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In coding theory, the Gilbert–Varshamov bound (due to Edgar Gilbert [1] and independently Rom Varshamov [2]) is a bound on the size of a (not necessarily linear) code. It is occasionally known as the Gilbert–Shannon–Varshamov bound (or the GSV bound), but the name "Gilbert–Varshamov bound" is by far the most popular. Varshamov proved ...
The Gilbert–Varshamov bound for linear codes is related to the general Gilbert–Varshamov bound, which gives a lower bound on the maximal number of elements in an error-correcting code of a given block length and minimum Hamming weight over a field. This may be translated into a statement about the maximum rate of a code with given length ...
An Improvement is done to the Gilbert-Varshamov bound already discussed above. Using the connection between permutation codes and independent sets in certain graphs one can improve the Gilbert–Varshamov bound asymptotically by a factor log ( n ) {\displaystyle \log(n)} , when the code length goes to infinity.
In coding theory, the Wozencraft ensemble is a set of linear codes in which most of codes satisfy the Gilbert-Varshamov bound. It is named after John Wozencraft, who proved its existence. The ensemble is described by Massey (1963), who attributes it to Wozencraft.
The first two of its ten chapters present background and introductory material, including Hamming distance, decoding methods including maximum likelihood and syndromes, sphere packing and the Hamming bound, the Singleton bound, and the Gilbert–Varshamov bound, and the Hamming(7,4) code.
The Gilbert–Varshamov bound, proved independently in 1952 by Gilbert and in 1957 by Rom Varshamov, [4] is a mathematical theorem that guarantees the existence of error-correcting codes that have a high transmission rate as a function of their length, alphabet size, and Hamming distance between codewords (a parameter that controls the number ...
These codes attracted interest in the coding theory community because they have the ability to surpass the Gilbert–Varshamov bound; at the time this was discovered, the Gilbert–Varshamov bound had not been broken in the 30 years since its discovery. [6]
Gilbert–Varshamov bound; Gilbert–Varshamov bound for linear codes; ... Second Johnson bound; Shannon's source coding theorem; Signed-digit representation;