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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 ...
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]
Theorem 1 (Johnson bound for ... is the floor function. Remark: Plugging the bound of Theorem 2 into the bound of Theorem 1 produces ... Gilbert–Varshamov bound ...
Such limitations often take the form of bounds that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. Examples of block codes are Reed–Solomon codes , Hamming codes , Hadamard codes , Expander codes , Golay codes , Reed–Muller codes and Polar codes .
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 ...
This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by the Łoś–Vaught test. More generally, any first-order theory with an infinite model has non-isomorphic, elementarily equivalent models, which can be obtained via the Löwenheim–Skolem theorem.