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Tanner proved the following bounds Let be the rate of the resulting linear code, let the degree of the digit nodes be and the degree of the subcode nodes be .If each subcode node is associated with a linear code (n,k) with rate r = k/n, then the rate of the code is bounded by
Below is a graph fragment of an example LDPC code using Forney's factor graph notation. In this graph, n variable nodes in the top of the graph are connected to (n−k) constraint nodes in the bottom of the graph. This is a popular way of graphically representing an (n, k) LDPC code.
Factor graphs and Tanner graphs are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors. [40]
A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K 5 and the complete bipartite graph K 3,3.
Zemor's algorithm is based on a type of expander graphs called Tanner graph. The construction of code was first proposed by Tanner. [ 3 ] The codes are based on double cover d {\displaystyle d} , regular expander G {\displaystyle G} , which is a bipartite graph.
Sparse graph code; Syndrome decoding; Tanner graph; Ternary Golay code; Tiger (hash function) Transverse redundancy check; Triple modular redundancy; Turbo code; UOWHF; Universal hashing; Universal Product Code; Verhoeff algorithm; Viterbi algorithm; Viterbi decoder; WHIRLPOOL
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. [1] If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. [2] For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3.
Tanner graph of a Sudoku. denotes the entries of the Sudoku in row-scan order. denotes the constraint functions: =, …, associated with rows, =, …, associated with columns and =, …, associated with the sub-grids of the Sudoku.. There are several possible decoding methods for sudoku codes. Some algorithms are very specific developments for Sudoku codes. Several methods are described in ...