## Daniel Kariuki Waweru , Fengfan Yang , Chunli Zhao , Lawrence Muthama Paul and Hongjun Xu## |

[TeX:] $$G(x)$$ | [TeX:] $$\Gamma(L, G(x), s)$$ | The weight spectrum of [TeX:] $$\Gamma(L, G(x), s)$$ |
---|---|---|

[TeX:] $$(x+\beta)^1$$ | [TeX:] $$[21,14,4]_2$$ | [1, 0, 0, 0, 84, 0, 924, 0, 2982, 0, 5796, 0, 4340, 0, 1956, 0, 273, 0, 28, 0, 0, 0]. |

[TeX:] $$(x+\beta)^3$$ | [TeX:] $$[21,11,6]_2$$ | [1, 0, 0, 0, 0, 0, 168, 0, 210, 0, 1008, 0, 280, 0, 360, 0, 21, 0, 0, 0, 0, 0]. |

[TeX:] $$(x+\beta)^5$$ | [TeX:] $$[21,5,10]_2$$ | [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 7, 0, 3, 0, 0, 0, 0, 0, 0, 0]. |

[TeX:] $$(x+\beta)^7$$ | [TeX:] $$[21,3,12]_2$$ | [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0]. |

[TeX:] $$(x+\beta)^11$$ | [TeX:] $$[73,27,20]_2$$ | [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26280, 0, 0, 0, 667293, 0, 0, 0, 6981136,0, 0, 0, 28840329, 0, 0, 0, 49665696, 0, 0, 0, 35826210, 0, 0, 0, 10837872, 0, 0, 0,1307430, 0, 0, 0, 64824, 0, 0, 0, 657, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. |

[TeX:] $$(x+\beta)^13$$ | [TeX:] $$[73,18,24]_2$$ | [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1533, 0, 0, 0, 12629, 0, 0, 0, 59130, 0, 0, 0, 92929, 0, 0, 0, 72927, 0, 0, 0, 20367, 0, 0, 0, 2409, 0, 0, 0, 219, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. |

TABLE II

Serial | [TeX:] $$C_S$$ | [TeX:] $$C_R$$ | [TeX:] $$C_D$$ | [TeX:] $$d_{\text {free }}$$ | Rate (R) |
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[TeX:] $$M_1$$ | [21, 5, 10] | [21, 3, 12] | [42, 5] | 10 | 0.1190 |

[TeX:] $$M_2$$ | [21, 14, 4] | [21, 11, 6] | [42, 14] | 4 | 0.3333 |

[TeX:] $$M_3$$ | [73, 27, 20] | [73, 18, 24] | [146, 27] | 20 | 0.1849 |

1) Cooperative scheme: Fig. 2 illustrates the single relay TDCG coding scheme. Two cyclic TDCG codes are used in the TDCG coding system, [TeX:] $$C_S$$ at the source and [TeX:] $$C_R$$ at the relay, and they both operate over a Rayleigh fading channel.

The message sequence [TeX:] $$\mathrm{m}_1$$ is encoded to the codeword [TeX:] $$\mathbf{c}_1 \in C_S$$ during the 1st time slot, and the source simultaneously broadcasts the coded modulated symbol sequences [TeX:] $$\mathbf{x}_{S, R}=\left[x_{S, R}^0, x_{S, R}^1, \cdots, x_{S, R}^{n_1-1}\right]$$ to the relay node and [TeX:] $$\mathbf{x}_{S, D}=\left[x_{S, D}^0, x_{S, D}^1, \cdots, x_{S, D}^{n_1-1}\right]$$ to the destination node. For each [TeX:] $$i=0,1, \cdots, n_1-1,$$ during the 1st time slot of the ith instant, i.e., [TeX:] $$i_1$$th time instant, the modulated symbols [TeX:] $$x_{S, R}^i$$ and [TeX:] $$x_{S, D}^i$$ are transmitted to their respective nodes over the Rayleigh fading channel. The received signals [TeX:] $$y_{S, R}^i$$ at the relay node and [TeX:] $$y_{S, D}^i$$ at the destination node are given by:

The [TeX:] $$h_{S, R}^i$$ (respectively, [TeX:] $$h_{S, D}^i$$) represents the channel coefficient owing to Rayleigh fading, a complex Gaussian variable with a zero mean and 0.5 variance per dimension. The [TeX:] $$n_{S, R}^i$$ (respectively, [TeX:] $$n_{S, D}^i$$) represents the additive white Gaussian noise, a complex Gaussian variable with a zero mean and 0.5 variance per dimension. After transmitting all the [TeX:] $$n_1$$ modulated symbols, the received signal sequences: At the relay node is denote by [TeX:] $$\mathbf{y}_{S, R}=\left[y_{S, R}^0, y_{S, R}^1, \cdots, y_{S, R}^{n_1-1}\right],$$ and at the destination node is denoted [TeX:] $$\mathbf{y}_{S, D}=\left[y_{S, D}^0, y_{S, D}^1, \cdots, y_{S, D}^{n_1-1}\right]$$.

During the 2nd time slot, the relay receives the estimated message [TeX:] $$\hat{\mathbf{m}}_1$$ from the source. The relay then selects the message sequence [TeX:] $$\mathbf{m}_2 \text { from } \hat{\mathbf{m}}_1$$, encodes it to the codeword [TeX:] $$\mathbf{c}_2 \in C_R$$ and broadcasts the coded modulated symbol sequence [TeX:] $$\mathbf{x}_{R, D}=\left[x_{R, D}^0, x_{R, D}^1, \cdots, x_{R, D}^{n_2-1}\right]$$ to the destination node. The received signal at the destination node during the [TeX:] $$i_2$$th time instant, for each [TeX:] $$i=0,1, \cdots, n_2-1$$, has the following form:

Here, [TeX:] $$h_{S, D}^i$$ and [TeX:] $$n_{R, D}^i$$ are defined similar to [TeX:] $$h_{S, R}^i$$ and [TeX:] $$n_{S, R}^i$$, respectively. After transmitting all the [TeX:] $$n_2$$ modulated symbols, the received signal sequence at the destination node is [TeX:] $$\mathbf{y}_{R, D}=\left[y_{R, D}^0, y_{R, D}^1, \cdots, y_{R, D}^{n_2-1}\right].$$

The overall received signal sequence at the destination node following the transmission of all the [TeX:] $$n_1+n_2$$ modulated symbols is given by [TeX:] $$\mathbf{y}=\left[\mathbf{y}_{S, D}, \mathbf{y}_{R, D}\right] .$$ Demodulating the signal [TeX:] $$\mathbf{y}$$ yields [TeX:] $$\hat{\mathbf{r}}=\left[\hat{\mathbf{r}}_1, \hat{\mathbf{r}}_2\right]$$, which is the initial likelihood of some codeword [TeX:] $$\left[\hat{\mathbf{c}}_1, \hat{\mathbf{c}}_2\right] \in C_D$$, with [TeX:] $$\hat{\mathbf{c}}_1 \in C_S$$ and [TeX:] $$\hat{\mathbf{c}}_2 \in C_R$$. If the relay selects its [TeX:] $$k_2$$ message sequence [TeX:] $$\mathbf{m}_2$$ independent of [TeX:] $$\hat{\mathbf{m}}_1$$ to generate the initial likelihood [TeX:] $$\hat{\mathbf{r}}_2^*$$, then through the joint construction, the destination obtains the initial likelihood [TeX:] $$\left[\hat{\mathbf{r}}_1, \hat{\mathbf{r}}_2^*\right]$$ of some codeword [TeX:] $$\left[\hat{\mathbf{c}}_1, \hat{\mathbf{c}}_2^*\right] \in \bar{C}_D$$. In this case, the optimized jointly-constructed destination code [TeX:] $$C_D$$ is a subcode of [TeX:] $$\bar{C}_D$$.

2) Non-cooperative scheme: Without a relay’s assistance, the source is transmitting the [TeX:] $$n_1$$ symbols in 5 to the destination. The received signal [TeX:] $$\mathbf{y}_{S, D}$$ is then demodulated into [TeX:] $$\hat{\mathbf{r}}_1$$, which represents the initial likelihood of some codeword [TeX:] $$\hat{\mathbf{c}}_1 \in C_S.$$

In this section, we describe an optimization design criterion and an efficient algorithm to achieve the design objective. In order to explicitly state the related positions of each of the [TeX:] $$k_2$$ bits chosen from [TeX:] $$k_1$$ information bits, we will refer to the patterns produced by the combination [TeX:] $$\left(\begin{array}{l} k_1 \\ k_2 \end{array}\right)$$ as having an index

where each j is in one-to-one correspondence to a distinct pattern, a [TeX:] $$k_2$$-tuple [TeX:] $$\boldsymbol{\Omega}_j=\left[l_1^{(j)}, \cdots, l_{k_2}^{(j)}\right],$$ that we will refer to as the jth selection pattern, where [TeX:] $$0 \leq l_1^{(j)}<\cdots<l_{k_2}^{(j)} \leq k_1-1$$. Thus, a set [TeX:] $$\Phi=\left\{\boldsymbol{\Omega}_j \mid j \in \Lambda\right\}$$ is formed by all of the selection patterns. The [TeX:] $$k_2$$ bits chosen from the [TeX:] $$k_1$$ decoded bits that correspond to each jth selection pattern [TeX:] $$\Omega_j$$ are denoted by the symbol [TeX:] $$\mathbf{m}_2^{(j)}=\left[m_0^{(j)}, m_1^{(j)}, \cdots, m_{k_2-1}^{(j)}\right],$$ with [TeX:] $$m_l^{(j)} \in \mathbb{F}_p$$. The jth selection pattern produces a destination code

whose minimum Hamming distance, which we call the free distance, is denoted as [TeX:] $$d_{\text {free}}^{(j)}.$$ For obtaining the optimal message bit selection pattern at the relay, the following optimization techniques are suggested.

Consider the TDCGCC scheme [TeX:] $$M_1$$ defined in Table II. We have [TeX:] $$k_1=5, k_2=3, \Lambda=\left(\begin{array}{l} 5 \\ 3 \end{array}\right)=10$$, and the set of information selection patterns is

[TeX:] $$\begin{aligned} \Phi= & \left\{\boldsymbol{\Omega}_1, \boldsymbol{\Omega}_{\mathbf{2}}, \boldsymbol{\Omega}_{\mathbf{3}}, \boldsymbol{\Omega}_{\mathbf{4}}, \boldsymbol{\Omega}_5, \boldsymbol{\Omega}_6, \boldsymbol{\Omega}_7, \boldsymbol{\Omega}_{\mathbf{8}}, \boldsymbol{\Omega}_{\mathbf{9}}, \boldsymbol{\Omega}_{10}\right\} \\ = & \{[0,1,2],[0,1,3],[0,1,4],[0,2,3],[0,2,4], \\ & {[0,3,4],[1,2,3],[1,2,4],[1,3,4],[2,3,4]\} . } \end{aligned}$$

Table III is a list of the destination codes [TeX:] $$C_D^{(j)}$$ corresponding to various jth selection patterns. For each [TeX:] $$j \in \Lambda$$, we determine that the maximum free distance [TeX:] $$d_{\text {free }}^{(j)}=10$$. We should also point out that, even though the minimum Hamming distance of the code [TeX:] $$C_D^{(j)}(42,5)$$ is not improved, the weight spectrum get optimized in the sense that codewords of low weight in [TeX:] $$C_D^{(j)}(42,5)$$ are greatly reduced for every j as compared to the codewords of [TeX:] $$C_S(21,5,10)$$.

Let [TeX:] $$d_{\text {free }}^{(\operatorname{Max})}$$ be defined as follows:

At the relay, we seek the unique optimal bit selection pattern that allows the destination code [TeX:] $$C_D^{(j)}\left(n_1+n_2, k_1\right)$$ to achieve the maximum free distance [TeX:] $$d_{\text {free }}^{(j)}=d_{\text {free }}^{(\operatorname{Max})}$$. Thus, the resulting code and its free distance are thus denoted as [TeX:] $$C_D\left(n_1+n_2, k_1, d_{\text {free}}\right)$$, by dropping the index j, and the search is complete. From Table III, an optimum code, [TeX:] $$C_D(42,5)$$, in terms of highest free distance, can not be determined uniquely because all the destination codes [TeX:] $$C_D^{(j)}(42,5)$$, for each [TeX:] $$j \in \Lambda,$$ achieve the maximum free distance [TeX:] $$d_{\text {free }}^{(Max)}=10$$. A few more steps are required. Starting with [TeX:] $$i=0, d_{\text {free }}^{(j)}=d_{\text {free }}^{(j)}+0,$$ and such that [TeX:] $$d_1 \leq d_{\text {free }}^{(j)}+i \leq n_1 \text {, let } N_{d_{\text {free }}^{(j)}+i}$$ be the number of codewords of Hamming weight [TeX:] $$d_{\text {free }}^{(j)}+i \text { in } C_D^{(j)}\left(n_1+n_2, k_1\right).$$ The unique optimum code [TeX:] $$C_D\left(n_1+n_2, k_1, d_{\text {free }}\right)$$ is obtained by maximizing the [TeX:] $$d_{\text {free }}^{(j)}+i$$ while minimizing the [TeX:] $$N_{d_{\text {free }}^{(j)}+i}$$ among all candidates [TeX:] $$C_D^{(j)}\left(n_1+n_2, k_1\right), j \in \Lambda .$$ In Table IV, we obtain all the destination codes [TeX:] $$C_D^{(j)}(14,5)$$ associated with various selection patterns in for which [TeX:] $$d_{\text {free }}^{(j)}+0=10$$ and [TeX:] $$N_{d_{\text {free }}^{(j)}+0}=2.$$ Because both codes in Table IV have the same Hamming weight spectrum, either of them is optimal and will produce a fully optimized relay channel code.

TABLE III

j | [TeX:] $$\Omega_j$$ | [TeX:] $$C_D^{(j)}(42,5)$$ | [TeX:] $$d_{\text {free }}^{(j)}$$ | The weight enumerator: [TeX:] $$w t_{(j)}(x)$$ |
---|---|---|---|---|

1 | [TeX:] $$\Omega_1$$ | [TeX:] $$C_D^{(1)}(42,5)$$ | 10 | [TeX:] $$1+3 x^{10}+0 x^{12}+18 x^{22}+7 x^{24}+3 x^{26}$$ |

2 | [TeX:] $$\Omega_2$$ | [TeX:] $$C_D^{(2)}(42,5)$$ | 10 | [TeX:] $$1+2 x^{10}+1 x^{12}+19 x^{22}+6 x^{24}+3 x^{26}$$ |

3 | [TeX:] $$\Omega_3$$ | [TeX:] $$C_D^{(3)}(42,5)$$ | 10 | [TeX:] $$1+3 x^{10}+0 x^{12}+18 x^{22}+7 x^{24}+3 x^{26}$$ |

4 | [TeX:] $$\Omega_4$$ | [TeX:] $$C_D^{(4)}(42,5)$$ | 10 | [TeX:] $$1+3 x^{10}+0 x^{12}+18 x^{22}+7 x^{24}+3 x^{26}$$ |

5 | [TeX:] $$\Omega_5$$ | [TeX:] $$C_D^{(5)}(42,5)$$ | 10 | [TeX:] $$1+3 x^{10}+0 x^{12}+18 x^{22}+7 x^{24}+3 x^{26}$$ |

6 | [TeX:] $$\Omega_6$$ | [TeX:] $$C_D^{(6)}(42,5)$$ | 10 | [TeX:] $$1+3 x^{10}+0 x^{12}+18 x^{22}+7 x^{24}+3 x^{26}$$ |

7 | [TeX:] $$\Omega_7$$ | [TeX:] $$C_D^{(7)}(42,5)$$ | 10 | [TeX:] $$1+3 x^{10}+0 x^{12}+18 x^{22}+7 x^{24}+3 x^{26}$$ |

8 | [TeX:] $$\Omega_8$$ | [TeX:] $$C_D^{(8)}(42,5)$$ | 10 | [TeX:] $$1+2 x^{10}+1 x^{12}+19 x^{22}+6 x^{24}+3 x^{26}$$ |

9 | [TeX:] $$\Omega_9$$ | [TeX:] $$C_D^{(9)}(42,5)$$ | 10 | [TeX:] $$1+3 x^{10}+0 x^{12}+18 x^{22}+7 x^{24}+3 x^{26}$$ |

10 | [TeX:] $$\Omega_10$$ | [TeX:] $$C_D^{(10)}(42,5)$$ | 10 | [TeX:] $$1+3 x^{10}+0 x^{12}+18 x^{22}+7 x^{24}+3 x^{26}$$ |

TABLE IV

j | [TeX:] $$\Omega_j$$ | [TeX:] $$C_D^{(j)}(42,5)$$ | [TeX:] $$d_{\text {free }}^{(j)}$$ | The weight enumerator: [TeX:] $$w t_{(j)}(x)$$ |
---|---|---|---|---|

2 | [TeX:] $$\Omega_2$$ | [TeX:] $$C_D^{(2)}(42,5)$$ | 10 | [TeX:] $$1+2 x^{10}+1 x^{12}+19 x^{22}+6 x^{24}+3 x^{26}$$ |

8 | [TeX:] $$\Omega_8$$ | [TeX:] $$C_D^{(8)}(42,5)$$ | 10 | [TeX:] $$1+2 x^{10}+1 x^{12}+19 x^{22}+6 x^{24}+3 x^{26}$$ |

In general, the exhaustive search method comprises the precise steps enumerated in Algorithm 1. Although an exhaustive search guarantees an optimally designed relay channel code, the efficiency of this method decreases as the value of [TeX:] $$k_1$$ increases because the processing complexity of Algorithm 1 also increases. It is more practical to conduct a partial search for promising candidates. We will now go over the partial search method in detail.

We suggest a division strategy that splits the [TeX:] $$k_1$$ decoded message bits at the relay into two nearly equal parts and two scenarios. This is illustrated in Fig. 3 for the TDCGCC scheme [TeX:] $$M_2$$. Define [TeX:] $$v=\left\lceil k_1 / 2\right\rceil=7$$ and let be an integer such that [TeX:] $$k_2 / 2<\mu \leq \min \left(v, k_2\right)$$. As a result, [TeX:] $$\mu \in\{6,7\}$$. In the division strategy, instead of selecting all [TeX:] $$k_2=11$$ bits from the [TeX:] $$k_1=14$$ bits available, which can be done in [TeX:] $$\left(\begin{array}{l} k_1 \\ k_2 \end{array}\right)=\left(\begin{array}{l} 14 \\ 11 \end{array}\right)=364$$ ways, we sample all [TeX:] $$\left(\begin{array}{l} v \\ \mu \end{array}\right)$$ possible choices of bits from first part, in scenario (i), and then sample all [TeX:] $$\left(\begin{array}{l} v \\ \mu \end{array}\right)$$ possible choices of bits from the last part, in scenario (ii). The remaining [TeX:] $$\left(k_2-\mu\right)$$ bits in each scenario are fixed randomly from the respective remaining parts. Let us denote an 11-tuple as

[TeX:] $$\boldsymbol{\Omega}_j=\left[l_1^{(j)}, l_2^{(j)}, l_3^{(j)}, l_4^{(j)}, l_5^{(j)}, l_6^{(j)}, l_7^{(j)}, l_8^{(j)}, l_9^{(j)}, l_{10}^{(j)}, l_{11}^{(j)}\right].$$

In scenario (i), first, we select = 7 bits from the first part, which can only be done in [TeX:] $$\left(\begin{array}{l} 7 \\ 7 \end{array}\right)=1$$ way, giving a pattern

[TeX:] $$\boldsymbol{\Omega}_1=\left[0,1,2,3,4,5,6, l_8^{(1)}, l_9^{(1)}, l_{10}^{(1)}, l_{11}^{(1)}\right],$$

where [TeX:] $$l_8^{(1)}, l_9^{(1)}, l_{10}^{(1)}, l_{11}^{(1)}$$ are to be fixed randomly from the last part. Setting [TeX:] $$l_8^{(1)}=7, l_9^{(1)}=8, l_{10}^{(1)}=9, l_{11}^{(1)}=10,$$ we obtain [TeX:] $$\boldsymbol{\Omega}_1=[0,1,2,3,4,5,6,7,8,9,10]$$. Secondly, we select [TeX:] $$\mu=6$$ out of the 7 bits from the first part, and this can be done in [TeX:] $$\left(\begin{array}{l} 7 \\ 6 \end{array}\right)=7$$ ways, giving the patterns

[TeX:] $$\begin{aligned} & \boldsymbol{\Omega}_2=\left[0,1,2,3,4,5, l_7^{(2)}, l_8^{(2)}, l_9^{(2)}, l_{10}^{(2)}, l_{11}^{(2)}\right], \\ & \boldsymbol{\Omega}_3=\left[0,1,2,3,4,6, l_7^{(3)}, l_8^{(3)}, l_9^{(3)}, l_{10}^{(3)}, l_{11}^{(3)}\right], \\ & \boldsymbol{\Omega}_4=\left[0,1,2,3,5,6, l_7^{(4)}, l_8^{(4)}, l_9^{(4)}, l_{10}^{(4)}, l_{11}^{(4)}\right], \\ & \boldsymbol{\Omega}_5=\left[0,1,2,4,5,6, l_7^{(5)}, l_8^{(5)}, l_9^{(5)}, l_{10}^{(5)}, l_{11}^{(5)}\right], \\ & \boldsymbol{\Omega}_6=\left[0,1,3,4,5,6, l_7^{(6)}, l_8^{(6)}, l_9^{(6)}, l_{10}^{(6)}, l_{11}^{(6)}\right], \\ & \boldsymbol{\Omega}_7=\left[0,2,3,4,5,6, l_7^{(7)}, l_8^{(7)}, l_9^{(7)}, l_{10}^{(7)}, l_{11}^{(7)}\right], \\ & \boldsymbol{\Omega}_8=\left[1,2,3,4,5,6, l_7^{(8)}, l_8^{(8)}, l_9^{(8)}, l_{10}^{(8)}, l_{11}^{(8)}\right], \end{aligned}$$

where [TeX:] $$l_7^{(j)}, l_8^{(j)}, l_9^{(j)}, l_{10}^{(j)}, l_{11}^{(j)}, j=2,3,4,5,6,7,8,$$ are to be fixed randomly from the last part of Fig. 3. For j = 2, 3, 4, 5, 6, 7, 8 we set that [TeX:] $$l_7^{(j)}=7, l_8^{(j)}=8, l_9^{(j)}=9, l_{10}^{(j)}=10, l_{11}^{(j)}=11$$ and get

[TeX:] $$\begin{aligned} & \boldsymbol{\Omega}_2=[0,1,2,3,4,5,7,8,9,10,11], \\ & \boldsymbol{\Omega}_3=[0,1,2,3,4,6,7,8,9,10,11], \\ & \boldsymbol{\Omega}_4=[0,1,2,3,5,6,7,8,9,10,11], \\ & \boldsymbol{\Omega}_5=[0,1,2,4,5,6,7,8,9,10,11], \\ & \boldsymbol{\Omega}_6=[0,1,3,4,5,6,7,8,9,10,11], \\ & \boldsymbol{\Omega}_7=[0,2,3,4,5,6,7,8,9,10,11], \\ & \boldsymbol{\Omega}_8=[1,2,3,4,5,6,7,8,9,10,11] . \end{aligned}$$

In symmetry, we repeat the above procedure now in scenario (ii) by reversing the roles played by the first and last parts of Fig. 3. So select [TeX:] $$\mu=6$$ of the 7 bits from the last-part to obtain 7 more patterns;

[TeX:] $$\begin{aligned} \boldsymbol{\Omega}_{9} & =\left[l_2^{(9)}, l_3^{(9)}, l_4^{(9)}, l_5^{(9)}, l_6^{(9)}, 7,8,9,10,11,12\right], \\ \boldsymbol{\Omega}_{10} & =\left[l_2^{(10)}, l_3^{(10)}, l_4^{(10)}, l_5^{(10)}, l_6^{(10)}, 7,8,9,10,11,13\right], \\ \boldsymbol{\Omega}_{11} & =\left[l_2^{(11)}, l_3^{(11)}, l_4^{(11)}, l_5^{(11)}, l_6^{(11)}, 7,8,9,10,12,13\right], \\ \boldsymbol{\Omega}_{12} & =\left[l_2^{(12)}, l_3^{(12)}, l_4^{(12)}, l_5^{(12)}, l_6^{(12)}, 7,8,9,11,12,13\right], \\ \boldsymbol{\Omega}_{13} & =\left[l_2^{(13)}, l_3^{(13)}, l_4^{(13)}, l_5^{(13)}, l_6^{(13)}, 7,8,10,11,12,13\right], \\ \boldsymbol{\Omega}_{14} & =\left[l_2^{(14)}, l_3^{(14)}, l_4^{(14)}, l_5^{(14)}, l_6^{(14)}, 7,9,10,11,12,13\right], \\ \boldsymbol{\Omega}_{15} & =\left[l_2^{(15)}, l_3^{(15)}, l_4^{(15)}, l_5^{(15)}, l_6^{(15)}, 8,9,10,11,12,13\right], \end{aligned}$$

where [TeX:] $$l_2^{(j)}, l_3^{(j)}, l_4^{(j)}, l_5^{(j)}, l_6^{(j)}$$ for j = 9, 10, 11, 12, 13, 14, 15 are fixed randomly from the first part. Thus for j = 9, 10, 11, 12, 13, 14, 15, we set [TeX:] $$l_2^{(j)}=2, l_3^{(j)}=3, l_4^{(j)}=4, l_5^{(j)}=5, l_6^{(j)}=6$$ and obtain

[TeX:] $$\begin{aligned} \boldsymbol{\Omega}_9 & =[2,3,4,5,6,7,8,9,10,11,12], \\ \boldsymbol{\Omega}_{10} & =[2,3,4,5,6,7,8,9,10,11,13], \\ \boldsymbol{\Omega}_{11} & =[2,3,4,5,6,7,8,9,10,12,13], \\ \boldsymbol{\Omega}_{12} & =[2,3,4,5,6,7,8,9,11,12,13], \\ \boldsymbol{\Omega}_{13} & =[2,3,4,5,6,7,8,10,11,12,13], \\ \boldsymbol{\Omega}_{14} & =[2,3,4,5,6,7,9,10,11,12,13], \\ \boldsymbol{\Omega}_{15} & =[2,3,4,5,6,8,9,10,11,12,13] . \end{aligned}$$

Lastly, we select all [TeX:] $$\mu=7$$ patterns from the last part, to obtain only one pattern

[TeX:] $$\boldsymbol{\Omega}_{16}=\left[l_3^{(16)}, l_4^{(16)}, l_5^{(16)}, l_6^{(16)}, 7,8,9,10,11,12,13\right] \text {, }$$

where [TeX:] $$l_3^{(16)}, l_4^{(16)}, l_5^{(16)}, l_6^{(16)}$$ are fixed randomly from the first part. Setting [TeX:] $$l_3^{(16)}=3, l_4^{(16)}=4, l_5^{(16)}=5, l_6^{(16)}=6 \text {, }$$ we obtain [TeX:] $$\boldsymbol{\Omega}_{16}=[3,4,5,6,7,8,9,10,11,12,13] .$$

Thus, the distinct set of information selection patterns is as follows:

The final step in the partial search is to apply Algorithm 1 after replacing after replacing [TeX:] $$\Lambda \text { with } \bar{\Lambda} \text { and } \Phi \text { with } \bar{\Phi}$$. This effectively reduces an exhaustive search to a partial search over 16 candidates rather than 364 in total.

To that end, using Algorithm 1, we obtain the maximum free distance [TeX:] $$d_{\text {free }}^{\text {(Max) }}=4, \bar{\Lambda}_0=\{1,2, \cdots, 16\}, \bar{\Phi}_0=\bar{\Phi}$$ and since [TeX:] $$\left|\bar{\Phi}_0\right|=16$$, the procedure proceeds to Algorithm 1’s iterative step 5. At iteration i = 0, for which [TeX:] $$d_{\text {free }}^{(j)}+0=4$$, we obtain [TeX:] $$N_{d_{\text {free }}^{(j)}+0}=1, \bar{\Phi}_0=\left\{\boldsymbol{\Omega}_3, \boldsymbol{\Omega}_7, \boldsymbol{\Omega}_{15}\right\} \text { and }\left|\bar{\Phi}_1\right|=3.$$ As a result, the procedure enters iteration i = 1. For iterations i = 1, 2, 3, 4, 5, [TeX:] $$d_{\text {free }}^{(j)}+i=5,6,7,8,9,$$ and we get [TeX:] $$N_{d_{\text {free }}^{(j)}+i}=0,6,0,0,0$$ and all the way through [TeX:] $$\left|\bar{\Phi}_i\right|=3$$ with every [TeX:] $$\bar{\Phi}_i=\bar{\Phi}_0$$. At iteration i = 6, we find that for [TeX:] $$d_{\text {free }}^{(j)}+6=10, N_{d_{\text {free }}^{(j)}+6}=50,\left|\bar{\Phi}_6\right|=1$$ where [TeX:] $$\bar{\Phi}_6=\left\{\Omega_7=[0,2,3,4,5,6,7,8,9,10,11]\right\}$$, and the search is complete.

In general, the partial search procedure entails a division strategy at the relay for dividing the [TeX:] $$k_1$$ decoded message bits into two nearly equal parts and two scenarios, as illustrated in Fig. 4. Algorithm 2 summarizes the specific steps of the partial search method. We will refer to a design at the relay by Algorithms 1 and 2 as optimal and sub-optimal, respectively. A randomly designed relay channel code is one in which the selected bits patterns are drawn at random from both the message and parity bits of the decoded codeword [TeX:] $$\hat{\mathbf{c}}_1 \in C_S$$, and it is never empirically optimal or sub-optimal.

In order to achieve the optimum destination codebook [TeX:] $$C_D^{(j)}$$ with respect to the largest free distance and improved spectrum, the relay must choose the best pattern [TeX:] $$\Omega_j$$ during the cooperation phase.

In the exhaustive search method, we apply Algorithm 1 to all [TeX:] $$\varphi=|\Phi|$$ codebooks where

Since the factorial function, i.e., [TeX:] $$f(n)=n !$$ increase asymptotically with the value of n, then except for short codes, or values of [TeX:] $$k_2$$ very close to those of [TeX:] $$k_1$$, [TeX:] $$\varphi$$ is a very large number. The partial search approach, on the other hand, apply Algorithm 1 to [TeX:] $$\bar{\varphi}=|\bar{\Phi}|$$ codebooks where

[TeX:] $$|\bar{\Phi}|=\sum_\mu\left(\begin{array}{l} v \\ \mu \end{array}\right)=\sum_\mu \frac{v !}{\mu !(v-\mu) !},$$

[TeX:] $$v=\left\lceil k_1 / 2\right\rceil, \text { and } k_2 / 2\lt \mu \leq \min \left(v, k_2\right) \text {. }$$ This method, in particular, produces results where the difference between the values of [TeX:] $$v \text { and } \mu$$ is much smaller than the difference in the values of [TeX:] $$k_1$$ and [TeX:] $$k_2$$. Hence, [TeX:] $$\bar{\varphi}\lt\lt\varphi .$$

To encode one message of block length [TeX:] $$k_1$$ using a generator matrix [TeX:] $$\mathbf{G}_S$$ of size [TeX:] $$k_1 \times n$$ at the source, the number of multiplication operations performed is [TeX:] $$O_s^{\times}=k_1 n$$, and the number of addition operations is [TeX:] $$O_s^{+}=\left(k_1-1\right) n$$. Hence, a total of [TeX:] $$O_s^{\times}+O_s^{+}=k_1 n+\left(k_1-1\right) n$$ elementary operations are involved in encoding one message block. The number of elementary operations then becomes

for message blocks encoded by the source. Similarly, the total number of elementary operations required to encode message blocks retransmitted by the relay node is

In the destination’s view, [TeX:] $$\Gamma n$$ direct sum operations of the two codewords generated by the source and relay are performed. Hence, the total number of elementary operations performed for each unique jth selection pattern [TeX:] $$\Omega_j$$ fixed at the relay is

[TeX:] $$\begin{aligned} O^{(1)} & \left.=\Gamma n\left(2 k_1-1\right)+\Gamma n\left(2 k_2-1\right)\right)+\Gamma n \\ & =\Gamma n\left(2\left(k_1+k_2\right)-1\right) . \end{aligned}$$

Consequently,

elementary operations are carried out in the exhaustive search method for [TeX:] $$\varphi$$ different selection patterns that are successively fixed at the relay. The number of elementary operations carried out in the partial search method is

for [TeX:] $$\bar{\varphi}$$ distinct selection patterns that are successively fixed at the relay. We obtain [TeX:] $$O^{\bar{\varphi}} \lt\lt O^{\varphi}$$.

We designate by [TeX:] $$C_S \text {-decoder and } C_R \text {-decoder }$$ the decoders related to the TDCG codes employed by the source and relay, respectively. Each decoder will utilize the maximum likelihood decoding algorithm.

We employ a joint decoding scheme based on two different algorithms, namely, the naive Algorithm 3 and smart Algorithm 4. In contrast to the naive algorithm, smart algorithm always fully exploits the resources at the destination from each time slot of the coded-cooperation. In every joint decoding scheme, the two decoders, [TeX:] $$C_S \text {-decoder and } C_R \text {-decoder }$$, will be applied at the destination to decode the two parts of noisecorrupted compound codeword [TeX:] $$\left[\hat{\mathbf{r}}_1, \hat{\mathbf{r}}_2\right]$$ jointly constructed by the source and relay, respectively.

TABLE V

Scheme | Selection Type | Selection pattern |
---|---|---|

[TeX:] $$M_1$$ | Optimal | [0, 1, 3] |

Sub-optimal | [1, 2, 4] | |

Random | [0, 1, 2] | |

[TeX:] $$M_2$$ | Optimal | [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13] |

Sub-optimal | [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] | |

Random | [0, 1, 2, 3, 4, 5, 6, 7, 10, 11,12] | |

[TeX:] $$M_3$$ | Sub-optimal | [0, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19] |

Random | [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] |

This section examines the bit-error-rate (BER) performances of the TDCGCC schemes in a flat fading channel under various channel conditions. The numerical simulation analysis has considered the TDCGCC schemes listed in Table II. Each TDCGCC system employs BPSK modulation with coherent detection over a fast Rayleigh fading channel. We represent the SNR for the channel between the (S-D), (S-R), and (R-D) links, respectively, by the symbols [TeX:] $$\lambda_{S, D}, \lambda_{S, R} \text {, and } \lambda_{R, D}$$. The total energy of each transmission scheme is taken to be unity. Each corresponding receiver has complete knowledge of the channel state. The relay always achieves SNR gain over the source because it is closer to the source than the destination is to the source. It is also assumed that [TeX:] $$\lambda_{R, D}=\lambda_{S, D}+2 \mathrm{~dB}$$

A performance comparison between optimized (i.e., optimal and sub-optimal) and un-optimized (i.e., random) designed relays in each TDCGCC scheme is investigated in this section. For a fair comparison, (S-R) link is assumed to be ideal, i.e., [TeX:] $$\left(\lambda_{S, R}=\infty\right)$$, and the joint decoder employs the smart algorithm. First, we look at the simulation results of the TDCGCC schemes [TeX:] $$M_1 \text { and } M_2$$ with both optimal and suboptimal relay designs plotted in Figs. 5 and 6, respectively. Both optimal and sub-optimal relay designs yield nearly the same BER performance at low to medium SNR, while at high SNR, the TDCGCC schemes with optimally designed relay nodes slightly outperform ones with the sub-optimally designed relay nodes. Since the sub-optimal design of a relay has a reduced processing complexity, it is then suggested and adopted for optimizing the TDCGCC schemes in all subsequent performance analyses.

Secondly, we look at the simulation results of the TDCGCC schemes [TeX:] $$M_1, M_2, \text { and } M_3$$ with un-optimized (i.e., random) and optimized (i.e., sub-optimal) designed relay designs plotted in Figs. 5, 6, and 7, respectively. The un-optimized designed relay lags behind the optimized designed relay and continuously gets degraded with an increase in the SNR. Evidently, optimized design in the relay node greatly improves the overall performance of a TDCG coding scheme. The loss in the BER performance of un-optimized design channel code is attributable to improper bit selection at the relay, which reduces the cooperation resources and hence the overall system performance.

We should use a decoding approach that can make use of all the resources that our optimized relay channel has to offer. For each of the three schemes [TeX:] $$M_1, M_2, \text { and } M_3$$, two decoding strategies the Naive Algorithm 3 and Smart Algorithm 4 joint decoding algorithms-are compared.We assume that each of the schemes uses an ideal (S-R) link and a sub-optimal designed relay. Figs. 8, 9, and 10 display the results from the Monte Carlos simulation. When the joint decoder uses the smart algorithm instead of the naive algorithm, the BER performance is better for each of the three TDCGCC schemes. This is due to the fact that the smart algorithm gains an advantage by feeding more reliable CR-decoder decoded bits into the CS-decoder during the joint decoding. For instance, utilizing the smart algorithm schemes [TeX:] $$M_1, M_2, \text { and } M_3$$, respectively, outperforms the naive algorithm scheme by approximately 0:8 dB, 1 dB, and 1:1 dB each at the [TeX:] $$\mathrm{BER}=10^{-2}$$, as shown in Figs. 8, 9, and 10, respectively. These results demonstrate the value of resources made available through optimized cooperation.

In the preceding simulation results in Sections VI-A and VI-B, the best conditions for a TDCGCC scheme to achieve full diversity are illustrated. These include a perfect (S-R) link, joint decoding based on a smart algorithm, and optimized relay channel code design. When comparing the BER performance of the non-cooperative (direct link) system and the TDCGCC scheme, we make use of these best conditions. Cooperation leads to improvements, as shown in Figs. 11, 12, and 13. Scheme [TeX:] $$M_1$$ has a 1:9 dB gain, for instance, when has a 1:9 dB gain, for instance, when [TeX:] $$\mathrm{BER}=1 \times 10^{-2}.$$ Scheme [TeX:] $$M_2$$ gain is approximately 2:7 dB at [TeX:] $$\mathrm{BER}=1.1 \times 10^{-2}$$, while Scheme [TeX:] $$M_3$$ gain is roughly 3 dB at [TeX:] $$\mathrm{BER}=1 \times 10^{-2}$$. We observe that the TDCGCC schemes post impressive performance as compared with corresponding noncooperative counterparts. These results validates the efficacy of the proposed TDCGCC scheme.

The perfect (S-R) channel, i.e., [TeX:] $$\lambda_{S, R}=\infty$$, is not possible for the majority of real-world applications. The following presumptions are made to evaluate the performance of each of the three TDCGCC schemes in such situations. We choose to set the non-ideal channel circumstances to either a poor (S-R) link, let’s say [TeX:] $$\lambda_{S, R}=0 \mathrm{~dB}$$, or a good (S-R) link, let’s say [TeX:] $$\lambda_{S, R}=15 \mathrm{~dB}$$. We also assume that no additional error propagation mitigation techniques, such as those proposed in [3], [30], are used at the relay and that the relay always takes part in the coded cooperation regardless of whether the relay terminal’s decoding is accurate or not. Fig. 14 shows that at [TeX:] $$\mathrm{BER}=1 \times 10^{-3}$$, the TDCGCC scheme [TeX:] $$M_1$$ with a (S-R) link of [TeX:] $$\lambda_{S, R}=15 \mathrm{~dB}$$ is only behind the ideal (S-R) link by about 2:2 dB. Similar results can be seen in Figs. 15 and 16, where the TDCGCC schemes under (S-R) link of [TeX:] $$\lambda_{S, R}=15 \mathrm{~dB}$$ are only 1:8 dB and 2:6 dB worse than that under the ideal (S-R) link at [TeX:] $$\mathrm{BER}=1 \times 10^{-2}$$ correspondingly. Therefore, the TDCGCC scheme performance approaches that of an ideal link when the (S-R) link become excellent.

However, the BER performance of TDCGCC schemes is severely hampered if the (S-R) link deteriorates, let’s say to [TeX:] $$\lambda_{S, R}=0 \mathrm{~dB}$$. Figs. 14, 15, and 16 demonstrate this, where the BER curves become practically flat at the (S-R) link of [TeX:] $$\lambda_{S, R}=0 \mathrm{~dB}$$. The relay node is unable to perform an accurate decoding due to the poor channel state between the source and relay nodes. As a result of the propagation of errors from the relay node, the destination node is unable to perform accurate joint-decoding, which lowers overall BER performance.

The Goppa code is a subfield subcode of a GRS code, just like the BCH code [31] is a subfield subcode of a RS code. Since their theory and application are reasonably simple, BCH codes are the most used in practice. Both codes are two most important subclasses of a broad class of Alternant codes [14]. BCH codes are well-known for performing well at short lengths, outperforming other codes including LDPC, RM, and polar codes [32]. As a fair comparison, first, we measure the TDCG code’s performance against a binary BCH code with a comparable length and code rate. The parameters of the BCH codes, which are used in schemes [TeX:] $$M_4, M_5, \text { and } M_6$$ to compare with TDCG codes in schemes [TeX:] $$M_1, M_2, \text { and } M_3$$, are described in Table VI. Since the BCH code only has one extra information bit than the comparable TDCG code, we assume the difference in data rates is negligible. Table VII lists the selection patterns associated to the various BCH coded-cooperative (BCHCC) schemes. Next, we present a BER performance comparison between the TDCGCC and the existing coded cooperation methods referenced. We have considered the following coded cooperation methods. RM codes, which employ [TeX:] $$C_S=R M(2,4) \text { and } C_R=R M(1,4),$$ and overall system rate of 11/32 [33], RS coded cooperation with adaptive cooperation level scheme with an overall system rate of 2/5 [6], and network-turbo code based distributed space-time block code (DSTBC-NTC) cooperative scheme with an overall system rate of 1/3 [34]. As such, we compare them with the TDCGCC scheme [TeX:] $$M_2$$, which has a similar overall system rate of 14/42.

TABLE VI

Scheme Serial | [TeX:] $$C_S\left(n_1, k_1, d_1\right)$$ | [TeX:] $$C_R\left(n_2, k_2, d_2\right)$$ | [TeX:] $$C_D\left(n_1+n_2, k_1\right)$$ | [TeX:] $$d_{\text {free }}$$ | Rate (R) |
---|---|---|---|---|---|

[TeX:] $$M_4$$ | [21,6,7] | [21,4,9] | [42, 6] | 7 | 0.1429 |

[TeX:] $$M_5$$ | [21,15,3] | [21,12,5] | [42, 15] | 3 | 0.3571 |

[TeX:] $$M_6$$ | [73,28,17] | [73,19,21] | [146,28] | 17 | 0.1918 |

TABLE VII

Scheme | Selection Type | Selection Pattern |
---|---|---|

[TeX:] $$M_4$$ | Sub-optimal | [0, 2, 3, 4] |

[TeX:] $$M_5$$ | Sub-optimal | [0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14] |

[TeX:] $$M_6$$ | Sub-optimal | 0, 1, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] |

Figs. 17, 18, and 19 show that, at [TeX:] $$\mathrm{BER}=1 \times 10^{-2}$$, the TDCGCC schemes [TeX:] $$M_1, M_2, \text { and } M_3$$ perform better than the BCHCC schemes [TeX:] $$M_4, M_5, \text { and } M_6$$, respectively, by about 1.1 dB, 0.8 dB, and 1.7 dB. Despite having one less information bit, a TDCGCC scheme offers a substantial advantage over a BCHCC scheme of a similar overall system rate. This enticing advantage can be attributed to the TDCG codes used at the source and relay nodes having a noticeably greater minimum Hamming distance, i.e., they have parameter such as those of the best-known linear codes. Therefore, when utilized in coded-cooperative coding schemes, TDCG codes remain to be superior to binary BCH codes.

The simulation results, which are also presented in Fig. 18, show how the proposed TDCGCC scheme outperforms existing ones like the RMCC in [33], the RS adaptive scheme (50% cooperative) in [6], and the DSTBC-NTC cooperative scheme in [34]. In comparison to the RMCC, RS adaptive (50% cooperative), and DSTBC-NTC cooperative schemes, the TDCGCC scheme, for instance, achieves improvements of around 0.9 dB, 2.5 dB and 2.8 dB at [TeX:] $$\mathrm{BER} \approx 1 \times 10^{-2}$$, respectively.

The main goal of the research is to investigate a novel induced-cyclic TDCG coded-cooperative diversity scheme. Since TDCG codes are among the best-known linear codes, they produce an excellent coded-cooperative system. The relay channel code is optimized using an effective method, which is shown to improve system performance. The simulation results showed that, when using joint decoding based on the smart algorithm and with an optimized relay, the TDCGCC scheme outperforms the non-cooperative method with outstanding results. The two factors that account for the TDCGCC schemes’ impressive performance are the joint decoding carried out at the destination, which successfully recovers the information from the source, and the added SNR gain provided by the optimized relay. Indeed, numerical simulation findings demonstrate that the proposed TDCGCC scheme performs better than the BCHCC and other existing coded-cooperation methods using the RM, RS, and DSTBC-NTC codes.

Daniel Kariuki Waweru received his B.Sc. degree in Mathematics and M.Sc. degree in Pure Mathematics from the University of Nairobi, Kenya, in 2015 and 2017, respectively. He is currently doing a Ph.D. from the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, China. His main interest is in algebraic coding theory, especially in the connections between codes, algebraic geometry and computer algebra tools. His most recent work are concerned with algebra-geometric codes and their application in wireless communications.

Fengfan Yang received the M.Sc. and Ph.D. degrees from the Northwestern Polytechnical University and Southeast University, China in 1993, and 1997, respectively, all in Electronic Engineering. Presently, he is working as a Professor in the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, China. His major research interests are information theory, channel coding and their applications for mobile and satellite communications.

Chunli Zhao received her B.S. degree in Electronic and Information Engineering from Henan Normal University Xinlian College, China, in 2014. She obtained her M.Sc. degree in Circuits and Systems at Henan Normal University, China, in 2017. She is currently doing a Ph.D. from the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, China. Her research interests are circuits and systems, signal processing, and channel coding.

Lawrence Muthama Paul obtained a B.Sc. degree in Mathematics in September, 2017 from the University of Nairobi, Kenya, and M.Sc. degree in Pure Mathematics in September, 2019 from the same University. From September, 2019 to September, 2020, he was a Graduate Assistant at the University of Nairobi. Currently, he is a Doctoral Student in the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, China. His current research interests are in algebra geometry coding theory and its applications in wireless communication.

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