A Trellis-Based Method for Removing Cycles From Bipartite Graphs and Construction of Low Density Parity Check Codes

Practical algorithms for tornado codes

“…Fig.2 Cycles in the incidence matrix In Fig.2, the combinations of "1" and their liner-transformations represent the cycles in tornado code graph, where the suspension points substitute the other matrix elements so that the cycles can be shown clearly. It is not difficult to find that the shortest cycle which has the most effect on decoding efficiency has four edges [8] , e.g., the special formation in Fig.2(a). Therefore, to improve the decoding efficiency, the shortest cycle should be avoided.…”

A Cycle Search Algorithm Based on a Message-Passing for the Design of Good LDPC Codes

“…The Mao's algorithm [3] and the PEG algorithm [4] are tree-based girth conditioning algorithms with different edge placement strategies and similar complexity. Also, the trellis-based algorithm [5] forms the list of paths that may form cycles, which is the identical information to that used in [3], [4]. Thus, it requires the same amount of memory as the tree-based algorithms do.…”

“…Actually, if the branches, which have not been passed by search paths, are removed from the trellis, the trellis becomes identical to what used in the tree-based search. Thus, the trellis-based algorithm [5] is computationally equivalent to the tree-based algorithms. Therefore, it is enough to compare with the algorithm in [3].…”

“…While the PEG algorithm has the different edge placement strategy that a new edge is progressively added to the graph to maximize girth in a best-effort way by checking cycles, both algorithms basically have the same computational complexity. Recently, a trellis-based search algorithm is proposed by constructing a trellis with repeated cascades of a Tanner graph and its mirror image [5]. However, this algorithm does not improve the number of required searches and the amount of required storage compared to the tree-based algorithms [3], [4] because the constructed trellis and corresponding path-finding algorithm used in [5] is actually equivalent to the tree-based algorithms.…”

“…Recently, a trellis-based search algorithm is proposed by constructing a trellis with repeated cascades of a Tanner graph and its mirror image [5]. However, this algorithm does not improve the number of required searches and the amount of required storage compared to the tree-based algorithms [3], [4] because the constructed trellis and corresponding path-finding algorithm used in [5] is actually equivalent to the tree-based algorithms. In this paper, we present a novel search algorithm to find the existence and exact length of cycles by using the message-passing algorithm, which requires only simple binary operations.…”