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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 ...
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space ), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [ 2] Indeed, if we fix three ...
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 BCH code with and higher has the generator polynomial. This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. It is also denoted as: (15, 1) BCH code. In fact, this code has only two codewords: 000000000000000 and 111111111111111 (a trivial repetition code ).
The Reed–Solomon code is a [n, k, n − k + 1] code; in other words, it is a linear block code of length n (over F) with dimension k and minimum Hamming distance = + The Reed–Solomon code is optimal in the sense that the minimum distance has the maximum value possible for a linear code of size ( n , k ); this is known as the Singleton bound .
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 asked to encode a message consisting of k bits, where holds, the RM ( r , m) code produces a codeword ...
Edit distance. In computational linguistics and computer science, edit distance is a string metric, i.e. a way of quantifying how dissimilar two strings (e.g., words) are to one another, that is measured by counting the minimum number of operations required to transform one string into the other. Edit distances find applications in natural ...
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 ...