1 ( τ1 ) same as c map 1 ( 1 ).This demonstrates a drawback of SM that the number of TA must be the power of 2 compared with GSM.Then, the  t × 1 SM symbol vector X S ( τ1 ) for each τ1 at the source node is written as where  1 ( τ1 ) is placed in the  1 ( τ1 )-th TA.Likewise, the / n) at the relay node is acquired by the same process of the SM mapper of X S ( τ1 ) during the phase-2.The received signal vectors Y S,R ( τ1 ), Y S,D ( τ1 ) and Y R,D ( τ2 ) at the respective phase can refer to (8), (9), and (11).Furthermore, the joint decoding approach is applicable to the OGCC-SM scheme as well.
In both OGCC-GSM and OGCC-SM schemes, the code rate is  CC = /( 1 +  2 ) from the perspective of the destination node.Additionally, the spectral efficiency for the OGCC-GSM scheme is represented as  GSM = ⌊log 2  t  a ⌋ + log 2  while that for the OGCC-SM is denoted as  SM = log 2  t + log 2 .Note that the OGCC-SM scheme requires more TAs than the OGCC-GSM scheme to achieve identical spectral efficiency.

Proposed RC-TAC Selection Algorithm Used in OGCC-GSM Scheme
For the OGCC-GSM scheme with  t TAs and  a activated TAs,  =  t  a valid TACs are available.However, only  ′ = 2 ⌊log 2 (  t a )⌋ TACs are utilized to map the message bits in the 'GSM Mapper' block of the Fig. 2. Thus, there are   ′ potential TAC index tables that are all stored in the set I, i.e., where I  denotes the TAC index table.Evidently, |I| = .Based on the CSI, Xiao el at.[15] proposed the optimum TAC selection algorithm that calculates all Euclidean distances (EDs) of different GSM vectors for each I  to select one table with the maximum MED.Unfortunately, the computational complexity is extremely high considering the large values of  t and  a .Therefore, the RC-TAC selection algorithm is proposed to search the promising candidates based on the channel modulus sorting and the corresponding EDs, instead of searching all  TAC index tables.

Algorithm Description
Taking the source-to-destination (S-D) transmission link as an example, the  r ×  t CSI matrix Z S,D = [z S,D 1 , z S,D 2 , ..., z S,D  t ] is known at the source where z S,D  ∈ C  r ×1 ( = 1, 2, ...,  t ) is the -th column of Z S,D .The specific selection algorithm includes six steps: Step 1 Determine the set Z of the modulus of the  t columns for Z S,D , given as Step 2 Sort Z in descending order with the corresponding TA index   is recorded sequentially and saved in the set Δ, where Δ = I 1 , I 2 , ..., I N t .
Step 3 Divide the sorted set Δ into two parts: first  u indexes and the remaining  t −  u indexes, stored in the Δ 1 and Δ 2 , respectively, where  u = ⌈  t +1 2 ⌉, and ⌈•⌉ represents the ceil operation.
Step 4 Perform a selection of prospective TAC index candidates.Preference is given to the indexes in the set Δ 1 .
To guarantee  ′ TACs, Δ 2 also need to be chosen.
The total number L is obtained as Then, save all reduced TAC index tables Ib, ( b = 1, 2, ..., Q) in the set Ī, where | Ī| = Q, and Q = L  ′ .
Step 5 For each table Ib, determine the MED  b min by calculating the EDs between different GSM vectors X S,1 and X S,2 at the source node, as [15]  b min = min where  ,  (  = 1, 2) denotes the index in the a single TAC I 1,  .
Step 6 Obtain the TAC index table I sel with a maximum  b min given as And, the selection is terminated.
Note that in Step 3, Δ 1 is viewed as the set of promising index candidates due to the large value of channel modulus.Thus,  a activated TA is selected first in the set Δ 1 .Then, incremental selections are made from the alternate index set Δ 2 to ensure  ′ different information vectors can be mapped one by one.

Complexity Analysis
In this manuscript, the measure of computational complexity is the number of floating-point operations (Flops), where a Flop represents a real-valued addition or multiplication operation.There are two components to the computational complexity of the proposed RC-TAC algorithm: the calculation of the channel modulus shown in ( 14) and the computation of the various EDs in (16).For the channel modulus, the single operation z S,D  2 F ( = 1, 2, ...,  t ) requires 4 r − 1 Flops.Thereby, the number of the Flops for  t channel modulus operations is  1 = 4 t  r −  t .Searching the MED in single table Ib needs 7 a  r  (4 r − 1) ( − 1) Flops.Since all Q tables are searched, the complexity  2 for determining the maximum MED is  2 = 7 a  r  Q(4 r − 1) ( − 1).Hence, the total Flops  RC of the proposed RC-TAC selection is given as Furthermore, the optimum TAC selection has the following complexity, given as For a more intuitive presentation, Table 4 compares the complexity of two selection algorithms by several groups of MIMO configurations.Compared to the optimal TAC selection, the proposed RC-TAC selection algorithm shows a dramatic reduction in complexity.When the values  t and  a are large, the advantage of the proposed RC-TAC algorithm becomes more apparent, and I sel may be selected more quickly.Tab. 4. The computational complexity comparison of the two selection algorithms.

Joint Decoding for the Proposed Schemes
To maximize the utilization of the two signals at the destination, and then enhance the system performance of the proposed OGCC-GSM and its benchmark OGCC-SM schemes, we propose a joint decoding approach.The flowchart can make reference to the 'Joint Goppa Decoding' block in Fig. 3 and the concrete steps are outlined below, Step 1 The detected vector c2 is first sent to the 'Decoder 2' of the optimized-Goppa code Γ 2 (L 2 , G opt,2 (x)) to attain the estimated information sequence ū from the relay.
Step 2 Through the 'Combiner' block, substitute the last  information bits in the systematic vector c1 with the newly acquired bits ū, while maintaining the initial first  1 −  parity-check bits p 1 of c1 .The mixed codeword ĉ1 is generated as ĉ1 = [p 1 , ū].
Note that 'joint' is embodied in Step 2 where the information bits in the relay are jointly used in the sequence at the source node.It is because the relay is positioned closer to the destination than the source in CC, resulting in more SNR gains.
In addition, Γ 2 (L 2 , G opt,2 (x)) at the relay contains more redundant bits, the detected information sequence ū is more reliable, and it is therefore totally utilized in the ĉ1 .Furthermore, Patterson algorithm [14] is applied in 'Decoder 1' and 'Decoder 2' blocks demonstrating a stronger error correction capability than the existing Euclidean algorithm [9], [10], which is verified in the next simulation section.

Simulation
In this section, Monte Carlo simulation results for analyzing the BER performance of the proposed OGCC-GSM scheme and other comparison schemes with the joint decoding over Rayleigh quasi-static fading channels are presented, where the number of simulated iterations reaches 10 5 .To generalize and evaluate the OGCC-GSM scheme more comprehensively, Goppa codes used in the proposed scheme are randomly chosen from three distinct fields, i.e., GF (2 4 ), GF(2 5 ) and GF(2 6 ), where ,  and  denote the primitive elements in respective Galois fields.The specific simulated parameters are summarized in Tab. 5, where different code rates ( CC = 1/4, 2/9, and 1/3) are considered for three scenarios.Define Υ S,D , Υ S,R , and Υ R,D stand for the signal noise ratio (SNR) per bit from the direct S-D, S-R, and R-D transmission links, respectively.Assume a 2 dB SNR enhancement at the relay compared to the source, i.e.Υ R,D = Υ S,D + 2 dB, and perfect CSI at all related receivers for the GSM detection.

BER Performance for the GCC-GSM Schemes Under Various Goppa Polynomials
Based on the proposed efficient selection approach for the optimum Goppa polynomials in Sec. 3, Table 6 provides multiple typical Goppa polynomials for the Goppa codes to exhibit several kinds of search results, including the optimal G opt,1 () and G opt,2 () under three scenarios as well as a few groups of random Goppa polynomials.The corresponding MHD  1 and  2 are also presented in Tab.6, where the number of the codewords with the weight equal to the maximum  1 and  2 , i.e.,  max( 1 ) ,  max( 2 ) , are also contained.The specific selection procedure can be referred to in Sec.3.2.As the search code space and the number of correctable errors are large, it is discovered that the optimal Goppa polynomial may be uniquely determined only by obtaining the minimum value of  max( 1 ) and  max( 2 ) for the source and relay, respectively, referring to the parameters in Scenarios 2 and 3 of Tab. 6.
Figures 4-6 demonstrate the BER performance of the GCC-GSM schemes for three scenarios to compare the influence of different Goppa polynomials, where  r =  t .Besides, the ideal source-to-relay transmission link (Υ S,R = ∞) and optimum TAC index table are applied in all three scenarios.From the simulated results, the GCC-GSM scheme using the optimal Goppa polynomials, i.e., OGCC-GSM scheme, exhibits a prominent performance gain over the original GCC-GSM schemes employing the random Goppa polynomials.For example, the OGCC-GSM scheme of Scenario 1 achieves the SNR gain of over 3.5 dB at BER = 3 × 10 −6 .Moreover, randomly selected Goppa polynomials may make the MHDs of Goppa codes utilized in the source and relay nodes very small, resulting in the occurrence of the error floor (EF) region and poor overall system performance, as shown in Scenarios 2 and 3 of Figs. 5 and 6.Additionally, the different  max( 1 ) and  max( 2 ) also affect BER performance, even if the Goppa codes with the random and optimal polynomials used have the same MHD.Clearly, the fewer codewords with a small weight, the stronger the error correction capability of the code, which is also confirmed by the simulation results.For instance, the OGCC-GSM scheme of Scenario 2 outperforms the GCC-GSM schemes using the polynomial Random-2-1 by a gain over 3.5 dB at BER = 1.3 × 10 −6 , where the same MHDs are selected at the source and relay for the fair comparison.In conclusion, the proposed effective Goppa polynomial selection algorithm has a substantial effect on the performance of the GCC-GSM, further indicating the necessity to propose the OGCC-GSM scheme.The optimal polynomials used in the OGCC-GSM scheme for the three scenarios in the simulations below are referred to in bold in Tab.6 and are no longer specified.Bit Error Rate (BER) OGCC-GSM, Optimal-3 GCC-GSM, Random-3-1 GCC-GSM, Random-3-2 GCC-GSM, Random-3-3 GCC-GSM, Random-3-4 Fig. 6.Performance of the GCC-GSM schemes with  r =  t = 7 and 16-QAM under different Goppa polynomials for Scenario 3. The optimal polynomials of the OGCC-GSM scheme under three scenarios are in bold.

Scenario
Tab. 6. Different Goppa polynomials for three simulated scenarios and the corresponding MHDs.

Performance of the OGCC-GSM Scheme Versus that of its Benchmark OGCC-SM Scheme
This subsection discusses the performance comparisons between the proposed OGCC-GSM scheme under the optimum TAC index table and the corresponding benchmark scheme for all three scenarios.Likewise, assume Υ S,R = ∞.Since the spectral efficiencies of the proposed OGCC-GSM scheme for three scenarios are  GSM = 3, 5, 9 bits/s/Hz, respectively, the spectral efficiencies of the benchmark OGCC-SM scheme must be set to match that of the OGCC-GSM scheme for an equitable comparison, i.e.,  SM =  GSM .Table 7 details the different numbers of TAs utilized by the OGCC-GSM and OGCC-SM schemes to attain identical spectral efficiency, including the required values  r and  for each scenario.
The performance comparison results of the OGCC-GSM and OGCC-SM schemes under the three scenarios are presented in Figs.7-9.From Fig. 7, the OGCC-GSM scheme can achieve a gain of 2 dB at BER ≈ 3.3 × 10 −6 compared to the benchmark OGCC-SM scheme in Scenario 1.Meanwhile, the OGCC-GSM scheme performs better than the OGCC-SM scheme with a margin of approximately 1.8 dB at BER ≈ 2.2 × 10 −6 under Scenario 2 (exhibited in Fig. 8) and 1.5 dB at BER ≈ 5×10 −5 under Scenario 3 (shown in Fig. 9), respectively.Consequently, the proposed OGCC-GSM is more applicable to engineering scenarios due to a substantial reduction in the number of TAs and a superior system performance compared to its benchmark OGCC-SM scheme.

Performance Comparison for the OGCC-GSM Employing Different TAC Selections
Figures 10-12 provide the BER performance of the OGCC-GSM scheme for three scenarios under the optimal TAC, RC-TAC, and random TAC selections, where Υ S,R = ∞,  r =  t are supposed in the simulations.For Scenario 1 of Fig. 10, the optimal TAC selection is equivalent to RC-TAC selection owing to the same search volume ( = Q = 15).Thereby, the OGCC-GSM scheme exhibits virtually identical performance under two distinct selection algorithms.Moreover, due to a small search volume,   the RC-TAC selection algorithm may pick out the same TAC index table as the optimal TAC selection algorithm, resulting in a nearly identical system performance, as depicted in Scenario 2 of Fig. 11.When the search volume is large as in Scenario 3 ( = 6545, Q = 561; see Tab. 4), the OGCC-GSM scheme under the proposed RC-TAC selection algorithm also demonstrates promising performance, with an SNR loss of only 0.6 dB compared to the scheme using the optimal selection at BER = 1.4 × 10 −6 .
From Figs. 10-12, the OGCC-GSM scheme under the proposed RC-TAC selection algorithm for three scenarios performs significantly better than that under the random TAC table.For example, a performance gain of 2 dB is acquired at BER ≈ 10 −6 for the OGCC-GSM under the RC-TAC selection compared to the random TAC table, while nearly no SNR loss compared to the scheme using the optimal TAC selection that requires large computational complexity, shown in Scenario 2 of Fig. 11.This indicates that the RC-TAC selection algorithm may provide a superior compromise between the search complexity and BER performance.

BER Performance Comparison of the OGCC-GSM Scheme Between the Practical and Ideal S-R Transmission Channels and its Non-Cooperative Scheme
All of the aforementioned simulations assume that the information between mobile users is accurately obtained, i.e., that the S-R transmission channel is ideal (Υ S,R = ∞), so as to achieve the purpose of controlling variables and ensure that more attention is paid to experimental parameters that need to be compared, such as the influence of different polynomials in Sec.7.1 on the proposed OGCC-GSM scheme, for more objective and unbiased results.Nevertheless, information transfer between users may not be ideal in actual situations.To make the proposed OGCC-GSM scheme more applicable, it is necessary to evaluate its efficacy under non-ideal conditions.
Hence, we present the performance comparisons for the OGCC-GSM scheme with the RC-TAC selection algorithm under multiple S-R transmission link states including the practical (Υ S,R ≠ ∞) and ideal (Υ S,R = ∞) links for three scenarios in Figs.13-15.In Scenario 1, Υ S,R = 8, 14 dBs are considered as the practical links.Besides, the non-ideal channels Υ S,R = 5, 12 dBs are preset for Scenario 2, and Υ S,R = 3, 10 dBs are evaluated in Scenario 3, respectively.It is observed that the OGCC-GSM scheme for three scenarios under the poor S-R channels, i.e., Υ S,R = 8, 5, and 3 dBs, respectively, demonstrates an approximative BER performance compared to the OGCC-GSM scheme under Υ S,R = ∞ at low SNR.However, at high SNR, the EF region occurs and the performance significantly deteriorates due to uncontrolled error propagation, which can be mitigated by the cyclic redundancy check (CRC) approach [16].In addition, in the case of a robust S-R link, the OGCC-GSM scheme for three scenarios exhibits nearly identical BER performance as the ideal   OGCC-GSM scheme.For example, when Υ S,R = 3 dB, the BER curve of OGCC-GSM scheme for Scenario 3 becomes flat from SNR = 4 dB, which performs worse than the OGCC-GSM scheme under Υ S,R = 10 dB by a margin of 7.5 dB at BER = 10 −4 .And the performance curve of the OGCC-GSM under Υ S,R = 10 dB almost coincides with that of the OGCC-GSM scheme under Υ S,R = ∞, as shown in Fig. 15.Therefore, the proposed OGCC-GSM scheme may achieve superior performance in practical application scenarios as the condition of the S-R channel is excellent.Moreover, from Figs. 13-15, the non-cooperative schemes of three scenarios all perform inferior to the cooperative scheme under the good S-R links by a margin of 2.2 dB loss at BER = 2 × 10 −5 for Scenario 1, 3 dB loss at BER = 2.8 × 10 −6 for Scenario 2, and, 1.6 dB loss at BER = 5 × 10 −6 for Scenario 3, respectively.These simulation results further indicate that the relay that possesses more SNR than the source not only enables the coded cooperative scheme to achieve not only more coding gain through the use of the other Goppa code with more redundant symbols but also more diversity gain through the system structure of two paths compared to the non-cooperative scheme.

Performance of the OGCC-GSM Scheme Versus that of the Existing Schemes
In this subsection, the performance of the proposed OGCC scheme is first compared to that of the general LDPC coded cooperation [5] and the existing relay-optimized GCC schemes, i.e., the GCC scheme [9] and the jointly designed GCC scheme [10], shown in Fig. 16.Then, the BER performance curves of the existing DGC-SM schemes [17] and DGC-GSM [18] are also plotted in Fig. 16 to compare with that of the proposed OGCC-SM and OGCC-GSM schemes.For a fair comparison, the code rate  CC = 1/4 is unified, and BPSK modulation and Rayleigh quasi-static fading are considered for all comparative schemes, where all simulated parameters of the proposed schemes refer to Scenario 1 of Tab. 5, and the good S-R link is set as Υ S,R = 14 dB.Moreover, to keep the spectral efficiency consistent as 3 bits/s/Hz, determine  t = 4,  r = 4 for both the OGCC-SM scheme and its counterpart DGC-SM scheme, and  t = 4,  a = 2, and  r = 4 for the OGCC-GSM and the comparative DGC-GSM schemes, respectively.Besides, the optimal TAC selection algorithm is utilized in both in both OGCC-GSM and DGC-GSM schemes.
From Fig. 16, the proposed OGCC scheme with the Patterson algorithm in each decoder outperforms the existing GCC schemes [9], [10] using the Euclidean algorithm and bit selection at the relay by a margin of over 4.2 dB at BER ≈ 2.8 × 10 −4 .Although the proposed benchmark OGCC-SM scheme only obtains a gain of approximately 0.5 dB over the existing DGC-SM schemes [17] and DGC-GSM [18] at BER ≈ 5.6×10 −6 , it avoids a significant amount of search caused by the relay information selection.Moreover, the proposed OGCC-GSM scheme performs substantially better than two comparable schemes under the same spectral efficiency by more than 4.2 dB at BER ≈ 5.6 × 10 −6 .
In addition, the proposed scheme also outperforms the LDPC coded cooperation with the belief propagation (BP) decoding [5] by a gain of about 4.5 dB at BER ≈ 2.2 × 10 −5 .
In fact, the LDPC code is a kind of competitive channel code for large-scale information transmission due to its parallel iterative decoding algorithm based on the low-density sparse parity check matrix.However, in this comparison, short-length LDPC codes are used in the coded cooperation that demonstrates poor performance compared to the Goppa codes since the parity-check matrix of short LDPC codes is prone to short girths, which greatly affects the decoding performance.Therefore, in the small-scale information transmission, Goppa codes may be a stronger contender than the LDPC codes.The simulated results verify the effectiveness of selecting the Goppa polynomials for the Goppa codes in the overall system and further reveal the dominance of the proposed OGCC-GSM scheme.

Conclusion
This manuscript first investigates a simple but effective selection approach from the multiple Goppa polynomials to determine the optimal one that results in an optimized Goppa code with the largest MHD and the improved weight distribution.Based on this approach, two optimized Goppa codes with the same information dimension are distributed in the source and relay nodes, respectively, to construct the novel OGCC scheme where the Goppa code at the relay contains more redundant bits.To achieve more diversity gain, the OGCC-GSM scheme is proposed with more than one transmit antenna activated in comparison to its benchmark OGCC-SM scheme.Moreover, an RC-TAC selection algorithm is developed with the aid of CSI based on the existing optimal TAC selection algorithm.Additionally, a joint decoding approach is presented to further improve the system performance, where each decoder employs the novel Patterson algorithm.Numerical simulations evaluate the performance of the proposed OGCC-GSM scheme from different perspectives over the Rayleigh quasi-static fading channel, demonstrating the supremacy and feasibility of the proposed scheme.In the following work, we will focus on the more efficient joint decoding algorithm at the destination and consider the low-complexity relay symbol selection for the proposed scheme with different information dimensions.
simultaneously to the relay and destination nodes via the  r ×  t MIMO Rayleigh quasi-static fading channels Z S,R and Z S,D , respectively, whose elements are complex independent and identically distributed (i.i.d.) Gaussian random variables (RVs) following CN (0, 1).Note that Z S,R and Z S,D keep constant for each  1 while varying independently from one Goppa code to the next due to the Rayleigh quai-static fading.The received  r × 1 signal vectors Y S,R ( 1 ) and Y S,D ( 1 ) are formulated as

Fig. 3 .
Fig. 3.The schematic diagram of the proposed joint decoding.

2 Fig. 4 .
Fig. 4. Performance of the GCC-GSM schemes with  r =  t = 4 and BPSK modulation under different Goppa polynomials for Scenario 1.