1 Review
2 GLDPC-Staircase code design
2.1 Code description
-
LDPC-Staircase code: this is the base code with length N L and dimension K. Let M L =N L −K and let H L =(H 1|H 2) be the associated parity check matrix1. From the LDPC-Staircase viewpoint, each row of H L defines the connections between the source and LDPC repair symbols. From the GLDPC-Staircase viewpoint, each row of H L defines the connections between the RS repair symbols and the source and LDPC repair symbols. Consequently, each LDPC-Staircase CN is represented as a powerful CN, called generalized check node, with GLDPC-Staircase codes.
-
RS codes: they are the outer codes (or components codes). A generalized check node of index m can generate e(m) extra-repair symbols from the RS point of view (plus one LDPC repair symbol if we use scheme A as we will see below). This is done with an RS (n m ,k m ) encoding over G F(2 b ) with 0≤e(m)≤E and m=1,...,M L . Here, E, k m , and n m are respectively the maximum number of extra-repair symbols per generalized check node, the RS code dimension and length for the generalized check node m.
-
the generalized check node that corresponds to RS codes;×
-
the variable nodes (VN) further divided into three categories:
-
source symbols;
-
LDPC repair symbols;
-
extra-repair symbols.
-
2.2 Schemes A and B
-
Scheme AFor row m>1, the source symbols (from the user viewpoint) involved in this row plus the LDPC repair symbol of row m−1 are considered as source symbols from the RS viewpoint. The new LDPC repair symbol plus the e(m) extra-repair symbols are considered as repair symbols from the RS viewpoint. Therefore, the LDPC repair symbol is also an RS repair symbol. For m=1, the only difference is that there is no previous repair symbol (beginning of the staircase).No matter the row, we have$$ n_{m} = k_{m} + 1 + e(m) ~ \text{and} ~ k_{m} = d_{r}(m) - 1, $$(1)where d r (m) is the degree of row m of H L . Due to the staircase structure of H 2, it follows$$ { d_{r}(m) = \left\lbrace \begin{array}{ccc} d_{r_{H_{1}}}+ 1 & \text{if} & m=1 \text{;} \left(d_{r_{H_{1}}} = \frac{N_{1}}{\frac{1}{r_{L}}-1}\right)\\ d_{r_{H_{1}}} + 2 & \text{if} & m>1 \text{;} \left(d_{r_{H_{1}}} = \frac{N_{1}}{\frac{1}{r_{L}}-1}\right). \end{array}\right. } $$(2)In order to fulfill the duality property of the LDPC repair symbols, we propose in [16] a specific construction of RS codes based on “quasi” Hankel matrix. The generator matrix G of these codes has the following form:$$ G =\left[ \begin{array}{cccccccccc} 1&0&\ldots&\ldots&0&\boldsymbol{1}&1&\ldots&\ldots&1 \\ 0&1&\ddots&\ddots&\vdots&\boldsymbol{1}&b_{1}&b_{2}&\ldots&b_{n_{m}-k_{m}} \\ 0&0&\ddots&\ddots&\vdots&\boldsymbol{1}&b_{2} &{{\cdot}{\raisebox{3pt}{$\cdot$}}{\raisebox{6pt}{$\cdot$}}}& {{\cdot}{\raisebox{3pt}{$\cdot$}}{\raisebox{6pt}{$\cdot$}}} &b_{n_{m}-k_{m}+1}\\ \vdots&\vdots&\ddots&\ddots&0&\vdots&\vdots&{{\cdot}{\raisebox{3pt}{$\cdot$}}{\raisebox{6pt}{$\cdot$}}}&{{\cdot}{\raisebox{3pt}{$\cdot$}}{\raisebox{6pt}{$\cdot$}}}&\vdots \\ 0&0&\ldots&0&1&\boldsymbol{1}&b_{k_{m}}&{{\cdot}{\raisebox{3pt}{$\cdot$}}{\raisebox{6pt}{$\cdot$}}}&{{\cdot}{\raisebox{3pt}{$\cdot$}}{\raisebox{6pt}{$\cdot$}}}&b_{n_{m}-1} \\ \end{array} \right] $$(3)where \(b_{i} = \frac {1} {1- {y}^{i}}\), for 1≤i≤q−1, y is an arbitrary primitive element of G F(q) and y i is computed over G F(q).Thanks to the column full of “1” in G for the first RS repair symbol, this latter can also be considered as an LDPC-Staircase symbol (it is the XOR sum of source symbols from the RS viewpoint).
-
Scheme BFor each row m, the various source symbols (from the user viewpoint) involved in this row plus the LDPC repair symbol(s) are considered as source symbols from the RS viewpoint. The e(m) extra-repair symbols are the only repair symbols from the RS viewpoint. No matter the row, we have$$ n_{m} = k_{m} + e(m)~\text{and}~k_{m} = d_{r}(m). $$(4)Here, any RS code (e.g., based on Hankel, Cauchy, or Vandermonde matrices) can be used.
2.3 Extra-repair symbol regular/irregular distributions
-
Regular distribution: f e =0 for \(e \in \{0,1,\dots,E-1\}\) and f E =1. Thus, each generalized check node m has the same number e(m)=E of extra-repair symbols and the rate of the extended code (GLDPC-Staircase code) is$$ r_{G} =\frac{r_{L}}{1+ (1 - r_{L})*E} $$(8)Figure 1 shows such a regular variant.
-
Irregular distribution: the generalized check nodes can have a different number of extra-repair symbols.Figure 2 shows such an irregular variant.×Cunche et al. [13] shows that there exists an irregular uniform distribution of extra-repair symbols which achieves performance close to the optimal irregular distribution. This irregular uniform distribution allows to allocate the extra-repair symbols with \(\bar {f} =\frac {E}{2}\) and \(f_{e}=\frac {1}{E+1}\) for \(e \in \{0,1,\dots,E\}\).
2.4 Encoding method
-
M L LDPC-Staircase repair symbols, (p 1, …\(p_{M_{L}}\)), and
-
\(M_{L}\bar {f}\) extra-repair symbols, ((e 1,1, … e 1,e(1)), …(\(e_{M_{L},1}\), …\(e_{M_{L},e(M_{L})}\))).
-
First row, using x=(S 1,S 2,S 4), produces$$ P_{1} = G^{4}_{rs}\times (S_{1}, S_{2}, S_{4}) $$$$ e_{1,1} = G^{5}_{rs} \times (S_{1}, S_{2}, S_{4}) $$$$ e_{1,2} = G^{6}_{rs} \times (S_{1}, S_{2}, S_{4}) $$
-
Second row, using x=(S 2,S 3), produces$$ P_{2} = G^{4}_{rs}\times (S_{2}, S_{3}, P_{1}) $$$$ e_{2,1} = G^{5}_{rs} \times (S_{2}, S_{3}, P_{1}) $$$$ e_{2,2} = G^{6}_{rs} \times (S_{2}, S_{3}, S_{1}) $$
-
Third row, using x=(S 1,S 4), produces$$ P_{3} = G^{4}_{rs}\times (S_{1}, S_{4}, P_{2}) $$$$ e_{3,1} = G^{5}_{rs} \times (S_{1}, S_{4}, P_{2}) $$$$ e_{3,2} = G^{6}_{rs} \times (S_{1}, S_{4}, P_{2}) $$
2.5 Decoding method
2.5.1 (IT + RS) decoding
-
the IT decoder over the LDPC-Staircase graph. Extra-repair symbols are ignored at this step. This decoder features a linear complexity but also sub-optimal erasure recovery capabilities;
-
the RS decoder over a given generalized check node. This is a classic RS decoding that takes into account the three types of symbols. This decoder features a higher complexity but is MDS;
2.5.2 Hybrid (IT/RS/ML) decoding
-
Binary ML decoder: extra-repair symbols are ignored at this step and instead it only considers binary equations, made of simple XOR sums, in order to reduce complexity. ML decoding can consist of simple Gaussian elimination (GE) on this sub-system.
-
Non-binary ML decoder: the full linear system is considered here and GE is performed on G F(2 b ). As in binary ML decoding, this step also features a quadratic complexity but operations are now significantly more complex (performed on G F(2 b ) instead of simple
XOR
). However, it allows reaching the maximum correction capabilities of the code.
3 Asymptotic analysis method
3.1 Preliminaries
3.2 Density evolution
3.2.1 Introduction
3.2.2 DE equations of GLDPC-Staircase codes
-
P ℓ , the probability of an LDPC symbol (source or repair) node sending an erasure at iteration ℓ to the connected generalized check nodes. Clearly, P 0 is equal to the channel erasure probability ε.
-
Q ℓ , the probability of a generalized check node sending an erasure (to an LDPC symbol-node) at iteration ℓ.
-
there are no erased symbols among \(v_{2},\dots,v_{d}\) (i.e., LDPC decoding);
-
there is at least one erased symbol among \(v_{2},\dots,v_{d}\), but the number of erasures among all the symbol-nodes \((v_{1},\dots,v_{d},e_{1,c},\dots,e_{e(c),c})\) is less than or equal to e(c)−1.
-
the number of erased symbols among \(v_{2},\dots,v_{d}\) is equal to i and the number of erased symbols among \(e_{1,c},\dots,e_{e(c),c}\) is equal to j, with \(1 \leq i \leq \min (d-1,e(c)-1)\) and 0≤j≤e(c)−1−i.
-
Rate: \(r_{G}=\frac {1}{2}\)
-
Base code: r L =0.8, N1=5$$ DD: \left\lbrace \begin{array}{l} \lambda(x) = 0.0909.x^{1}+0.9091.x^{4}, \rho(x) = x^{21}\\ L(x) = 0.2.x^{2}+0.8.x^{5}, R(x) =x^{22} \\ \end{array}\right. $$(18)
-
E=3 (regular distribution of extra-repair symbols).
3.3 EXIT functions of GLDPC-Staircase codes
3.3.1 Introduction
3.3.2 (IT + RS) EXIT function: h (IT+RS) (ε)
3.3.3 (IT + RS) EXIT curve
-
r L =0.6, N1=5
-
DD:$$ \left\lbrace \begin{array}{l} \lambda(x) = 0.2105x^{1} + 0.7895x^{4}, \rho(x) = x^{9}, \\ L(x) = 0.4x^{2} + 0.6x^{5}, R(x) = x^{10} \\ \end{array}\right. $$(22)
3.3.4 Upper bound on the ML decoding threshold
-
Step 1: Plot the (IT + RS) EXIT curve as parametrized in Eq. (21).
-
Step 2: Determine the \(\bar {\epsilon }^{ML}\) by integrate backwards from the right end of the curve where ε=1. The integration process stops at \(\bar {\varepsilon }^{ML}\) where it assures Eq. (25). This gives the upper bound \(\bar {\varepsilon }^{ML}\) of the GLDPC-Staircase codes.
-
Step 3: The ML EXIT curve is now the curve which is zero at the left of the upper bound on the ML decoding threshold and equals to the (IT + RS) EXIT curve to the right of this decoding threshold (i.e., the (IT + RS) EXIT and the ML EXIT curves coincide above \(\bar {\varepsilon }^{ML}\)).
4 Optimization of GLDPC-Staircase codes
4.1 Description
-
Scheme A has the property that on each generalized check node, the repair symbol generated by the LDPC code is also an RS repair symbol.
-
On the opposite, with scheme B the generated LDPC repair symbol, on each generalized check node, is an RS source symbol.
-
the extra-repair symbols distribution across the H L rows: regular distribution or irregular uniform distribution,
-
the N1 parameter of the base code: degree of source variable nodes in H L ,
-
the base code rate r L .
4.2 Experimental conditions
N1 | edge DD (λ(x), ρ(x)) | node DD (L(x),R(x)) |
---|---|---|
3 |
λ(x)=0.25.x
1+0.75.x
2, ρ(x)=x
7
|
L(x)=0.3333.x
2+0.6667.x
3, R(x)=x
8
|
4 |
λ(x)=0.2.x
1+0.8.x
3, ρ(x)=x
9
|
L(x)=0.3333.x
2+0.6667.x
4, R(x)=x
10
|
5 |
λ(x)=0.1666.x
1+0.8333.x
4, ρ(x)=x
11
|
L(x)=0.3333.x
2+0.6667.x
5, R(x)=x
12
|
6 |
λ(x)=0.1429.x
1+0.8571.x
5, ρ(x)=x
13
|
L(x)=0.3333.x
2+0.6667.x
6, R(x)=x
14
|
r
L
| edge DD (λ(x), ρ(x)) | node DD (L(x),R(x)) |
---|---|---|
\(\frac {1}{2}\)
|
λ(x)=0.2857.x
1+0.7143.x
4, ρ(x)=x
6
|
L(x)=0.5000.x
2+0.5000.x
5, R(x)=x
7
|
\(\frac {2}{3}\)
|
λ(x)=0.1666.x
1+0.8333.x
4, ρ(x)=x
11
|
L(x)=0.3333.x
2+0.6667.x
5, R(x) = x
12
|
0.75 |
λ(x)=0.1176.x
1+0.8824.x
4, ρ(x)=x
16
|
L(x)=0.2500.x
2+0.75.x
5, R(x)=x
17
|
0.8 |
λ(x)=0.0909.x
1+0.9091.x
4, ρ(x)=x
21
|
L(x)=0.2.x
2+0.8.x
5, R(x)=x
22
|
4.3 Best coding scheme for GLDPC-Staircase codes
4.3.1 Asymptotic results
r
G
| Decoding threshold |
N1 | Scheme A | Scheme B | |
---|---|---|---|---|---|
\(\frac {1}{2}\) (E=1) |
ε
(IT+RS)
| 3 | 0.4571 | 0.2905 | |
4 | 0.4269 | 0.2867 | |||
5 | 0.3943 | 0.2709 | |||
6 | 0.3646 | 0.2536 | |||
\(\bar {\varepsilon }^{\text {ML}}\)
| 3 | 0.4900 | 0.4985 | ||
4 | 0.4976 | 0.4996 | |||
5 | 0.4993 | 0.4997 | |||
6 | 0.4998 | 0.4999 | |||
\(\frac {2}{5}\) (E=2) |
ε
(IT+RS)
| 3 | 0.5532 | 0.4041 | |
4 | 0.5136 | 0.3878 | |||
5 | 0.4744 | 0.3639 | |||
6 | 0.4392 | 0.3401 | |||
\(\bar {\varepsilon }^{\text {ML}}\)
| 3 | 0.5946 | 0.5994 | ||
4 | 0.5990 | 0.5998 | |||
5 | 0.5998 | 0.5999 | |||
6 | 0.5999 | 0.5999 |
Decoding threshold |
E
|
r
L
| Scheme A | Scheme B |
---|---|---|---|---|
ε
(IT+RS)
| 0 |
\(\frac {1}{2}\)
| 0.4380 | 0.4380 |
1 |
\(\frac {2}{3}\)
| 0.3943 | 0.2709 | |
2 | 0.75 | 0.3647 | 0.2773 | |
3 | 0.8 | 0.3443 | 0.2819 | |
\(\bar {\varepsilon }^{\text {\scriptsize ML}}\)
| 0 |
\(\frac {1}{2}\)
| 0.4946 | 0.4946 |
1 |
\(\frac {2}{3}\)
| 0.4993 | 0.4997 | |
2 | 0.75 | 0.4999 | 0.4999 | |
3 | 0.8 | 0.4999 | 0.4999 |
Decoding threshold |
f(%) |
r
L
| Scheme A | Scheme B |
---|---|---|---|---|
ε
(IT+RS)
| [0] |
\(\frac {1}{2}\)
| 0.4380 | 0.4380 |
[25 50 25] |
\(\frac {2}{3}\)
| 0.4064 | 0.3029 | |
[0 50 25 0 25] | 0.75 | 0.3874 | 0.3089 | |
[5 10 15 20 50] | 0.8 | 0.3589 | 0.3007 | |
\(\bar {\varepsilon }^{\text {\scriptsize ML}}\)
| [0] |
\(\frac {1}{2}\)
| 0.4946 | 0.4946 |
[25 50 25] |
\(\frac {2}{3}\)
| 0.4993 | 0.4997 | |
[0 50 25 0 25] | 0.75 | 0.4998 | 0.4998 | |
[5 10 15 20 50] | 0.8 | 0.4998 | 0.4998 |
Decoding threshold |
r
G
|
E
| Scheme A | Scheme B | Shannon limit |
---|---|---|---|---|---|
ε
(IT+RS)
|
\(\frac {2}{3}\)
|
E=0 | 0.2709 | 0.2709 | 0.3333 |
\(\frac {1}{2}\)
|
E=1 | 0.3943 | 0.2709 | 0.5 | |
\(\frac {2}{5}\)
|
E=2 | 0.4744 | 0.3639 | 0.6 | |
\(\frac {1}{3}\)
|
E=3 | 0.5332 | 0.4394 | 0.6667 | |
\(\bar {\varepsilon }^{\text {\scriptsize ML}}\)
|
\(\frac {2}{3}\)
|
E=0 | 0.3301 | 0.3301 | 0.3333 |
\(\frac {1}{2}\)
|
E=1 | 0.4993 | 0.4997 | 0.5 | |
\(\frac {2}{5}\)
|
E=2 | 0.5998 | 0.5999 | 0.6 | |
\(\frac {1}{3}\)
|
E=3 | 0.6666 | 0.6666 | 0.6667 |
Decoding threshold |
r
G
|
f(%) | Scheme A | Scheme B | Shannon limit |
---|---|---|---|---|---|
ε
(IT+RS)
|
\(\frac {2}{3}\)
| [0] | 0.2709 | 0.2709 | 0.3333 |
\(\frac {1}{2}\)
| [25 50 25] | 0.4064 | 0.3029 | 0.5 | |
\(\frac {2}{5}\)
| [0 25 50 25] | 0.4819 | 0.3765 | 0.6 | |
\(\frac {1}{3}\)
| [0 25 25 10 15 15 10] | 0.5598 | 0.4768 | 0.6667 | |
\(\bar {\varepsilon }^{\text {ML}}\)
|
\(\frac {2}{3}\)
| [0] | 0.3301 | 0.3301 | 0.3333 |
\(\frac {1}{2}\)
| [25 50 25] | 0.4993 | 0.4997 | 0.5 | |
\(\frac {2}{5}\)
| [0 25 50 25] | 0.5997 | 0.5999 | 0.6 | |
\(\frac {1}{3}\)
| [0 25 25 10 15 15 10] | 0.6665 | 0.6666 | 0.6667 |
4.3.2 Finite length results
4.3.3 Conclusion of the analysis
4.4 Tuning internal parameters of GLDPC-Staircase codes
4.4.1 The extra-repair symbol distribution
Decoding type |
r
G
|
E
|
Δ
1
|
f(%) |
Δ
2
|
---|---|---|---|---|---|
(IT + RS) |
\(\frac {2}{3}\)
|
E=0 | 0.06243 | [0] | 0.06243 |
\(\frac {1}{2}\)
|
E=1 | 0.1057 | [25 50 25] | 0.0936 | |
\(\frac {2}{5}\)
|
E=2 | 0.1256 | [0 25 50 25] | 0.1181 | |
\(\frac {1}{3}\)
|
E=3 | 0.13346 | [0 25 25 10 15 15 10] | 0.1068 | |
\(\frac {1}{3.5}\)
|
E=4 | 0.1350 | [0 25 0 0 0 75] | 0.1167 | |
\(\frac {1}{5}\)
|
E=7 | 0.1268 | [0 0 0 0 0 0 10 80 10] | 0.1262 | |
ML |
\(\frac {2}{3}\)
|
E=0 | 0.00323 | [0] | 0.00323 |
\(\frac {1}{2}\)
|
E=1 | 0.0007 | [25 50 25] | 0.0007 | |
\(\frac {2}{5}\)
|
E=2 | 0.0002 | [0 25 50 25] | 0.0003 | |
\(\frac {1}{3}\)
|
E=3 | 0.0001 | [0 25 25 10 15 15 10] | 0.0002 | |
\(\frac {1}{3.5}\)
|
E=4 | 0.000081 | [0 25 0 0 0 75] | 0.0000857 | |
\(\frac {1}{5}\)
|
E=7 | 0.00007 | [0 0 0 0 0 0 10 80 10] | 0.00008 |
Decoding type |
r
L
|
E
|
Δ
1
|
f(%) |
Δ
2
|
---|---|---|---|---|---|
(IT+RS) |
\(\frac {1}{2}\)
| 0 | 0.062 | [0] | 0.062 |
\(\frac {2}{3}\)
| 1 | 0.1057 | [25 50 25] | 0.0936 | |
0.75 | 2 | 0.1353 | [0 50 25 0 25] | 0.1126 | |
0.8 | 3 | 0.1557 | [5 10 15 20 50] | 0.1411 | |
ML |
\(\frac {1}{2}\)
| 0 | 0.0054 | [0] | 0.0054 |
\(\frac {2}{3}\)
| 1 | 0.0007 | [25 50 25] | 0.0007 | |
0.75 | 2 | 0.0001 | [0 50 25 0 25] | 0.0002 | |
0.8 | 3 | 0.0001 | [5 10 15 20 50] | 0.0002 |
Distribution type | Decoding |
N1 |
\(r_{G}=\frac {1}{2}\)
|
\(r_{G}=\frac {2}{5}\)
|
\(r_{G}=\frac {1}{3}\)
| |
---|---|---|---|---|---|---|
threshold | ||||||
Regular distribution |
ε
(IT+RS)
| 3 | 0.4571 | 0.5532 | 0.6184 | |
4 | 0.4269 | 0.5136 | 0.5753 | |||
5 | 0.3943 | 0.4744 | 0.5332 | |||
6 | 0.3646 | 0.4392 | 0.4955 | |||
\(\bar {\varepsilon }^{\text {ML}}\)
| 3 | 0.49 | 0.5946 | 0.6633 | ||
4 | 0.4976 | 0.5990 | 0.6661 | |||
5 | 0.4993 | 0.5998 | 0.6666 | |||
6 | 0.4998 | 0.5999 | 0.6666 | |||
Irregular uniform | ||||||
distribution |
ε
(IT+RS)
| 3 | 0.4605 | 0.5563 | 0.6287 | |
4 | 0.4365 | 0.5196 | 0.5959 | |||
5 | 0.4064 | 0.4819 | 0.5598 | |||
6 | 0.3779 | 0.4476 | 0.5257 | |||
\(\bar {\varepsilon }^{\text {ML}}\)
| 3 | 0.4863 | 0.5935 | 0.6600 | ||
4 | 0.4966 | 0.5987 | 0.6654 | |||
5 | 0.4993 | 0.5997 | 0.6666 | |||
6 | 0.4997 | 0.5998 | 0.6666 |
4.4.2 N1 parameter
r
G
|
N1 | (IT + RS) decoding | ML decoding |
---|---|---|---|
\(\frac {2}{3}\) (E=0) | 3 | 1.06669 | 1.04225 |
5 | 1.09682 | 1.00636 | |
6 | 1.12217 | 1.00396 | |
7 | 1.14624 | 1.00250 | |
\(\frac {1}{2}\) (E=1) | 3 | 1.10487 | 1.01627 |
5 | 1.22160 | 1.00097 | |
6 | 1.27825 | 1.00040 | |
7 | 1.32857 | 1.00009 | |
\(\frac {1}{3}\) (E=3) | 3 | 1.17963 | 1.00668 |
5 | 1.42080 | 1.00019 | |
6 | 1.53045 | 1.00007 | |
7 | 1.62660 | 1.00001 |
r
L
| Decoding threshold |
N1 |
E=2 |
E=3 | |
---|---|---|---|---|---|
\(r_{L}=\frac {2}{3}\)
|
ε
(IT+RS)
| 3 | 0.5532 | 0.6184 | |
4 | 0.5136 | 0.5753 | |||
5 | 0.4744 | 0.5332 | |||
6 | 0.4392 | 0.4955 | |||
\(\bar {\varepsilon }^{\text {ML}}\)
| 3 | 0.5946 | 0.6633 | ||
4 | 0.5990 | 0.6661 | |||
5 | 0.5998 | 0.6666 | |||
6 | 0.5999 | 0.6666 | |||
\(r_{L}=\frac {1}{2}\)
|
ε
(IT+RS)
| 3 | 0.7236 | 0.7769 | |
4 | 0.7002 | 0.7530 | |||
5 | 0.66698 | 0.7232 | |||
6 | 0.6387 | 0.6927 | |||
\(\bar {\varepsilon }^{\text {ML}}\)
| 3 | 0.7403 | 0.7933 | ||
4 | 0.7476 | 0.7985 | |||
5 | 0.7493 | 0.7996 | |||
6 | 0.7498 | 0.7999 |
4.4.3 The base code rate r L
r
G
|
E=0 |
E=1 |
E=2 |
E=3 |
E=4 |
E=5 |
δ
s
h
|
---|---|---|---|---|---|---|---|
1/3.5 | 0.7054 | 0.7124 | 0.7138 |
0.7141
| 0.7142 | 0.7142 | 0.7142 |
1/3 | 0.6634 | 0.6652 |
0.6664
| 0.6665 | 0.6666 | 0.6666 | 0.6667 |
1/2 | 0.4946 | 0.4993 |
0.4999
| 0.4999 | 0.4999 | 0.4999 | 0.5000 |
2/3 | 0.3301 | 0.3330 |
0.3333
| 0.3333 | 0.3333 | 0.3333 | 0.3333 |
3/4 | 0.2484 |
0.2498
| 0.2499 | 0.2499 | 0.2499 | 0.2499 | 0.2500 |
9/10 | 0.0991 |
0.0999
| 0.0999 | 0.0999 | 0.0999 | 0.0999 | 0.1000 |
5 Performance evaluation
5.1 Achieved performance
5.1.1 (IT + RS) versus hybrid (IT/RS/ML) decoding
5.1.2 Detailed hybrid (IT/RS/ML) decoding inefficiency ratio results
5.1.3 Hybrid (IT/RS/ML) decoding failure probability results
K
| Reception overhead | Average decoding failure prob. |
---|---|---|
32 | 0 | 0.0305 |
1 | 4.2×10−3
| |
2 | 1.1×10−4
| |
3 | 4×10−5
| |
4 | 8×10−6
| |
5 | 7×10−6
| |
6 | 2×10−6
| |
256 | 0 | 0.22 |
1 | 0.0351 | |
2 | 1.18×10−3
| |
3 | 7.39×10−4
| |
4 | 4.97×10−4
| |
5 | 3.35×10−4
| |
6 | 1.37×10−4
| |
1000 | 0 | 0.6967 |
1 | 0.2725 | |
2 | 0.0494 | |
3 | 0.0262 | |
4 | 2.68×10−4
| |
5 | 6.96×10−5
| |
6 | 9×10−6
|
5.2 Comparison with other erasure correcting codes
5.2.1 Comparison with LDPC-Staircase codes
5.2.2 Comparison with another GLDPC code [26]
Code type | Decoding threshold |
---|---|
GLDPC-Staircase, Hankel-RS | 0.4993 |
GLDPC, BCH | 0.4967 |
5.2.3 Comparison with Raptor and RaptorQ codes
-
Case 1: K=32, N=128;
-
Case 2: K=256, N=1024;
-
Case 3: K=1024, N=3072.