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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 words a, b and c, then whenever there is a ...
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.
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 ...
In signal processing and statistics, a window function (also known as an apodization function or tapering function[ 1]) is a mathematical function that is zero-valued outside of some chosen interval. Typically, window functions are symmetric around the middle of the interval, approach a maximum in the middle, and taper away from the middle.
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.
The Hamming scheme, named after Richard Hamming, is also known as the hyper-cubic association scheme, and it is the most important example for coding theory. [1][2][3]In this scheme X=Fn,{\displaystyle X={\mathcal {F}}^{n},}the set of binary vectors of length n,{\displaystyle n,}and two vectors x,y∈Fn{\displaystyle x,y\in {\mathcal {F}}^{n ...
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 signals and transmission. More generally, a Hamming space can be defined over any alphabet (set) Q as the set of words of a fixed length N with ...