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Griesmer Bound
In the mathematics of coding theory, the Griesmer bound, named after James Hugo Griesmer, is a bound on the length of linear binary codes of dimension ''k'' and minimum distance ''d''. There is also a very similar version for non-binary codes. Statement of the bound For a binary linear code, the Griesmer bound is: : n\geqslant \sum_^ \left\lceil\frac\right\rceil. Proof Let N(k,d) denote the minimum length of a binary code of dimension ''k'' and distance ''d''. Let ''C'' be such a code. We want to show that : N(k,d)\geqslant \sum_^ \left\lceil\frac\right\rceil. Let ''G'' be a generator matrix of ''C''. We can always suppose that the first row of ''G'' is of the form ''r'' = (1, ..., 1, 0, ..., 0) with weight ''d''. : G= \begin 1 & \dots & 1 & 0 & \dots & 0 \\ \ast & \ast & \ast & & G' & \\ \end The matrix G' generates a code C', which is called the residual code of C. C' obviously has dimension k'=k-1 and length n'=N(k,d)-d. C' has a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Coding
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 encodings fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 polynomi ... [...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] |
Hamming Bound
In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible words. It gives an important limitation on the efficiency with which any error-correcting code can utilize the space in which its code words are embedded. A code that attains the Hamming bound is said to be a perfect code. Background on error-correcting codes An original message and an encoded version are both composed in an alphabet of ''q'' letters. Each code word contains ''n'' letters. The original message (of length ''m'') is shorter than ''n'' letters. The message is converted into an ''n''-letter codeword by an encoding algorithm, transmitted over a noisy channel, and finally decoded by the receiver. The decoding process interprets a garbled codeword, referred to as si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Bound
In applied mathematics, the Johnson bound (named after Selmer Martin Johnson) is a limit on the size of error-correcting codes, as used in coding theory for data transmission or communications. Definition Let C be a ''q''-ary code of length n, i.e. a subset of \mathbb_q^n. Let d be the minimum distance of C, i.e. :d = \min_ d(x,y), where d(x,y) is the Hamming distance between x and y. Let C_q(n,d) be the set of all ''q''-ary codes with length n and minimum distance d and let C_q(n,d,w) denote the set of codes in C_q(n,d) such that every element has exactly w nonzero entries. Denote by , C, the number of elements in C. Then, we define A_q(n,d) to be the largest size of a code with length n and minimum distance d: : A_q(n,d) = \max_ , C, . Similarly, we define A_q(n,d,w) to be the largest size of a code in C_q(n,d,w): : A_q(n,d,w) = \max_ , C, . Theorem 1 (Johnson bound for A_q(n,d)): If d=2t+1, : A_q(n,d) \leq \frac. If d=2t+2, : A_q(n,d) \leq \frac. Theorem 2 (Joh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plotkin Bound
In the mathematics of coding theory, the Plotkin bound, named after Morris Plotkin, is a limit (or bound) on the maximum possible number of codewords in binary codes of given length ''n'' and given minimum distance ''d''. Statement of the bound A code is considered "binary" if the codewords use symbols from the binary alphabet \. In particular, if all codewords have a fixed length ''n'', then the binary code has length ''n''. Equivalently, in this case the codewords can be considered elements of vector space \mathbb_2^n over the finite field \mathbb_2. Let d be the minimum distance of C, i.e. :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 binary code of length n and minimum distance d. The Plotkin bound places a limit on this expression. Theorem (Plotkin bound): i) If d is even and 2d > n , then : A_(n,d) \leq 2 \left\lfloor\frac\right\rfloor. ii) If d is odd and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias Bassalygo Bound
The Elias Bassalygo bound is a mathematical limit used in coding theory for error correction during data transmission or communications. Definition Let C be a q-ary code of length n, i.e. a subset of n.Each q-ary block code of length n is a subset of the strings of \mathcal_q^n, where the alphabet set \mathcal_q has q elements. Let R be the ''rate'' of C, \delta the ''relative distance'' and :B_q(y, \rho n) = \left \ be the '' Hamming ball'' of radius \rho n centered at y. Let \text_q(y, \rho n) = , B_q(y, \rho n), be the ''volume'' of the Hamming ball of radius \rho n . It is obvious that the volume of a Hamming Ball is translation-invariant, i.e. indifferent to y. In particular, , B_q(y, \rho n), =, B_q(0, \rho n), . With large enough n, the ''rate'' R and the ''relative distance'' \delta satisfy the Elias-Bassalygo bound: :R \leqslant 1 - H_q ( J_q(\delta))+o(1), where : H_q(x)\equiv_\text -x\log_q \left ( \right)-(1-x)\log_q is the ''q''-ary entropy func ... [...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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
