A construction of the high-rate regular quasi-cyclic LDPC codes

This section gives a brief review of FG-LDPC codes and introduces some definitions as well as notations that are used throughout the rest of the paper.

An m-dimensional finite Euclidean geometry over the field GF(2^s) with m and s being the two positive integers, denoted EG(m,2^s), is a set of 2^sm-tuples (a₀,a₁,⋯,a_m−1) where a_i∈GF(2^s) for each i=0,1,…,m−1.

lines passing through it. The lines of EG(m,2^s) can be partitioned into k cosets where any two lines of EG(m,2^s) belong to the same coset if and only if they are parallel to each other.

Recall that α is a primitive element of EG(m,2^s). The parity-check matrix H_EG of the Euclidean geometry code is a k×(2^ms−1) matrix whose columns correspond to the ordered \(1=\alpha ^{0}\,,\,\alpha \,,\,\ldots \,,\,\alpha ^{2^{ms}-2}\phantom {\dot {i}\!}\). The rows of H_EG correspond to the cosets in EG(m,2^s) which do not contain the lines passing through the origin. That is, the (i,j) entry in H_EG is equal to 1 if the ith coset contains a line passing the jth point, and is equal to 0 otherwise.

In this section, we proceed with the procedure of the construction of the high-rate regular (QC-LDPC) codes through the following three steps: (1) Construct an n×n EG code and transform it into a proper form. (2) Utilize the transformed EG code matrix to construct an auxiliary matrix for identifying 6-cycles of it. (3) Eliminate the 6-cycles.

3.2 Identifying the 6-cycles

In this section, we develop a procedure to eliminate 6-cycles in (11). For this purpose, without loss generality, we assume that we are given:

$$ \mathbf{H}_{\text{qc}}' = [\mathrm{\Psi}(\mathbf{v}_{0}), \mathrm{\Psi}(\mathbf{v}_{1}), \ldots, \mathrm{\Psi}(\mathbf{v}_{k-1})], $$

(12)

where the row weight r_i of Ψ(v_i) for each i satisfying 0≤i≤k−1 is at least λ which is described in the previous section. The vector v_i with 0≤i≤k−1 is:

$$ \mathbf{v}_{i} = \left(\nu_{0}^{i}, \nu_{1}^{i}, \ldots, \nu_{b-1}^{i}\right)\,. $$

(13)

The first row of (12) is therefore equal to:

$$ \mathbf{h} = [\mathbf{v}_{0}, \mathbf{v}_{1}, \ldots, \mathbf{v}_{k-1}] \,. $$

(14)

In order to eliminate 6-cycles, we transform matrix (12) into a new matrix of form (17). This leads us to introduce the operation φ which is defined as:

$$ \varphi(\mathbf{v}_{i})= \left(\nu_{\frac{b-1}{2}+1}^{i}\,,\ldots, \nu_{b-1}^{i}, \nu_{0}^{i}, \nu_{1}^{i}, \ldots, \nu_{\frac{b-1}{2}}^{i}\right)\,. $$

(15)

Therefore, we transform (14) into:

$$ \hat{\mathbf{h}} = [\varphi(\mathbf{v}_{0})\,,\varphi(\mathbf{v}_{1})\,,\ldots\,,\varphi(\mathbf{v}_{k-1})]\,. $$

(16)

The transformation \(\mathcal {T}\) maps \(\hat {\mathbf {h}}\) in (16) to a b×(kb+b−1) matrix is defined as:

$$ \begin{array}{ll} &\mathcal{T}(\hat{\mathbf{h}}) = \left[ \begin{array}{cccccc} \varphi(\mathbf{v}_{0}) & \varphi(\mathbf{v}_{1}) & \cdots & \varphi(\mathbf{v}_{k-1}) & 0 & \mathbf{0}_{1 \times (b-2)}\\ 0 & \varphi(\mathbf{v}_{0}) & \varphi(\mathbf{v}_{1}) & \cdots & \varphi(\mathbf{v}_{k-1}) & \mathbf{0}_{1 \times (b-2)}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 0 & \mathbf{0}_{1 \times (b-2)} & \varphi(\mathbf{v}_{0}) & \varphi(\mathbf{v}_{1}) & \cdots & \varphi(\mathbf{v}_{k-1}) \\ \end{array} \right]_{b \times kb}\\ &= \left[ \begin{array}{cccccccccccccc} \nu_{\frac{b-1}{2}+1}^{0} & \cdots & \nu_{0}^{0} & \cdots & \nu_{\frac{b-1}{2}}^{0} & \nu_{\frac{b-1}{2}+1}^{1} & \cdots & \nu_{\frac{b-1}{2}}^{1} & \cdots & \nu_{\frac{b-1}{2}}^{k-1} & 0 & 0 & \cdots & 0 \\ 0 & \nu_{\frac{b-1}{2}+1}^{0} & \cdots & \nu_{0}^{0} & \cdots & \nu_{\frac{b-1}{2}}^{0} & \nu_{\frac{b-1}{2}+1}^{1} & \cdots & \nu_{\frac{b-1}{2}}^{1} & \cdots & \nu_{\frac{b-1}{2}}^{k-1} & 0 &\cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 0 & 0 & \cdots & 0 &\nu_{\frac{b-1}{2}+1}^{0} & \cdots & \nu_{0}^{0} & \cdots & \nu_{\frac{b-1}{2}}^{0} & \nu_{\frac{b-1}{2}+1}^{1} & \cdots & \nu_{\frac{b-1}{2}}^{1} & \cdots & \nu_{\frac{b-1}{2}}^{k-1}\\ \end{array} \right] \end{array} $$

(17)

Examing (17), we see that the first row of \(\mathcal {T}(\hat {\mathbf {h}})\) is obtained through adding b−1 zeros to the end of the vector \(\hat {\mathbf {h}}\). Each of the rest rows of \(\mathcal {T}(\hat {\mathbf {h}})\) is the circular shift of the row above it.

The set \(L(\hat {\mathbf {h}})\) comprised of the indices corresponding to the nonzero entries of \(\hat {\mathbf {h}}\) in (16) is defined as:

$$ L(\hat{\mathbf{h}}) = \{p_{0}, p_{1}, \ldots, p_{\rho-1}\}, $$

(18)

where we have \(\rho = \sum \limits _{i=0}^{k-1}r_{i}\) with r_i representing the row weight of Ψ(v_i). We define an oblique line segment \(l_{p_{i}}\) which starts from p_i and forms a descending diagonal from left to right in \(\mathcal {T}(\hat {\mathbf {h}})\),i.e., \(l_{p_{i}}\) constituting of the set of the entries of \(\mathcal {T}(\hat {\mathbf {h}})\) as follows:

$$ l_{p_{i}} = \left\{\mathcal{T}(\hat{\mathbf{h}})_{1,p_{i}}, \mathcal{T}(\hat{\mathbf{h}})_{2,p_{i}+1}, \ldots, \mathcal{T}(\hat{\mathbf{h}})_{b,p_{i}+b-1}\right\}\,. $$

(19)

Clearly, from the definition, the total number of the oblique line segments of the form (19) is ρ. Thus, the set of all the oblique line segments is:

$$ \left\{l_{p_{0}}\,,l_{p_{1}}\,,\ldots\,,l_{p_{\rho-1}}\right\}\,. $$

(20)

We define the distance between two oblique lines as:

$$ |l_{p_{i_{0}}}-l_{p_{i_{1}}}| = |p_{i_{0}}-p_{i_{1}}|\,. $$

(21)

From [15], it follows that the possible of 6-cycles appeared in a parity-check matrix is one of the following forms shown in Fig. 1.

Case 1: Assume that 6-cycle is comprised of three oblique line segments from (20). Therefore, from (20), the total number of subsets comprised of three oblique line segments is \(\dbinom {\rho }{3}\). For each subset comprised of three oblique line segments \(\left \{l_{p_{i_{0}}}, l_{p_{i_{1}}}, l_{p_{i_{2}}}\right \}\) from (20), we employ (21) to compute:

$$\begin{array}{@{}rcl@{}} \left\{ \begin{aligned} d_{0} &=|p_{i_{1}}-p_{i_{0}}| \\ d_{1} &=|p_{i_{2}}-p_{i_{1}}| \,.\\ \end{aligned} \right. \end{array} $$

(22)

If \({d_{0}\over {d_{1}}}\) is not equal to either \(1\over {2}\) or 2, then we delete such a subset comprised of these three oblique line segments. The remaining subsets constitute Ω₃ which can be represented as:

$$ \mathrm{\Omega}_{3} = \left\{ \left\{l_{p_{i_{0}}}, l_{p_{i_{1}}}, l_{p_{i_{2}}}\right\}, \left\{l_{p_{i_{3}}}, l_{p_{i_{4}}}, l_{p_{i_{5}}}\right\}, \ldots\right\}\,. $$

(23)

Clearly, each subset of three oblique line segments \(\{l_{p_{i_{j}}}, l_{p_{i_{j+1}}}, l_{p_{i_{j+2}}}\}\) in Ω₃ satisfies:

$$ {|l_{p_{i_{j+1}}}-l_{p_{i_{j}}}|\over{|l_{p_{i_{j+2}}}-l_{p_{i_{j+1}}}}|}={|p_{i_{j+1}}-p_{i_{j}}|\over{|p_{i_{j+2}}-p_{i_{j+1}}|}}={1\over{2}}\,\,\,\,\rm{or}\,\,\,2\,. $$

(24)

Figure 2 shows that 6-cycles are constituted by three oblique lines.

Case 2: Assume that 6-cycle is comprised of four oblique line segments from (20). Therefore, from (20), the total number of subsets comprised of four oblique line segments is \(\dbinom {\rho }{4}\). For each subset comprised of four oblique line segments \(\{l_{p_{i_{0}}}, l_{p_{i_{1}}}, l_{p_{i_{2}}}, l_{p_{i_{3}}}\}\) from (20), we employ (21) to compute:

$$\begin{array}{@{}rcl@{}} \left\{ \begin{aligned} d_{0} &=|p_{i_{1}}-p_{i_{0}}| \\ d_{1} &=|p_{i_{2}}-p_{i_{1}}| \\ d_{2} &=|p_{i_{3}}-p_{i_{2}}| \,.\\ \end{aligned} \right. \end{array} $$

(25)

If the computed distance d₀,d₁, and d₂ are not satisfied to any of these equations listed in the following set:

$$\begin{array}{@{}rcl@{}} \begin{array}{cc} &\left\{ \begin{array}{lll} d_{0} = d_{1}+d_{2}, & d_{1} = d_{0}+d_{2}, & d_{2} = d_{0}+d_{1}, \\ d_{0} = 2d_{2}, & d_{2} = 2d_{0}, & d_{0} = d_{1}+2d_{2}, \\ d_{2} = 2d_{0}+d_{1}\,,& d_{0} = d_{1}, & d_{1} = d_{2} \\ \end{array} \right\}, \end{array} \end{array} $$

(26)

then we delete such a subset comprised of these four oblique line segments. The remaining subsets constitute Ω₄ which can be represented as:

$$ \mathrm{\Omega}_{4} = \left\{ \left\{l_{p_{i_{0}}}, l_{p_{i_{1}}}, l_{p_{i_{2}}}, l_{p_{i_{3}}}\right\}, \left\{l_{p_{i_{4}}}, l_{p_{i_{5}}}, l_{p_{i_{6}}}, l_{p_{i_{7}}}\right\}, \ldots\right\}\,. $$

(27)

Figure 3 shows that 6-cycles are constituted by four oblique line segments.

Case 3: Assume that 6-cycle is comprised of five oblique line segments from (20). Therefore, from (20), the total number of subsets comprised of five oblique line segments is \(\dbinom {\rho }{5}\). For each subset comprised of five oblique line segments \(\{l_{p_{i_{0}}}, l_{p_{i_{1}}}, l_{p_{i_{2}}}, l_{p_{i_{3}}}, l_{p_{i_{4}}}\}\) from (20), we employ (21) to compute:

$$\begin{array}{@{}rcl@{}} \left\{ \begin{aligned} d_{0} &=|p_{i_{1}}-p_{i_{0}}| \\ d_{1} &=|p_{i_{2}}-p_{i_{1}}| \\ d_{2} &=|p_{i_{3}}-p_{i_{2}}| \\ d_{3} &=|p_{i_{4}}-p_{i_{3}}|\,. \\ \end{aligned} \right. \end{array} $$

(28)

If the computed distance d₀,d₁,d₂, and d₃ are not satisfied to any of these equations listed in the following set:

$$\begin{array}{*{20}l} \begin{array}{cc} &\left\{ \begin{array}{lll} d_{0} = d_{1}+d_{3}, & d_{3} = d_{0}+d_{1}, & d_{0} = 2d_{2}+d_{3}, \\ d_{1} = d_{0}+d_{3}, & d_{0} = d_{2}+d_{3}, & d_{0} = d_{2}+2d_{3}, \\ d_{2} = d_{0}+d_{3}, & d_{0} = d_{1}+2d_{2}+d_{3}, & d_{3} = 2d_{0}+d_{1}, \\ d_{3} = d_{0}+d_{2}, & d_{3} = d_{0}+2d_{1}+d_{2}, & d_{3} = d_{0}+2d_{1} \\ \end{array} \right\}, \end{array} \end{array} $$

(29)

then we delete such a subset of these five oblique line segments. The remaining subsets constitute Ω₅ which can be represented as:

$$ \mathrm{\Omega}_{5} \,=\, \left\{ \left\{l_{p_{i_{0}}}, l_{p_{i_{1}}}, l_{p_{i_{2}}}, l_{p_{i_{3}}}, l_{p_{i_{4}}}\!\right\},\! \left\{l_{p_{i_{5}}}, l_{p_{i_{6}}}, l_{p_{i_{7}}}, l_{p_{i_{8}}}, l_{p_{i_{9}}}\!\right\}, \ldots\right\}\!. $$

(30)

Figure 4 shows that 6-cycles are constituted by five oblique line segments.

In example 1 of this section, we utilize the procedure from (4) to (37) to construct a regular quasi-cyclic (4095,3781) LDPC code as well as (38) to construct regular quasi-cyclic (7560,6931) and (8190,7561) LDPC codes where the coding rates of all these three codes are 12/13.

In example 2, we utilize the procedure from (4) to (37) to construct a regular quasi-cyclic (2925,2341) LDPC code and a regular quasi-cyclic (4095,3781) LDPC code where the coding rates of all these two codes are 4/5 and 11/12, respectively.

In example 3 and example 4, the procedure of (4)–(38) is also employed to separately construct a regular quasi-cyclic (6930,6301) LDPC code and a regular quasi-cyclic (7280,6370) LDPC code with the coding rate 10/11 and 7/8, respectively. As a comparison of the performance, we also utilize the schemes from [16–18] to construct six (4095,3771), (4095,3729), (16383,14923), (8176,7156), (1020,935), and (2850,2280) QC-LDPC codes. The simulation results in the following examples show that the procedure from (4) to (38) is an effective scheme to yield the high-rate regular QC-LDPC codes with moderate code lengths.

Example 1: Following the scheme of [9], we first construct a regular (4095,3367) EG-LDPC code over the field EG(2,2⁶). Since we have 4095=c×b with c=13 and b=315, the procedure from (4) to (8) suggests that the parity-check matrix of the constructed (4095,3367) EG-LDPC code can be decomposed into 13 submatrices of size b×b shown in (8). Of the 13 submatrices of size b×b, choose the parameter λ=4 to perform (9) to (11) where parameter λ is introduced above Eq. (9). Then, we select the first k=13 submatrices and perform (12) to (37) and (38) separately which yields a regular quasi-cyclic 315×4095 parity-check matrix \(\mathbf {H}_{\text {qc}}^{(1)}\) defined in (37) as well as a 630×8190 parity-check matrix \(\mathbf {H}_{\text {qc}}^{(2)}\) defined in (38) respectively. The null spaces of \(\mathbf {H}_{\mathrm { qc}}^{(1)}\) and \(\mathbf {H}_{\text {qc}}^{(2)}\) render a regular quasi-cyclic (4095,3781) LDPC code as well as a (8190,7561) LDPC code with the coding rate 12/13. Similarily, we also choose the first k=12 submatrices and perform (12) to (38) which yields a regular quasi-cyclic 630×7560 parity-check matrix \(\mathbf {H}_{\text {qc}}^{(2)}\) defined in (38). The null space of this \(\mathbf {H}_{\text {qc}}^{(2)}\) also renders a regular quasi-cyclic (7560,6931) LDPC code with the coding rate 11/12. As a comparison, we also employ the procedure of [16] to construct the (4095,3771) QC-LDPC code with the coding rate being 0.92.

Figure 5 shows the performances of these codes where it (a) represents the bit error rate (BER) and (b) depicts the frame error rate (FER). In these figures, the circle, box, and diamond solid lines respectively represent the performances of the regular quasi-cyclic (4095,3781), (7560,6931), and (8190,5161) LDPC codes. The star dotted line corresponds to the performance of the (4095,3771) QC-LDPC code generated by [16].

Example 2: Following the scheme of [9], we first construct a regular (4095,3367) EG-LDPC code over the field EG(2,2⁶). Since we have 4095=c×b with c=7, b=585, and c=45, b=91, the procedure from (4) to (8) suggests that the parity-check matrix of the constructed (4095,3367) EG-LDPC code can be decomposed into 7 and 45 submatrices of size b×b shown in (8). Of 7 and 45 submatrices of size b×b, choose the parameters λ=4 and λ=3 to perform (9) to (11) where parameter λ is introduced above Eq. (9). Then, we select separately the first k=5 and k=12 submatrices then perform (12) to (37) which yields a regular quasi-cyclic 585×2925 and 91×1091 parity-check matrix \(\mathbf {H}_{\text {qc}}^{(1)}\) both defined in (37). As a comparison, we also employ the procedure of [17, 18] construct the (2850,2280) and(1020,935) QC-LDPC code with the coding rate being 4/5 and 11/12, respectively.

Figure 6 shows the performances of these codes where it (a) represents the bit error rate (BER) and (b) depicts the frame error rate (FER). In these figures, the circle and box solid lines respectively represent the performances of the regular quasi-cyclic (2925,2341) and (1092,1001) LDPC codes. The circle and box dotted line corresponds to the performance of the (2850,2280) and 1020,935 QC-LDPC code generated by [17, 18].

Example 3: Following the scheme of [9], we first construct a regular (4095,3367) EG-LDPC code over the field EG(2,2⁶). Since we have 4095=c×b with c=13 and b=315, the procedure from (4) to (8) suggests that the parity-check matrix of the constructed (4095,3367) EG-LDPC code can be decomposed into 13 submatrices of size b×b shown in (8). Of 13 submatrices of size b×b, choose the parameter λ=4 to perform (9) to (11) where parameter λ is introduced above Eq. (9). Then, we select the first k=11 submatrices and perform (12) to (38) which yields a regular quasi-cyclic 630×6930 parity-check matrix \(\mathbf {H}_{\text {qc}}^{(2)}\) defined in (38). The null space of \(\mathbf {H}_{\text {qc}}^{(2)}\) renders a regular quasi-cyclic (6930,6301) LDPC code with the coding rate 10/11. As a comparison, we also employ the procedure of [16] to construct the (4095,3729) and (16383,14923) QC-LDPC code with the coding rates of both codes being 0.91.

Figure 7 shows the performances of these codes where it (a) represents the bit error rate (BER) and (b) depicts the frame error rate (FER). In these figures, the circle solid line represents the performance of the regular quasi-cyclic (6930,6301) LDPC code constructed from the procedure from (4) to (38). The box and diamond dotted lines respectively correspond to the performances of the (4095,3729) and (16383,14923) QC-LDPC codes yielded from [16].

Example 4: Following the scheme of [9], we first construct a regular (4095,3367) EG-LDPC code over the field EG(2,2⁶). Since we have 4095=c×b with c=9 and b=455, the procedure from (4) to (8) suggests that the parity-check matrix of the constructed (4095,3367) EG-LDPC code can be decomposed into nine submatrices of size b×b shown in (8). Of the nine submatrices of size b×b, choose the parameter λ=4 to perform (9) to (11) where parameter λ is introduced above Eq. (9). Then, we select the first k=8 submatrices and perform the procedure from (12) to (38) which yields a regular quasi-cyclic 910×7280 parity-check matrix \(\mathbf {H}_{\text {qc}}^{(2)}\) defined in (38). The null space of this \(\mathbf {H}_{\text {qc}}^{(2)}\) renders a regular quasi-cyclic (7280,6371) LDPC code with the coding rate 7/8. As a comparison, we also employ the procedure of [16] to construct the (8176,7156) QC-LDPC code with the coding rate being 0.875.

Figure 8 shows the performances of these codes where it (a) represents the bit error rate (BER) and (b) depicts the frame error rate (FER). In these figures, the circle solid line represents the performance of the regular quasi-cyclic (7280,6371) LDPC code yielded from the procedure of (4)–(38). The box dotted line corresponds to the performance of the (8176,7156) QC-LDPC code (example 7 of [16]) generated by the scheme of [16].