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Hamming also noticed the problems with flipping two or more bits, and described this as the "distance" (it is now called the Hamming distance, after him). Parity has a distance of 2, so one bit flip can be detected but not corrected, and any two bit flips will be invisible.
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 ...
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 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 =, the set of binary vectors of length , and two vectors , are -th associates if they are Hamming distance apart.
One strategy for solving this version of the hat problem employs Hamming codes, which are commonly used to detect and correct errors in data transmission. The probability for winning will be much higher than 50%, depending on the number of players in the puzzle configuration: for example, a winning probability of 87.5% for 7 players.
The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. [5] FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in multicast.
The minimum distance of a set of codewords of length is defined as = {,:} (,) where (,) is the Hamming distance between and .The expression (,) represents the maximum number of possible codewords in a -ary block code of length and minimum distance .
In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two.It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.