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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.
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 ...
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]
We suppose that the inner code meets the Gilbert–Varshamov bound, i.e. it has rate and relative distance satisfying + (). Random linear codes are known to satisfy this property with high probability, and an explicit linear code satisfying the property can be found by brute-force search (which requires time polynomial in the size of the ...
Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler.The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem.
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.
The description above is given for what is now called a serially concatenated code. Turbo codes, as described first in 1993, implemented a parallel concatenation of two convolutional codes, with an interleaver between the two codes and an iterative decoder that passes information forth and back between the codes. [6]
Ward defended the utility of the five ways (for instance, on the fourth argument he states that all possible smells must pre-exist in the mind of God, but that God, being by his nature non-physical, does not himself stink) whilst pointing out that they only constitute a proof of God if one first begins with a proposition that the universe can ...