Finding Dominant Trapping Sets of Irregular LDPC codes
3.2 Definitions
The rest of the chapter is organized as follows. We present the definitions of various types of the TSs and the ASs in Section 3.2. Section 3.3 reviews some of the popular techniques to obtain the dominant TSs. One popular approach to find the dominant TSs known as the hierarchical approach, is further illustrated in detail in this section. Section 3.4 presents the proposed method of finding the dominant TSs of an irregular LDPC code. The results of the enumerations of the TSs and the ASs of various commonly considered irregular codes are presented in Section 3.5. The chapter is concluded with Section 3.6.
3.2 Definitions
Suppose H is the parity-check matrix of an LDPC code. The corresponding Tanner graph GH = (V ∪U, E) contains the sets V of the VNs, U of the CNs and E of the edges. Consider a subset T ⊂ V. Let N (T) ⊂ U denote the set of the neighboring CNs connected to the VNs in T. The subgraph ST induced by the VNs in T contains the nodes T ∪ N(T) and the edges {(v, c)∈E:v ∈T, c∈ N(T)}. Let E(T) andO(T) denote the even-degree and odd-degree CNs re- spectively present inST. With these notations, we first present the commonly considered definition of a TS [33, 41, 52].
Definition 3.1. A subset T of V is an (a, b) trapping set (TS) if |T| = a and |O(T)| = b. T is called elementary if all CNs in O(T) and E(T) are of degree 1 and 2, respectively (the degrees being measured in the context of the subgraph ST).
2 10 14 44 59
8 17 18 26 28 64 24 37 55
T
z }| {
| {z }
E(T)
| {z }
O(T)
Figure 3.1: A (5,3) TS from theN = 120, M = 64 code (120.64.3.109) [48]
A (5,3) TS from theN = 120, M = 64 code (120.64.3.109) available at [48] is shown in Figure 3.1.
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The sets T, E(T) and O(T) are also shown in the figure. This TS is of elementary type as all the CNs in O(T) and E(T) are of degree 1 and 2, respectively. The number of VNs present in a TS is usually referred to as the size of the TS. For example, the size of the (5,3) TS is 5. Although a TS denotes a set of VNs, we will use the term “trapping set” for both the set of VNs and the subgraph induced by the VNs interchangeably.
If all the bits or VNs in the TS are detected incorrectly then the even-degree CNs will be mis- satisfied and the odd-degree CNs will be unsatisfied. During the decoding process, the mis-satisfied CNs will send wrong-signed messages and the unsatisfied CNs will send correct-signed messages to- wards the neighboring VNs. If the numberbof unsatisfied CNs is very small compared to the number of mis-satisfied CNs, then a large number of incorrect messages will be flowing within the TS. In that case, the decoder will never be able to come out of the incorrect state even if the number of iterations is increased. In this way, if aand bare small and all the VNs of the TS are detected incorrectly, then it is almost impossible for the iterative decoder to correct the decoding decisions [33,39,40]. Such TSs with relatively small values of aand bare usually considered as the dominant ones.
Definition 3.1 specifies only the number of VNs |T| = a and the number of odd-degree CNs
|O(T)|=b. The even-degree CNs in E(T) also play a crucial role in determining the harmfulness of a TS. In order to consider the effect of the even-degree CNs, a modified definition of a TS has been used in [41].
Definition 3.2. A subsetT of V is an (a, b) TS if the following conditions are satisfied:
(i) |T|=a.
(ii) |O(T)|=b.
(iii) Each VN of T with a degree more than 2 is connected to at least two CNs in E(T).
(iv) Each VN of T with a degree equal to 2 is connected to at least one CN inE(T).
The (5,3) TS shown in Figure 3.1 is a legitimate TS according to Definition 3.2 as all the VNs in T are of degree 3 and are connected to at least two CNs in E(T). In the rest of the chapter, Definition 3.2 will be used for a TS unless otherwise specified.
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3.2 Definitions
Dolecek et al. have formulated the concept of the absorbing set (AS) [24] which is defined in the following:
Definition 3.3. An(a, b) trapping setT is called an absorbing set (AS) if each VN inT is connected to strictly more CNs in E(T) than in O(T). An absorbing set T is called a fully absorbing set (FAS) if each VN in V \T is connected to strictly more CNs in U \ O(T) than in O(T).
The (5,3) TS in Figure 3.1 is an AS as each VN is connected to strictly more even-degree CNs than odd-degree CNs. To check whether an AS is an FAS or not, the whole Tanner graph needs to be examined. Note that any AS is a TS but the converse is not always true. The ASs form a subset of the collection of TSs. Each VN in an AS receives strictly more incorrect messages than correct messages. Consequently, the chances of the reversal of the erroneous decisions at those VNs become low. Hence, the ASs are the most detrimental classes of TSs.
C8 C6
C10 24
14 17 2
28
26
10
37
18 59 44
55
8 64
Figure 3.2: A different representation of the (5,3) TS in Figure 3.1
Almost each dominant TS contains one or more short cycles [41, 79]. For example, consider the (5,3) TS shown in Figure 3.1. For better visualization of the cycles, the TS is redrawn in Figure 3.2.
It can be observed that the (5,3) TS contains two cycles: C6 of length 6 (2-28-10-26-59-64) and C8 of length 8 (14-17-2-64-59-18-44-8). Observe that the subgraph in Figure 3.2 contains an outer cycle C10 of length 10 (14-17-2-28-10-26-59-18-44-8). However, the nodes and the edges inC10 are already included in C6 and C8. Therefore, the TS is considered to be constituted by only C6 and C8.
Note that any codeword of weight a can be considered as an (a,0) TS as there is no odd-degree CN involved. Therefore, the codewords can be considered as special type of TSs. The next section briefly reviews the various methods available in the literature to find the dominant TSs.