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In mathematical terms, Hamming codes are a class of binary linear code. For each integer r ≥ 2 there is a code-word with block length n = 2r − 1 and message length k = 2r − r − 1. Hence the rate of Hamming codes is R = k / n = 1 − r / (2r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal ...
Hamming (7,4) In coding theory, Hamming (7,4) is a linear error-correcting code that encodes four bits of data into seven bits by adding three parity bits. It is a member of a larger family of Hamming codes, but the term Hamming code often refers to this specific code that Richard W. Hamming introduced in 1950.
The block code and its parameters. Error-correcting codes are used to reliably transmit digital data over unreliable communication channels subject to channel noise. When a sender wants to transmit a possibly very long data stream using a block code, the sender breaks the stream up into pieces of some fixed size.
The Hamming space of binary strings of length 3. The distance between vertices in the cube graph equals the Hamming distance between the strings. In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all binary strings of length N. [1][2] It is used in the theory of coding ...
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.
There are many different erasure coding schemes. The most popular erasure codes are Reed-Solomon coding, Low-density parity-check code (LDPC codes), and Turbo codes. [1] As of 2023, modern data storage systems can be designed to tolerate the complete failure of a few disks without data loss, using one of 3 approaches: [2] [3] [4] Replication; RAID
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 with block length , size and minimum distance . It is also known as the Joshibound. [1] proved by Joshi (1958) and even earlier by Komamiya (1953).
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. [1] Linear codes allow for more efficient encoding and ...