In coding theory, a Tanner graph, named after Michael Tanner, is a bipartite graph used to state constraints or equations which specify error correcting codes. In coding theory, Tanner graphs are used to construct longer codes from smaller ones. Both encoders and decoders employ these graphs extensively.
Tanner graphs were proposed by Michael Tanner [1] as a means to create larger error correcting codes from smaller ones using recursive techniques. He generalized the techniques of Elias for product codes.
Tanner discussed lower bounds on the codes obtained from these graphs irrespective of the specific characteristics of the codes which were being used to construct larger codes.
Tanner graphs are partitioned into subcode nodes and digit nodes. For linear block codes, the subcode nodes denote rows of the parity-check matrix H. The digit nodes represent the columns of the matrix H. An edge connects a subcode node to a digit node if a nonzero entry exists in the intersection of the corresponding row and column.
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
The advantage of these recursive techniques is that they are computationally tractable. The coding algorithm for Tanner graphs is extremely efficient in practice, although it is not guaranteed to converge except for cycle-free graphs, which are known not to admit asymptotically good codes. [2]
Zemor's decoding algorithm, which is a recursive low-complexity approach to code construction, is based on Tanner graphs.
In telecommunication, a convolutional code is a type of error-correcting code that generates parity symbols via the sliding application of a boolean polynomial function to a data stream. The sliding application represents the 'convolution' of the encoder over the data, which gives rise to the term 'convolutional coding'. The sliding nature of the convolutional codes facilitates trellis decoding using a time-invariant trellis. Time invariant trellis decoding allows convolutional codes to be maximum-likelihood soft-decision decoded with reasonable complexity.
In computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three. Richard W. Hamming invented Hamming codes in 1950 as a way of automatically correcting errors introduced by punched card readers. In his original paper, Hamming elaborated his general idea, but specifically focused on the Hamming(7,4) code which adds three parity bits to four bits of data.
In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
In information theory, a low-density parity-check (LDPC) code is a linear error correcting code, a method of transmitting a message over a noisy transmission channel. An LDPC code is constructed using a sparse Tanner graph. LDPC codes are capacity-approaching codes, which means that practical constructions exist that allow the noise threshold to be set very close to the theoretical maximum for a symmetric memoryless channel. The noise threshold defines an upper bound for the channel noise, up to which the probability of lost information can be made as small as desired. Using iterative belief propagation techniques, LDPC codes can be decoded in time linear in their block length.
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics.
In cryptography, the McEliece cryptosystem is an asymmetric encryption algorithm developed in 1978 by Robert McEliece. It was the first such scheme to use randomization in the encryption process. The algorithm has never gained much acceptance in the cryptographic community, but is a candidate for "post-quantum cryptography", as it is immune to attacks using Shor's algorithm and – more generally – measuring coset states using Fourier sampling.
Belief propagation, also known as sum–product message passing, is a message-passing algorithm for performing inference on graphical models, such as Bayesian networks and Markov random fields. It calculates the marginal distribution for each unobserved node, conditional on any observed nodes. Belief propagation is commonly used in artificial intelligence and information theory, and has demonstrated empirical success in numerous applications, including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability.
A Viterbi decoder uses the Viterbi algorithm for decoding a bitstream that has been encoded using a convolutional code or trellis code.
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. Linear codes allow for more efficient encoding and decoding algorithms than other codes.
In coding theory, decoding is the process of translating received messages into codewords of a given code. There have been many common methods of mapping messages to codewords. These are often used to recover messages sent over a noisy channel, such as a binary symmetric channel.
The Hadamard code is an error-correcting code named after Jacques Hadamard that is used for error detection and correction when transmitting messages over very noisy or unreliable channels. In 1971, the code was used to transmit photos of Mars back to Earth from the NASA space probe Mariner 9. Because of its unique mathematical properties, the Hadamard code is not only used by engineers, but also intensely studied in coding theory, mathematics, and theoretical computer science. The Hadamard code is also known under the names Walsh code, Walsh family, and Walsh–Hadamard code in recognition of the American mathematician Joseph Leonard Walsh.
In coding theory, a parity-check matrix of a linear block code C is a matrix which describes the linear relations that the components of a codeword must satisfy. It can be used to decide whether a particular vector is a codeword and is also used in decoding algorithms.
In coding theory, concatenated codes form a class of error-correcting codes that are derived by combining an inner code and an outer code. They were conceived in 1966 by Dave Forney as a solution to the problem of finding a code that has both exponentially decreasing error probability with increasing block length and polynomial-time decoding complexity. Concatenated codes became widely used in space communications in the 1970s.
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of vertices from an n-vertex graph can partition the graph into disjoint subgraphs each of which has at most vertices.
In mathematics, more specifically in linear algebra, the spark of a matrix is the smallest integer such that there exists a set of columns in which are linearly dependent. If all the columns are linearly independent, is usually defined to be 1 more than the number of rows. The concept of matrix spark finds applications in error-correction codes, compressive sensing, and matroid theory, and provides a simple criterion for maximal sparsity of solutions to a system of linear equations.
In coding theory, expander codes form a class of error-correcting codes that are constructed from bipartite expander graphs. Along with Justesen codes, expander codes are of particular interest since they have a constant positive rate, a constant positive relative distance, and a constant alphabet size. In fact, the alphabet contains only two elements, so expander codes belong to the class of binary codes. Furthermore, expander codes can be both encoded and decoded in time proportional to the block length of the code.
In coding theory, folded Reed–Solomon codes are like Reed–Solomon codes, which are obtained by mapping Reed–Solomon codewords over a larger alphabet by careful bundling of codeword symbols.
In coding theory, Zemor's algorithm, designed and developed by Gilles Zemor, is a recursive low-complexity approach to code construction. It is an improvement over the algorithm of Sipser and Spielman.
Sudoku codes are non-linear forward error correcting codes following rules of sudoku puzzles designed for an erasure channel. Based on this model, the transmitter sends a sequence of all symbols of a solved sudoku. The receiver either receives a symbol correctly or an erasure symbol to indicate that the symbol was not received. The decoder gets a matrix with missing entries and uses the constraints of sudoku puzzles to reconstruct a limited amount of erased symbols.
A galactic algorithm is one with record-breaking theoretical (asymptotic) performance, but which is never used in practice. Typical reasons are that the performance gains only appear for problems that are so large they never occur, or the algorithm's complexity outweighs a relatively small gain in performance. Galactic algorithms were so named by Richard Lipton and Ken Regan, because they will never be used on any data sets on Earth.
