Algebraic QC-LDPC Codes with Girth 6 and Free of Small Size - PowerPoint PPT Presentation

Algebraic QC-LDPC Codes with Girth 6 and Free of Small Size Elementary Trapping Sets Daniel Panario School of Mathematics and Statistics Carleton University daniel@math.carleton.ca Joint work with Farzane Amirzade (Shahrood University of Technology) and Mohammad-Reza Sadeghi (Amirkabir University of Technology) LAWCI – July 27, 2018

Introduction LDPC codes A low-density parity-check (LDPC) code is a linear code whose parity-check matrix is sparse. Tanner graph A Tanner graph is a bipartite graph with vertex sets formed by the set of variable nodes (VNs) and the set of check nodes (CNs). The adjacency matrix of the Tanner graph is the parity-check matrix of the code. If for each variable node v and each check node c we have deg ( v ) = γ and deg ( c ) = n , then the Tanner graph gives a ( γ, n ) -regular LDPC code. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 2 / 26

Example  1 0 0 0 1 1 0  H = 0 1 0 1 0 1 0  .  0 0 1 0 1 0 1 Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 3 / 26

A Tanner graph is an important representation of a code. The girth, that is, the length of the shortest cycles of the Tanner graph, has been known to influence the code performance. Another phenomenon that significantly influences the performance of LDPC codes is known as trapping sets. Empirical results in the literature show that among all trapping sets, the most harmful ones are the elementary trapping sets (ETSs) Trapping set An ( a , b ) trapping set of size a is an induced subgraph of the Tanner graph on a variable nodes and b check nodes of odd degrees. If the check nodes have degrees 1 or 2, then the subgraph is an elementary trapping set (ETS). Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 4 / 26

Variable Node graph In an ( a , b ) ETS, by removing all 1-degree check nodes and replacing every 2-degree check node with an edge, we obtain a graph with a vertices: the Variable Node (VN) graph. Example Suppose γ = 3. We denote variable nodes with circles and check nodes with squares. We depict a (5,3) ETS and its corresponding VN graph: Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 5 / 26

Quasi-cyclic LDPC codes Quasi-cyclic low-density parity-check codes (QC-LDPC codes) are an important category of LDPC codes. These codes are practical and have simple implementations. There is a large body of work using algebraic-based constructions for LDPC and QC-LDPC codes. A technique to avoid small trapping sets and to reduce the error floor of QC-LDPC codes is masking by which some non-zero elements of an exponent matrix (to be defined next) are turned into zero. Thus, the number of edges of the Tanner graph and the number of short cycles are reduced, and so some trapping sets are also removed. Our construction of these codes does not use masking or PEG algorithm. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 6 / 26

Quasi-cyclic LDPC codes Let N be a positive integer. Consider the following exponent matrix B = [ b ij ] , where b ij ∈ { 0 , 1 , . . . , N − 1 } or b ij = ∞ ,   b 00 b 01 b 0 ( n − 1 ) · · ·     b 10 b 11 b 1 ( n − 1 )  · · ·   . . .  B = ... (1)  . . .  .  . . .        b ( m − 1 ) 0 b ( m − 1 ) 1 b ( m − 1 )( n − 1 ) · · · We replace b ij � = ∞ by the N × N circulant permutation matrix (CPM) such that the nonzero 1-component of the top row is in the b ij -th position. Elements b ij = ∞ are replaced by the N × N zero matrix. The null space of this parity-check matrix gives a QC-LDPC code. If B contains no ∞ , then we have a fully connected QC-LDPC code. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 7 / 26

Characterizing trapping sets and their properties S. Laendner et al., IEEE Int. Symp. Turbo Codes and Iterative Inf. Process. (2010) A characterization of ( a , b ) trapping sets of ( 3 , n ) -regular LDPC codes from Steiner triple systems. Q. Huang et al., IEEE Int. Symp. Inform. Theory (ISIT) (2011) A binary ( γ, n ) -regular LDPC code with girth 6 contains no ( a , b ) trapping sets of size a ≤ γ , where b a < 1. F. Amirzade and M.R. Sadeghi, IEEE Trans. Commun. (2018) Tight lower bounds on the size of ETSs of LDPC codes with different girths were presented. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 8 / 26

Constructing LDPC codes free of small size trapping sets D. V. Nguyen et al., IEEE Int. Symp. Inform. Theory (ISIT) (2012) A progressive-edge-growth (PEG) algorithm to construct QC-LDPC codes free of some small trapping sets. M. Diouf et al., IEEE Int. Symp. Inf. Theory (ISIT) (2015) An improved PEG algorithm to construct ( 3 , n ) -regular LDPC codes with girth 8 whose Tanner graph is free of ( 5 , 3 ) trapping sets and contains a minimum number of ( 6 , 4 ) trapping sets. X. Tao et al., IEEE Commun. Letters (2017) Avoiding 8-cycles results in a fully-connected ( 3 , n ) -regular QC-LDPC codes with girth 8 and without ( a , b ) ETSs, where a ≤ 8 and b ≤ 3. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 9 / 26

Our results We give an algebraic construction of QC-LDPC codes with column weight 3 and girth 6 whose Tanner graphs are free of small ETSs. First, we present a new construction for the exponent matrix of QC-LDPC codes with girth at least 6 based on three multiplicative cyclic subgroups of the finite field F q . Then, we give a submatrix of the exponent matrix where we can prove that its Tanner graph is free of ( 4 , 0 ) and ( 4 , 2 ) ETSs. We also determine the existence of ETSs in the Tanner graph using edge-coloring. We show that the Tanner graph of a fully connected ( 3 , n ) -regular QC-LDPC code with girth 6 contains no ( 5 , 1 ) ETS. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 10 / 26

Edge-coloring of a Graph An edge coloring of a graph G is an assignment of colors (labels) to the edges of the graph so that no two adjacent edges have the same color (label). The minimum required number of colors for the edges of a given graph is the chromatic index of the graph denoted by X ′ ; it indicates that the graph has an X ′ -edge-coloring. Vizing’s Theorem If ∆( G ) is the maximum degree of a graph G , then ∆( G ) ≤ X ′ ≤ ∆( G ) + 1 . Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 11 / 26

Let us consider three VN graphs corresponding to ( 4 , 0 ) , ( 4 , 2 ) and ( 5 , 1 ) ETSs in an LDPC code with column weight 3: The maximum degree of each VN graph is 3. We use three colors u , v , w to color the edges. The ( 4 , 0 ) , ( 4 , 2 ) ETSs are 3-edge-coloring. The ( 5 , 1 ) ETS has no 3-edge-coloring. Vizing’s Theorem proves that it is 4-edge-coloring. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 12 / 26

Basic ideas of the proof: The degree of each vertex of the VN graph determines the number of rows of B which are involved in an ETS. Each edge of the VN graph is characterized by a row index of B . The existence of two adjacent edges with the same color indicates that a column of the parity-check matrix contains two 1-components which belong to a CPM which is impossible. Proposition Given a fully connected ( γ, n ) -regular QC-LDPC code, a necessary condition for the Tanner graph to contain an ( a , b ) ETS is that the VN graph has a γ -edge-coloring. Corollary Tanner graph of a fully connected ( 3 , n ) -regular QC-LDPC code with girth 6 contains no ( 5 , 1 ) ETS. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 13 / 26

Consider a fully connected QC-LDPC code with column weight 3. Every vertex of a VN graph corresponds to a column of B and each edge of a VN graph corresponds to a row of B . Example Let an exponent matrix B with N = 5 and row indices u , v , w :   0 0 0 0 B = 0 2 3 4  .  0 3 2 1 In the parity-check matrix H , 1-components related to a 6-cycle with variable nodes v 0 , v 11 , v 16 , v 0 and the check nodes c 0 , c 7 , c 10 have 3 labels u , v , w as their indices (see the next figure). Also, the edges of the VN graph are colored by these 3 labels. Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 14 / 26

 0 0 0 0  N = 5 , B = 0 2 3 4  ,  0 3 2 1   1 u .... 1 .... 1 u .... 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 ....     H = .. 1 .. .... 1 1 v .... . 1 v ... .     ... 1 . 1 .... . 1 ... .. 1 ..     .... 1 . 1 ... .. 1 .. ... 1 .     1 w .... ... 1 . .. 1 .. . 1 w ...     . 1 ... .... 1 ... 1 . .. 1 ..     .. 1 .. 1 .... .... 1 ... 1 .     ... 1 . . 1 ... 1 .... .... 1   .... 1 .. 1 .. . 1 ... 1 .... Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 15 / 26

Daniel Panario QC-LDPC Codes Free of ETS LAWCI – July 27, 2018 16 / 26