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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 is the best known in terms of relative distance for codes over alphabets of size less than 49. [citation needed] For larger alphabets, algebraic geometry codes sometimes achieve an asymptotically better rate vs. distance tradeoff than is given by the Gilbert–Varshamov bound. [1]
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.
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.
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.
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]
In 1957 he proved the Gilbert-Varshamov bound for linear codes (independently of Edgar Gilbert who proved the non-linear part). From 1968 he worked in Yerevan and was director of the Computer Centre (now Institute for Informatics and Automation Problems [1]) of the Academy of Sciences of the Armenian SSR.
Theoretical interest in LDPC codes also follows from their amenability to mathematical analysis. In his dissertation, Gallager showed that LDPC codes achieve the Gilbert–Varshamov bound for linear codes over binary fields with high probability.