Analysis of Absorbing Sets Using Cosets and Syndromes IEEE - - PowerPoint PPT Presentation

Analysis of Absorbing Sets Using Cosets and Syndromes

IEEE International Symposium on Information Theory

Emily McMillon∗ Allison Beemer† Christine A. Kelley∗

∗University of Nebraska – Lincoln †University of Wisconsin – Eau Claire

June 2020

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 1 / 28

Outline

1

Background

2

Coset and subspace characterization

3

Syndrome weight characterization

4

Fully absorbing set search algorithm

5

Conclusions

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 2 / 28

Background

LDPC codes and decoding

Low-density parity-check (LDPC) codes [Gallager, ’62] Characterized by sparse parity-check matrices

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 3 / 28

Background

LDPC codes and decoding

Low-density parity-check (LDPC) codes [Gallager, ’62] Characterized by sparse parity-check matrices Have sparse bipartite graph representations [Tanner, ’81]

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 3 / 28

Background

LDPC codes and decoding

Let C be the [6, 3] binary linear code given by the parity check matrix H below left. Then the corresponding Tanner graph is the below right graph. H =           1 1 1 1 1 1 1 1 1          

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 4 / 28

Background

LDPC codes and decoding

Low-density parity-check (LDPC) codes [Gallager, ’62] Characterized by sparse parity-check matrices Have sparse bipartite graph representations [Tanner, ’81] May be decoded efficiently using iterative decoding algorithms

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 5 / 28

Background

LDPC codes and decoding

Low-density parity-check (LDPC) codes [Gallager, ’62] Characterized by sparse parity-check matrices Have sparse bipartite graph representations [Tanner, ’81] May be decoded efficiently using iterative decoding algorithms These algorithms are suboptimal due to certain substructures in the code’s graph representation [Richardson ’01, Di et al. ’02]

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 5 / 28

Background

Absorbing sets

Have been shown to characterize decoder failure on BSC and AWGN channels [Dolecek et al, ’07].

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 6 / 28

Background

Absorbing sets

Have been shown to characterize decoder failure on BSC and AWGN channels [Dolecek et al, ’07]. Definition An (a, b)-absorbing set is a subset D ⊆ V of variable nodes in a code’s Tanner graph G = (V, W; E) such that

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 6 / 28

Background

Absorbing sets

Have been shown to characterize decoder failure on BSC and AWGN channels [Dolecek et al, ’07]. Definition An (a, b)-absorbing set is a subset D ⊆ V of variable nodes in a code’s Tanner graph G = (V, W; E) such that #D = a,

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 6 / 28

Background

Absorbing sets

Have been shown to characterize decoder failure on BSC and AWGN channels [Dolecek et al, ’07]. Definition An (a, b)-absorbing set is a subset D ⊆ V of variable nodes in a code’s Tanner graph G = (V, W; E) such that #D = a, #O(D) = b, where O(D) is the subset of check nodes of odd degree in the subgraph induced by D ∪ N(D)

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 6 / 28

Background

Absorbing sets

Have been shown to characterize decoder failure on BSC and AWGN channels [Dolecek et al, ’07]. Definition An (a, b)-absorbing set is a subset D ⊆ V of variable nodes in a code’s Tanner graph G = (V, W; E) such that #D = a, #O(D) = b, where O(D) is the subset of check nodes of odd degree in the subgraph induced by D ∪ N(D) and each variable node in D has strictly fewer neighbors in O(D) than W \ O(D).

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 6 / 28

Background

Absorbing sets – example

Below is an example of a subgraph induced by a (4, 3)-absorbing set.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 7 / 28

Background

Cosets and syndromes

Recall the following definitions.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 8 / 28

Background

Cosets and syndromes

Recall the following definitions. Definition Given an [n, k] linear code C over Fq, its distinct cosets x + C partition Fn

q into qn−k sets of size qk. The weight of a coset is the smallest

weight of a vector in a coset, and any vector of this smallest weight in the coset is called a coset leader.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 8 / 28

Background

Cosets and syndromes

Recall the following definitions. Definition Given an [n, k] linear code C over Fq, its distinct cosets x + C partition Fn

q into qn−k sets of size qk. The weight of a coset is the smallest

weight of a vector in a coset, and any vector of this smallest weight in the coset is called a coset leader. Definition Given a parity check matrix H for an [n, k] linear code C over Fq, the syndrome of a vector x in Fn

q is the vector in Fn−k q

defined by syn(x) = HxT.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 8 / 28

Background

Standard array

Definition Given a parity check matrix H for an [n, k] binary linear code C, its standard array is a 2k column by 2n−k row table such that

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 9 / 28

Background

Standard array

Definition Given a parity check matrix H for an [n, k] binary linear code C, its standard array is a 2k column by 2n−k row table such that the first row is all the codewords of C,

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 9 / 28

Background

Standard array

Definition Given a parity check matrix H for an [n, k] binary linear code C, its standard array is a 2k column by 2n−k row table such that the first row is all the codewords of C, the first column is made up of coset leaders of C,

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 9 / 28

Background

Standard array

Definition Given a parity check matrix H for an [n, k] binary linear code C, its standard array is a 2k column by 2n−k row table such that the first row is all the codewords of C, the first column is made up of coset leaders of C, and each of the remaining entries is the sum of the row’s coset leader and the column’s codeword.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 9 / 28

Background

Standard array – example

Let C be the [6, 3] binary linear code given by the parity check matrix H below left. Then the corresponding Tanner graph is the below right graph. H =           1 1 1 1 1 1 1 1 1          

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 10 / 28

Background

Standard array – example

The code defined on the previous slide has standard array as shown below.

000000 001110 010101 011011 100011 101101 110110 111000 000 000001 001111 010100 011010 100010 101100 110111 111001 110 000010 001100 010111 011001 100001 101111 110100 111010 101 000100 001010 010001 011111 100111 101001 110010 111100 011 001000 000110 011101 010011 101011 100101 111110 110000 001 010000 011110 000101 001011 110011 111101 100110 101000 010 100000 101110 110101 111011 000011 001101 010110 011000 100 001001 000111 011100 010010 101010 100100 111111 110001 111

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 11 / 28

Background

Standard array – example

The code defined on the previous slide has standard array as shown

• below. This code has 23 = 8 total coset leaders.

000000 001110 010101 011011 100011 101101 110110 111000 000 000001 001111 010100 011010 100010 101100 110111 111001 110 000010 001100 010111 011001 100001 101111 110100 111010 101 000100 001010 010001 011111 100111 101001 110010 111100 011 001000 000110 011101 010011 101011 100101 111110 110000 001 010000 011110 000101 001011 110011 111101 100110 101000 010 100000 101110 110101 111011 000011 001101 010110 011000 100 001001 000111 011100 010010 101010 100100 111111 110001 111

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 11 / 28

Background

Standard array – example

The code defined on the previous slide has standard array as shown

• below. This code has 23 = 8 total coset leaders.

000000 001110 010101 011011 100011 101101 110110 111000 000 000001 001111 010100 011010 100010 101100 110111 111001 110 000010 001100 010111 011001 100001 101111 110100 111010 101 000100 001010 010001 011111 100111 101001 110010 111100 011 001000 000110 011101 010011 101011 100101 111110 110000 001 010000 011110 000101 001011 110011 111101 100110 101000 010 100000 101110 110101 111011 000011 001101 010110 011000 100 001001 000111 011100 010010 101010 100100 111111 110001 111

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 11 / 28

Background

Standard array – example

The code defined on the previous slide has standard array as shown

• below. This code has 23 = 8 total coset leaders.

000000 001110 010101 011011 100011 101101 110110 111000 000 000001 001111 010100 011010 100010 101100 110111 111001 110 000010 001100 010111 011001 100001 101111 110100 111010 101 000100 001010 010001 011111 100111 101001 110010 111100 011 001000 000110 011101 010011 101011 100101 111110 110000 001 010000 011110 000101 001011 110011 111101 100110 101000 010 100000 101110 110101 111011 000011 001101 010110 011000 100 001001 000111 011100 010010 101010 100100 111111 110001 111

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 11 / 28

Background

Standard array – example

The code defined on the previous slide has standard array as shown

• below. This code has 23 = 8 total coset leaders.

000000 001110 010101 011011 100011 101101 110110 111000 000 000001 001111 010100 011010 100010 101100 110111 111001 110 000010 001100 010111 011001 100001 101111 110100 111010 101 000100 001010 010001 011111 100111 101001 110010 111100 011 001000 000110 011101 010011 101011 100101 111110 110000 001 010000 011110 000101 001011 110011 111101 100110 101000 010 100000 101110 110101 111011 000011 001101 010110 011000 100 001001 000111 011100 010010 101010 100100 111111 110001 111

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 11 / 28

Background

Standard array – example

The code defined on the previous slide has standard array as shown

• below. This code has 23 = 8 total coset leaders.

000000 001110 010101 011011 100011 101101 110110 111000 000 000001 001111 010100 011010 100010 101100 110111 111001 110 000010 001100 010111 011001 100001 101111 110100 111010 101 000100 001010 010001 011111 100111 101001 110010 111100 011 001000 000110 011101 010011 101011 100101 111110 110000 001 010000 011110 000101 001011 110011 111101 100110 101000 010 100000 101110 110101 111011 000011 001101 010110 011000 100 001001 000111 011100 010010 101010 100100 111111 110001 111

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 11 / 28

Background

Absorbing set support vector

Definition For a subset S of variable nodes, we define xS ∈ Fn

2 to be the word

whose support is equal to S; we refer to xS as the support vector of S. So xD will denote the support vector of an absorbing set D.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 12 / 28

Background

Absorbing set support vector

Definition For a subset S of variable nodes, we define xS ∈ Fn

2 to be the word

whose support is equal to S; we refer to xS as the support vector of S. So xD will denote the support vector of an absorbing set D. Goal: Use the syndrome and coset of an absorbing set support vector to find and classify them.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 12 / 28

Background

Example

Let C be the [9, 3] binary linear code given by the parity check matrix H below. H =                           1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                          

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 13 / 28

Background

Example

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 14 / 28

Background

Example

Let D be the (3, 1)-absorbing set below.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 14 / 28

Background

Example

Let D be the (3, 1)-absorbing set below. 1 1 1 The support vector of D is xD = 000111000.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 14 / 28

Background

Example

Let D be the (3, 1)-absorbing set below. 1 1 1 The support vector of D is xD = 000111000. The syndrome HxT

D = 100000 and corresponds exactly to the

unsatisfied check nodes in the graph.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 14 / 28

Background

Example

Let D be the (3, 1)-absorbing set below. 1 1 1 The support vector of D is xD = 000111000. The syndrome HxT

D = 100000 and corresponds exactly to the

unsatisfied check nodes in the graph. Note that b = wt(HxT

D).

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 14 / 28

Coset and subspace characterization

Coset and subspace characterization

Theorem The set of absorbing set support vectors that belong to a single coset

• f a code C is either empty or has the form xD + D, where D is a

subspace of C, and D is any absorbing set in that coset.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 15 / 28

Coset and subspace characterization

Coset and subspace characterization

Theorem The set of absorbing set support vectors that belong to a single coset

• f a code C is either empty or has the form xD + D, where D is a

subspace of C, and D is any absorbing set in that coset. Corollary For binary codes, the number of absorbing set support vectors appearing in any coset is either zero or 2i for some i ≤ k.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 15 / 28

Coset and subspace characterization

Especially harmful absorbing sets

Definition An (a, b)-absorbing set is

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 16 / 28

Coset and subspace characterization

Especially harmful absorbing sets

Definition An (a, b)-absorbing set is fully absorbing if it has the property that each variable node not in D has strictly fewer neighbors in O(D) than in W \ O(D).

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 16 / 28

Coset and subspace characterization

Especially harmful absorbing sets

Definition An (a, b)-absorbing set is fully absorbing if it has the property that each variable node not in D has strictly fewer neighbors in O(D) than in W \ O(D). elementary if it is an absorbing set such that all check nodes in N(D) have degree 1 or 2 in the subgraph induced by D ∪ N(D).

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 16 / 28

Coset and subspace characterization

Especially harmful absorbing sets

Definition An (a, b)-absorbing set is fully absorbing if it has the property that each variable node not in D has strictly fewer neighbors in O(D) than in W \ O(D). elementary if it is an absorbing set such that all check nodes in N(D) have degree 1 or 2 in the subgraph induced by D ∪ N(D). low-weight if a < dmin(C).

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 16 / 28

Coset and subspace characterization

Coset characterization of fully absorbing sets

Theorem If D is a fully absorbing set, then the support of every word in the coset containing xD is also a fully absorbing set.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 17 / 28

Coset and subspace characterization

Coset characterization of fully absorbing sets

Theorem If D is a fully absorbing set, then the support of every word in the coset containing xD is also a fully absorbing set. Corollary An absorbing set D is fully absorbing if and only if all vectors in xD + C correspond to fully absorbing sets.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 17 / 28

Syndrome weight characterization

Syndrome weight characterization

A syndrome condition on fully absorbing sets is given in [Dolecek et al ’10].

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 18 / 28

Syndrome weight characterization

Syndrome weight characterization

A syndrome condition on fully absorbing sets is given in [Dolecek et al ’10]. We first give the following straightforward generalization. Theorem A subset D ⊆ V is an absorbing set if and only if for all v ∈ D, wt(HxT

D\v) > wt(HxT D) = b.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 18 / 28

Syndrome weight characterization

Syndrome weight bounds for low weight and fully

Let dv denote the degree of variable node v. Let dc denote the degree of check node c.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 19 / 28

Syndrome weight characterization

Syndrome weight bounds for low weight and fully

Let dv denote the degree of variable node v. Let dc denote the degree of check node c. Lemma Any vector corresponding to a low-weight absorbing set has syndrome weight at most maxv dv − 1 2

• (dmin(C) − 1).

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 19 / 28

Syndrome weight characterization

Syndrome weight bounds for low weight and fully

Let dv denote the degree of variable node v. Let dc denote the degree of check node c. Lemma Any vector corresponding to a low-weight absorbing set has syndrome weight at most maxv dv − 1 2

• (dmin(C) − 1).

Lemma Any vector corresponding to a fully absorbing set has syndrome weight at most maxv dv − 1 2

• ·

n minc dc .

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 19 / 28

Syndrome weight characterization

Sufficient condition to be fully absorbing

For small enough syndrome weights, we can guarantee fully absorbing set vectors.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 20 / 28

Syndrome weight characterization

Sufficient condition to be fully absorbing

For small enough syndrome weights, we can guarantee fully absorbing set vectors. Theorem If s is a syndrome vector, and wt(s) ≤ minv dv − 1 2

• ,

where dv is the degree of variable node v, then the support of every word in the corresponding coset is fully absorbing.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 20 / 28

Syndrome weight characterization

Example

Consider again the [9, 3] binary linear code C given by the parity check matrix H below. (Same example as previously.) H =                           1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                          

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 21 / 28

Syndrome weight characterization

Standard array observations

Blue Fill Absorbing Set Green Fill Elementary Absorbing Set Red Fill Fully Absorbing Set Purple Fill Fully and Elementary Absorbing Set White Text Low-Weight Absorbing Set

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 22 / 28

Fully absorbing set search algorithm

Searching for fully absorbing sets

We provide an algorithm for finding fully absorbing sets using the syndromes and coset analysis.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 23 / 28

Fully absorbing set search algorithm

Searching for fully absorbing sets

We provide an algorithm for finding fully absorbing sets using the syndromes and coset analysis. The following result restricts the search space during the algorithm.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 23 / 28

Fully absorbing set search algorithm

Searching for fully absorbing sets

We provide an algorithm for finding fully absorbing sets using the syndromes and coset analysis. The following result restricts the search space during the algorithm. Theorem If a coset with syndrome s is not fully absorbing, then any absorbing set with syndrome support containing supp(s) is also not fully absorbing.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 23 / 28

Fully absorbing set search algorithm

Algorithm

Algorithm 1: Fully Absorbing Set Search Algorithm

Input : H ∈ Fm×n

2

Output: L = {ys ∈ Fn

2 :

ys is a fully absorbing set}

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 24 / 28

Fully absorbing set search algorithm

Algorithm

Algorithm 1: Fully Absorbing Set Search Algorithm

Input : H ∈ Fm×n

2

Output: L = {ys ∈ Fn

2 :

ys is a fully absorbing set}

1 Initialization: 2

L ← ∅, B ← ∅

3

t1 ← (mini{wt(hi)} − 1) /2

4

t2 ← (maxi{wt(hi)} − 1) /2 (n/mini{wt(ri)})

5

i ← 1

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 24 / 28

Fully absorbing set search algorithm

Algorithm

Algorithm 1: Fully Absorbing Set Search Algorithm

Input : H ∈ Fm×n

2

Output: L = {ys ∈ Fn

2 :

ys is a fully absorbing set}

1 Initialization: 2

L ← ∅, B ← ∅

3

t1 ← (mini{wt(hi)} − 1) /2

4

t2 ← (maxi{wt(hi)} − 1) /2 (n/mini{wt(ri)})

5

i ← 1

6 while i ≤ t1 do 7

foreach s ∈ Fm

2 s.t. wt(s) = i do 8

L ← L ∪ {ys}

9

end

10

i ← i + 1

11 end Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 24 / 28

Fully absorbing set search algorithm

Algorithm

Algorithm 1: Fully Absorbing Set Search Algorithm

Input : H ∈ Fm×n

2

Output: L = {ys ∈ Fn

2 :

ys is a fully absorbing set}

1 Initialization: 2

L ← ∅, B ← ∅

3

t1 ← (mini{wt(hi)} − 1) /2

4

t2 ← (maxi{wt(hi)} − 1) /2 (n/mini{wt(ri)})

5

i ← 1

6 while i ≤ t1 do 7

foreach s ∈ Fm

2 s.t. wt(s) = i do 8

L ← L ∪ {ys}

9

end

10

i ← i + 1

11 end 12 while i ≤ t2 do 13

foreach s ∈ Fm

2 s.t. wt(s) = i, s B

do

14

if ys fully absorbing then

15

L ← L ∪ {ys}

16

end

17

else

18

B ← B ∪ {s} ∪ {s′ ∈ Fm

2 :

supp(s) ⊆ supp(s′)}

19

end

20

end

21

i ← i + 1

22 end Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 24 / 28

Conclusions

Conclusions

We have provided:

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 25 / 28

Conclusions

Conclusions

We have provided: Subspace, coset, and syndrome weight characterizations that give a new avenue for finding absorbing sets, including special classes

• f absorbing sets.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 25 / 28

Conclusions

Conclusions

We have provided: Subspace, coset, and syndrome weight characterizations that give a new avenue for finding absorbing sets, including special classes

• f absorbing sets.

An initial search algorithm for finding fully absorbing sets based on these methods.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 25 / 28

Conclusions

Current/future work

We are currently working on: Examining how redundancy affects the presence/location of various types of absorbing sets in the array.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 26 / 28

Conclusions

Current/future work

We are currently working on: Examining how redundancy affects the presence/location of various types of absorbing sets in the array. Further characterization of where low-weight absorbing sets occur in the array.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 26 / 28

Conclusions

Current/future work

We are currently working on: Examining how redundancy affects the presence/location of various types of absorbing sets in the array. Further characterization of where low-weight absorbing sets occur in the array. Characterizing the subspaces of a code that correspond to cosets containing absorbing set vectors.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 26 / 28

Conclusions

Current/future work

We are currently working on: Examining how redundancy affects the presence/location of various types of absorbing sets in the array. Further characterization of where low-weight absorbing sets occur in the array. Characterizing the subspaces of a code that correspond to cosets containing absorbing set vectors. Determining classes of codes for which the proposed search algorithm is practical.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 26 / 28

Conclusions

Current/future work

We are currently working on: Examining how redundancy affects the presence/location of various types of absorbing sets in the array. Further characterization of where low-weight absorbing sets occur in the array. Characterizing the subspaces of a code that correspond to cosets containing absorbing set vectors. Determining classes of codes for which the proposed search algorithm is practical. In other work, we give upper bounds on b that may be used to

• btain a range of syndrome values that need to be considered.

[M., Beemer, Kelley, submitted]

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 26 / 28

Conclusions

Selected references

1

Gallager, R., “Low-density parity-check codes,” 1962.

2

Tanner, R., “A recursive approach to low complexity codes,” 1981.

3

Di et al., “Finite-length analysis of low-density parity-check codes on the binary erasure channel,” 2002.

4

Richardson, T., “Error floors of LDPC codes,” 2003.

5

Dolecek et al., “Analysis of absorbing sets for array-based LDPC codes,” 2007.

6

Dolecek et al., “Analysis of absorbing sets and fully absorbing sets of array-based LDPC codes,” 2010.

7

Beemer, A., McMillon, E., Kelley, C. A., “Extremal absorbing sets in low-density parity-check codes,” Submitted.

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 27 / 28

Conclusions

Thank you!

Emily McMillon Absorbing Sets, Cosets, and Syndromes June 2020 28 / 28