On the Equality of the ACE and the EMD of a Cycle for ACE Spectrum
4.3 Sufficient Conditions for the Equality of the ACE and the EMD of a Cycle for the ACE Spectrum Constrained Codes
As described in Section 4.1, for the ACE spectrum constrained LDPC codes, the cycles of length 2i (i= 1, . . . , dACE) are constrained to have the ACE values at leastη2i. These constraints prevent certain cycles from containing sub-cycles. Analyzing these constraints, we derive three sufficient conditions for the equality of the ACE and the EMD of a cycle for the ACE spectrum constrained codes.
A cycle with its ACE not equal to EMD, contains some sub-cycles. These sub-cycles have the following two constraints:
(i) Girth constraint: The length of each sub-cycle must be greater than or equal to the girth.
(ii) ACE constraint: The ACE values of the sub-cycles must be greater than or equal to their corresponding ACE constraints.
Using the girth constraint first, a sufficient condition for the equality of the ACE and the EMD of a cycle is derived as follows.
Theorem 4.1. For an LDPC code with girth g, any cycle C2i of length2i has equal ACE and EMD if i < g−2.
v1
Cg1 c1 Cg2
v2
C2imin (outer cycle)
Figure 4.4: A cycle containing two sub-cycles of minimum possible length, i.e., girthg Proof. Consider a cycle C2imin involving the minimum number of VNs such that it contains two legitimate sub-cycles. The minimum number of VNs that can be present in a legitimate cycle is g/2,
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4.3 Sufficient Conditions for the Equality of the ACE and the EMD of a Cycle for the ACE Spectrum Constrained Codes
where g is the girth of the Tanner graph. This situation is depicted in Figure 4.4. The two VNs v1 andv2 share a CNc1 which is not present inC2imin. This results in the formation of two sub-cyclesCg1 and Cg2 of lengthg each. The number of VNs present in each ofCg1 and Cg2 is g/2. The total number of VNs present in the outer cycle C2imin is now given by
imin= g 2 +g
2 −2
=g−2
Therefore, a cycle C2i of length 2i, cannot contain any sub-cycle if i < imin = g−2. In other words, the ACE and EMD ofC2i will be equal ifi < g−2.
Note that the sufficient condition in Theorem 4.1 is valid for any general LDPC code as it is related to the girth only. Recall from Section 2.5.2 that, for the ACE constrained codes, if the number of degree-2 VNs is less than the total number of the CNs, then any cycle must contain at least one VN with degree more than two [70]. In that case, the minimum possible ACE of any cycle for the ACE constrained codes is 1. The following theorem says that a cycle with the minimum possible ACE has the same ACE and EMD.
Theorem 4.2. For any cycleC2i of length 2i, ACE (C2i) is equal to EMD (C2i) if ACE (C2i) = 1.
Proof. Figure 4.5 shows a cycle C2i whose ACE value is equal to 1. Observe thatC2i contains (i−1) v1
v2 c1
C2i (outer cycle)
Figure 4.5: A cycle with ACE equal to 1
number of degree-2 VNs and only one VN v1 of degree 3. In that case, the CN c1 connected to v1
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(but not involved in the cycle) cannot be shared by any of the remaining VNs like v2 as they are of degree 2. Thus, C2i cannot contain any sub-cycle which, in turn, implies that ACE (C2i) is equal to the EMD (C2i).
Next we derive the sufficient condition for ACE=EMD of a cycle for the ACE spectrum constrained LDPC codes using the ACE constraints of the sub-cycles. Let ACE(s)min(C2i) denote the minimum ACE required for C2i to have some VNs sharing s number of CNs outside C2i. ACE(s)min(C2i) satisfies the following lemma which says that ACE(s)min(C2i) is a non-decreasing function of s.
Lemma 4.2. ACE(s)min(C2i)≤ACE(s+1)min (C2i)
Proof. Consider the cycle C2i drawn in the left of Figure 4.6. It does not show the CNs present in the cycle. Therefore, the entire cycle is drawn in a dashed form. The VNs in C2i share (s+ 1)
C2i c1 c2
cs
cs+1 v1
v2
C2i
c1 c2 cs
cs+2
v1
v2 Removal ofcs+1
cs+3
b b b b b b b b b b b b
Figure 4.6: Removal of a CN shared outside of a cycle by two VNs involved in the cycle
number of CNs (c1, c2, . . . , cs+1) outside C2i with the minimum possible ACE value, ACE(s+1)min (C2i).
The configuration ofC2i can be changed to have snumber of CNs shared outside the cycle by deleting any one CN out of the (s+ 1) CNs. Let this CN becs+1and let it be originally connected to the VNs v1 and v2 involved inC2i. The removal of cs+1 can be done as follows: delete cs+1 and put the edges fromv1 tocs+2 and fromv2 tocs+3, wherecs+2 andcs+3are any two CNs not present in the subgraph induced by the VNs in C2i. After the removal of cs+1, we get a modified subgraph as shown in the right of Figure 4.6. The removal of cs+1 will delete some of the sub-cycles with no further creation of new cycles in the subgraph. Moreover, as the number of edges coming out of each VN involved
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4.3 Sufficient Conditions for the Equality of the ACE and the EMD of a Cycle for the ACE Spectrum Constrained Codes
in the cycle is unaltered, the degree of each of the VNs remains fixed. Hence, the ACE values of all the remaining cycles in the modified subgraph will remain the same according to (4.1). Consequently, they will continue to satisfy their corresponding ACE constraints even after the removal ofcs+1. This fact ensures that the cycle C2i with the modified subgraph is a valid one for the code with the given ACE spectrum. The ACE ofC2i with the modified subgraph is the same as that of the cycle with the original subgraph, i.e., ACE(s+1)min (C2i). Hence, ACE(s)min(C2i) has to be less than or equal to the ACE of the cycle with the present configuration, i.e.,
ACE(s)min(C2i)≤ACE(s+1)min (C2i)
Using Lemma 4.2, a sufficient condition for the equality of the ACE and the EMD of a cycle for the ACE spectrum constrained codes is derived in the following. This condition will be relevant only when the sufficient conditions in Theorem 4.1 and Theorem 4.2 are not satisfied.
Theorem 4.3. For an ACE spectrum constrained code with the ACE spectrum (η2, η4,· · · , η2dACE) and girth g, ACE (C2i) is equal to EMD (C2i), i≥g−2, if
ACE (C2i)< min
j∈{g2,g2+1,···,⌈i−12 ⌉+1}
η2j+η2(i−j+2)
2 . (4.2)
Proof. The ACE of a cycle is exactly equal to its EMD if the cycle does not contain any sub-cycle as per Lemma 4.1. In that case, no VNs in the cycle share any CN outside the cycle. According to Lemma 4.2, if ACE (C2i) < ACE(1)min(C2i), then ACE (C2i) < ACE(s)min(C2i),∀s ≥ 1. It means that if ACE (C2i)<ACE(1)min(C2i), then the VNs in the cycle cannot share any CN outside the cycle and consequently we must have ACE (C2i) = EMD (C2i). Hence, the sufficient condition for the equality of the ACE and the EMD of C2i is
ACE (C2i)<ACE(1)min(C2i). (4.3) In the following, the expression for ACE(1)min(C2i) will be derived.
Let the VNs involved in the cycle C2i be denoted byv1, v2, . . . , vi−1 and vi. Consider a situation where two VNs in C2i share a CN outside C2i and other VNs do not. Without loss of generality, let these two VNs bev1 and vj. The situation is shown in Figure 4.7. The two sub-cycles produced in this case can have lengths 2j and 2(i−j+ 2) for j ∈ ng2,g2 + 1,· · ·,li−12 m+ 1o. Let these two
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C2i (outer cycle)
C2(i−j+2)
C2j v1
v2
vj
vi
Figure 4.7: Formation of two sub-cycles due to the sharing of one CN outside a cycle of length 2i
cycles be denoted by C2j and C2(i−j+2). C2j and C2(i−j+2) contain the VNs {v1, v2, . . . , vj} and {v1, vj, vj+1, . . . , vi} respectively.
Using (4.1), the ACE values of the cyclesC2j and C2(i−j+2) can be written as
ACE (C2j) = Xj
k=1
(dvk−2) (4.4)
and ACEC2(i−j+2)= (dv1 −2) + Xi
k=j
(dvk−2). (4.5)
The cyclesC2j and C2(i−j+2) must satisfy their corresponding ACE constraints. Therefore, Xj
k=1
(dvk −2)≥η2j (4.6)
and (dv1−2) + Xi
k=j
(dvk −2)≥η2(i−j+2). (4.7)
The VNs v1 and vj involved in C2i share a CN outside C2i. Therefore, the degrees of v1 and vj are at least 3. For the other VNs not sharing any CN outsideC2i, the degree is greater than or equal to 2. Therefore,
(dvr −2)≥0 ∀r∈ {2, . . . , i} \ {j}. (4.8)
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4.3 Sufficient Conditions for the Equality of the ACE and the EMD of a Cycle for the ACE Spectrum Constrained Codes
Adding (4.6), (4.7) and (4.8) gives
2 Xi
k=1
(dvk−2)≥η2j +η2(i−j+2)
⇒ACE (C2i)≥
η2j+η2(i−j+2) 2
. (4.9)
Therefore, any two VNs possibly can share a CN outside C2i only if (4.9) is satisfied for any j∈ng2,g2 + 1,· · · ,li−12 m+ 1o.
Thus, ACE(1)min(C2i) is given by
ACE(1)min(C2i) = min
j∈{g2,g2+1,···,⌈i−12 ⌉+1}
η2j+η2(i−j+2) 2
. (4.10)
Using (4.10) in (4.3), the sufficient condition for the equality of the ACE and the EMD can be written as
ACE (C2i)< min
j∈{g2,g2+1,···,⌈i−12 ⌉+1}
η2j+η2(i−j+2) 2
. (4.11)
As the ACE values are integers only, we have equivalently ACE (C2i)< min
j∈{g2,g2+1,···,⌈i−12 ⌉+1}
η2j+η2(i−j+2)
2 . (4.12)
Using Theorem 4.3, the following result is derived.
Result 4.1. For an(∞,∞, η6, η8, η10) code,ACE (C8) andACE (C10)will be equal to the correspond- ing EMDs if ACE (C8)< η6 and ACE (C10)<(η6+η8)/2.
Proof. In this case, the girth g = 6. Therefore, in (4.2), j can take only one value which is g/2 = 3.
Then, the proof directly follows from (4.2).
In the following, we explain the sufficient conditions with the help of some examples.
Example 4.3. Consider an LDPC code of girth,g= 10. All the cycles of length10 (i= 5),12 (i= 6) and 14 (i= 7) satisfy i < g−2. Hence, these cycles have equal respective ACE and EMD according to Theorem 4.1.
Example 4.4. Suppose, for a cycle C20 of length 20, ACE (C20) = 1. Then, by Theorem 4.2, ACE (C20) = EMD (C2i).
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Example 4.5. Consider an (∞,∞,13,9,4) code. All the cycles of length 8 with the ACE value less than 13, will have equal ACE and EMD according to Result 4.1. For all the cycles of length10 having ACE value less than 11 ((13 + 9)/2), the ACE will be equal to the EMD.
Remark 4.1. The EMD of a cycle measures its connectivity with the rest of the Tanner graph. If the EMD is low, the cycle is harmful to iterative decoding. The calculation of the EMD is not easy as we have to go for the extra step of examining the entire subgraph induced by the VNs in the cycle. On the other hand, the ACE can be easily computed from the degrees of the VNs in the cycle. Sometimes, the ACE is not equal to the EMD and in those cases, ACE may not reflect the detrimental effect correctly. Thus, the sufficient conditions presented in this section can help one to assertively asses the detrimental effect of a cycle based on its ACE value.