Concept
Hamming distance
Related concepts (22)
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.
Decoding methods
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. is considered a binary code with the length ; shall be elements of ; and is the distance between those elements. One may be given the message , then ideal observer decoding generates the codeword .
String metric
In mathematics and computer science, a string metric (also known as a string similarity metric or string distance function) is a metric that measures distance ("inverse similarity") between two text strings for approximate string matching or comparison and in fuzzy string searching. A requirement for a string metric (e.g. in contrast to string matching) is fulfillment of the triangle inequality. For example, the strings "Sam" and "Samuel" can be considered to be close.
Error correction code
In computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The central idea is that the sender encodes the message in a redundant way, most often by using an error correction code or error correcting code (ECC). The redundancy allows the receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors.
Hamming weight
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string, or the digit sum of the binary representation of a given number and the l1 norm of a bit vector. In this binary case, it is also called the population count, popcount, sideways sum, or bit summation.
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.
Lee distance
In coding theory, the Lee distance is a distance between two strings and of equal length n over the q-ary alphabet {0, 1, ..., q − 1} of size q ≥ 2. It is a metric defined as If q = 2 or q = 3 the Lee distance coincides with the Hamming distance, because both distances are 0 for two single equal symbols and 1 for two single non-equal symbols. For q > 3 this is not the case anymore; the Lee distance between single letters can become bigger than 1. However, there exists a Gray isometry (weight-preserving bijection) between with the Lee weight and with the Hamming weight.
Gray code
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representation of the decimal value "1" in binary would normally be "" and "2" would be "". In Gray code, these values are represented as "" and "". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.
Hypercube graph
In graph theory, the hypercube graph Q_n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q_3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q_n has 2^n vertices, 2^n – 1n edges, and is a regular graph with n edges touching each vertex. The hypercube graph Q_n may also be constructed by creating a vertex for each subset of an n-element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each n-digit binary number, with two vertices adjacent when their binary representations differ in a single digit.
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector.