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Parity-check Matrix
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. Definition Formally, a parity check matrix ''H'' of a linear code ''C'' is a generator matrix of the dual code, ''C''⊥. This means that a codeword c is in ''C ''if and only if the matrix-vector product (some authors would write this in an equivalent form, c''H''⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. That is, they show how linear combinations of certain digits (components) of each codeword equal zero. For example, the parity check matrix :H = \left \begin 0&0&1&1\\ 1&1&0&0 \end \right, compactly represents the parity check equations, :\begin c_3 + c_4 &= 0 \\ c_1 + c_2 &= 0 \end, that must be satisfied for the vector (c_1, c_2, c_3, c_4) to be a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coding Theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied by various scientific disciplines—such as information theory, electrical engineering, mathematics, linguistics, and computer science—for the purpose of designing efficient and reliable data transmission methods. This typically involves the removal of redundancy and the correction or detection of errors in the transmitted data. There are four types of coding: # Data compression (or ''source coding'') # Error control (or ''channel coding'') # Cryptographic coding # Line coding Data compression attempts to remove unwanted redundancy from the data from a source in order to transmit it more efficiently. For example, ZIP data compression makes data files smaller, for purposes such as to reduce Internet traffic. Data compression and er ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Block Code
In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of ''all'' block codes in a unified way. Such limitations often take the form of ''bounds'' that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay codes, and Reed–Muller codes. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes, because they can be generated using boolean polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Codeword
In communication, a code word is an element of a standardized code or protocol. Each code word is assembled in accordance with the specific rules of the code and assigned a unique meaning. Code words are typically used for reasons of reliability, clarity, brevity, or secrecy. See also * Code word (figure of speech) * Coded set * Commercial code (communications) * Compartmentalization (information security) * Duress code * Error correction and detection * Marine VHF radio * Password * Safeword * Spelling alphabet A spelling alphabet ( also called by various other names) is a set of words used to represent the letters of an alphabet in oral communication, especially over a two-way radio or telephone. The words chosen to represent the letters sound sufficien ... References * * *UNHCR Procedure for Radio Communication External links UNHCR Procedure for Radio Communication Data transmission Cryptography {{crypto-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generator Matrix
In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. The codewords are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix. Terminology If G is a matrix, it generates the codewords of a linear code ''C'' by : w=sG where w is a codeword of the linear code ''C'', and s is any input vector. Both w and s are assumed to be row vectors. A generator matrix for a linear , k, dq-code has format k \times n, where ''n'' is the length of a codeword, ''k'' is the number of information bits (the dimension of ''C'' as a vector subspace), ''d'' is the minimum distance of the code, and ''q'' is size of the finite field, that is, the number of symbols in the alphabet (thus, ''q'' = 2 indicates a binary code, etc.). The number of redundant bits is denoted by r = n - k. The ''standard'' form for a generator matrix is, : G = \begin I_k , P \end, where I_k is the k \times k identity m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Code
In coding theory, the dual code of a linear code :C\subset\mathbb_q^n is the linear code defined by :C^\perp = \ where :\langle x, c \rangle = \sum_^n x_i c_i is a scalar product. In linear algebra terms, the dual code is the annihilator of ''C'' with respect to the bilinear form \langle\cdot\rangle. The dimension of ''C'' and its dual always add up to the length ''n'': :\dim C + \dim C^\perp = n. A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code. Self-dual codes A self-dual code is one which is its own dual. This implies that ''n'' is even and dim ''C'' = ''n''/2. If a self-dual code is such that each codeword's weight is a multiple of some constant c > 1, then it is of one of the following four types: *Type I codes are binary self-dual codes which are not doubly even. Type I codes are always even (every codeword has even Hamming weight). *Type II codes are b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Code
A binary code represents text, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often "0" and "1" from the binary number system. The binary code assigns a pattern of binary digits, also known as bits, to each character, instruction, etc. For example, a binary string of eight bits (which is also called a byte) can represent any of 256 possible values and can, therefore, represent a wide variety of different items. In computing and telecommunications, binary codes are used for various methods of encoding data, such as character strings, into bit strings. Those methods may use fixed-width or variable-width strings. In a fixed-width binary code, each letter, digit, or other character is represented by a bit string of the same length; that bit string, interpreted as a binary number, is usually displayed in code tables in octal, decimal or hexadecimal notation. There are many character sets and many character encodin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syndrome Decoding
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. Notation C \subset \mathbb_2^n is considered a binary code with the length n; x,y shall be elements of \mathbb_2^n; and d(x,y) is the distance between those elements. Ideal observer decoding One may be given the message x \in \mathbb_2^n, then ideal observer decoding generates the codeword y \in C. The process results in this solution: :\mathbb(y \mbox \mid x \mbox) For example, a person can choose the codeword y that is most likely to be received as the message x after transmission. Decoding conventions Each codeword does not have an expected possibility: there may be more than one codeword with an equal likelihood of mutating into the received message. In such a case, the sender and receiver(s) must ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syndrome Decoding
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. Notation C \subset \mathbb_2^n is considered a binary code with the length n; x,y shall be elements of \mathbb_2^n; and d(x,y) is the distance between those elements. Ideal observer decoding One may be given the message x \in \mathbb_2^n, then ideal observer decoding generates the codeword y \in C. The process results in this solution: :\mathbb(y \mbox \mid x \mbox) For example, a person can choose the codeword y that is most likely to be received as the message x after transmission. Decoding conventions Each codeword does not have an expected possibility: there may be more than one codeword with an equal likelihood of mutating into the received message. In such a case, the sender and receiver(s) must ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Code
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 mathematical terms, Hamming codes are a class of binary linear code. For each integer there is a code-word with block length and message length . Hence the rate of Hamming codes is , which is the highest p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books by decree in 1586, it is the second oldest university press after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics known as the Delegates of the Press, who are appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 500 years, OUP has primarily focused on the publication of pedagogical texts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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