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 Background 1 Coset and subspace characterization 2 Syndrome weight characterization 3 Fully absorbing set search algorithm 4 Conclusions 5 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.  1 0 0 0 1 1        H = 0 1 0 1 0 1           0 0 1 1 1 0   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 F q , its distinct cosets x + C partition q into q n − k sets of size q k . The weight of a coset is the smallest F n 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 F q , its distinct cosets x + C partition q into q n − k sets of size q k . The weight of a coset is the smallest F n 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 F q , the syndrome of a vector x in F n q is the vector in F n − k defined by q syn ( x ) = H x T . 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 2 k column by 2 n − 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 2 k column by 2 n − 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 2 k column by 2 n − 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 2 k column by 2 n − 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.  1 0 0 0 1 1        H = 0 1 0 1 0 1           0 0 1 1 1 0   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 2 3 = 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 2 3 = 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