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. It gives an important limitation on the efficiency with which any error-correcting code can utilize the space in which its code words are embedded. A code that attains the Hamming bound is said to be a perfect code.
theory, the Hammingbound 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...
include the Hamming code (which makes use of a Hamming matrix), the Hamming window, Hamming numbers, sphere-packing (or Hammingbound), Hamming graph concepts...
computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors...
Sphere packing on the corners of a hypercube (with the spheres defined by Hamming distance) corresponds to designing error-correcting codes: if the spheres...
block codes. The first error-correcting code was the Hamming(7,4) code, developed by Richard W. Hamming in 1950. This code transforms a message consisting...
\rho n)|} be the volume of the Hamming ball of radius ρn{\displaystyle \rho n}. It is obvious that the volume of a Hamming Ball is translation-invariant...
1+2C_{11}^{1}+2^{2}C_{11}^{2}=3^{5}} situations, i.e. the constructed WA lies on the Hammingbound for t = 2 {\displaystyle t=2} and in this sense is perfect. To date...
proof of the bound. Diamond code Elias Bassalygo bound Gilbert–Varshamov bound Griesmer boundHammingbound Johnson bound Singleton bound Plotkin, Morris...
In a cube In a cylinder Close-packing Kissing number Sphere-packing (Hamming) bound Other 2-D packing Square packing Other 3-D packing Tetrahedron Ellipsoid...
Gilbert–Varshamov bound Griesmer boundHammingbound Plotkin bound Singleton bound Johnson, Selmer Martin (April 1962). "A new upper bound for error-correcting...
communication" (PDF). U.S. patent no. 2,632,058 Morgan 1998, pp. 973–975. Hamming 1950, pp. 147–160. Ling & Xing 2004, pp. 82–88. Pless 1982, pp. 21–24....
omitted. Elias Bassalygo bound Gilbert-Varshamov boundHammingbound Johnson bound Plotkin bound Singleton bound J. H. Griesmer, "A bound for error-correcting...
salad): the primary ingredient, ham, is mixed with smaller amounts of chopped vegetables or relishes, and the whole is bound with liberal amounts of a mayonnaise...
Calderbank, Peter Shor and Andrew Steane. According to the quantum Hammingbound, encoding a single logical qubit and providing for arbitrary error correction...
2 {\displaystyle [2^{r}-1,2^{r}-r-1,3]_{2}} Hamming code. Since d = 3 {\displaystyle d=3} , this Hamming code can correct a 1-bit error. Example : The...
-\mathbf {x_{2}} )=0} . Since the minimum Hamming weight of ( 7 , 4 , 3 ) {\displaystyle (7,4,3)} Hamming Code is 3, d H ( x 1 , x 2 ) ≥ 3 {\displaystyle...
mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. FEC can be...
fields, including communication complexity (e.g., in proofs of the gap Hamming problem) and graph theory. An interesting anti-concentration inequality...
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