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Linear 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 (cf. syndrome decoding). Linear codes are used in forward error correction and are applied in methods for transmitting symbols (e.g., bits) on a communications channel so that, if errors occur in the communication, some errors can be corrected or detected by the recipient of a message block. The codewords in a linear block code are blocks of symbols that are encoded using more symbols than the original value to be sent. A linear code of length ''n'' transmits blocks containing ''n'' symbols. For example, the ,4,3 Hamming code is a linear binary code which represents 4-bit messages using 7-bit codewords. Two distinct ... [...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] |
Hamming Distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to change one string into the other, or the minimum number of ''errors'' that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences. It is named after the American mathematician Richard Hamming. A major application is in coding theory, more specifically to block codes, in which the equal-length strings are vectors over a finite field. Definition The Hamming distance between two equal-length strings of symbols is the number of positions at which the corresponding symbols are different. Examples The symbols may be letters, bits, or decimal digits, among other possibilities. For example, the Hamming distance betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton Bound
In coding theory, the Singleton bound, named after Richard Collom Singleton, is a relatively crude upper bound on the size of an arbitrary block code C with block length n, size M and minimum distance d. It is also known as the Joshibound. proved by and even earlier by . Statement of the bound The minimum distance of a set C of codewords of length n is defined as d = \min_ d(x,y) where d(x,y) is the Hamming distance between x and y. The expression A_(n,d) represents the maximum number of possible codewords in a q-ary block code of length n and minimum distance d. Then the Singleton bound states that A_q(n,d) \leq q^. Proof First observe that the number of q-ary words of length n is q^n, since each letter in such a word may take one of q different values, independently of the remaining letters. Now let C be an arbitrary q-ary block code of minimum distance d. Clearly, all codewords c \in C are distinct. If we puncture the code by deleting the first d-1 letters of each code ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Code
In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communication channel or storage in a storage medium. An early example is an invention of language, which enabled a person, through speech, to communicate what they thought, saw, heard, or felt to others. But speech limits the range of communication to the distance a voice can carry and limits the audience to those present when the speech is uttered. The invention of writing, which converted spoken language into visual symbols, extended the range of communication across space and time. The process of encoding converts information from a source into symbols for communication or storage. Decoding is the reverse process, converting code symbols back into a form that the recipient understands, such as English or/and Spanish. One reason for coding is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reed–Muller Code
Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication. Moreover, the proposed 5G standard relies on the closely related polar codes for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied in theoretical computer science. Reed–Muller codes generalize the Reed–Solomon codes and the Walsh–Hadamard code. Reed–Muller codes are linear block codes that are locally testable, locally decodable, and list decodable. These properties make them particularly useful in the design of probabilistically checkable proofs. Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When ''r'' and ''m'' are integers with 0 ≤ ''r'' ≤ ''m'', the Reed–Muller code with parameters ''r'' and ''m'' is denoted as RM(''r'', ''m''). When aske ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Code
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. The Hadamard code is an example of a linear code of length 2^m over a binary alphabet. Unfortunately, this term is somewhat ambiguous as some references assume a message length k = m while others assume a message length of k = m+1. In this article, the first case is called the Hadamard code while the second is called the augmented Hadam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Element (mathematics)
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets of . Sets can themselves be elements. For example, consider the set B = \. The elements of are ''not'' 1, 2, 3, and 4. Rather, there are only three elements of , namely the numbers 1 and 2, and the set \. The elements of a set can be anything. For example, C = \ is the set whose elements are the colors , and . Notation and terminology The relation "is an element of", also called set membership, is denoted by the symbol "∈". Writing :x \in A means that "''x'' is an element of ''A''". Equivalent expressions are "''x'' is a member of ''A''", "''x'' belongs to ''A''", "''x'' is in ''A''" and "''x'' lies in ''A''". The expressions "''A'' includes ''x''" and "''A'' contains ''x''" are also used to mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the Domain of a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: :\ker(L) = \left\ . Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same Image (mathematics), image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows that the image of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Kernel (matrix)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: :\ker(L) = \left\ . Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows that the image of is isomorphic to the quotient of by ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
