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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 ...
Codes in general are often denoted by the letter C, and a code of length n and of rank k (i.e., having n code words in its basis and k rows in its generating matrix) is generally referred to as an (n, k) code. Linear block codes are frequently denoted as [n, k, d] codes, where d refers to the code's minimum Hamming distance between any two code ...
Using a [,,] linear block code, one can prove that there exists a permutation code in the symmetric group of degree , having minimum distance at least and large cardinality. [9] A lower bound for permutation codes that provides asymptotic improvements in certain regimes of length and distance of the permutation code [ 9 ] is discussed below.
A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...
In the linear code case a different proof of the Singleton bound can be obtained by observing that rank of the parity check matrix is . [4] Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at most n − k + 1 {\displaystyle n-k+1} .
The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function : is called "bounded" then this usually means that its image is a bounded subset of its codomain.
The bound is obtained by considering the range of parameters that are obtainable by concatenating a "good" outer code with a "good" inner code . Specifically, we suppose that the outer code meets the Singleton bound , i.e. it has rate r o u t {\displaystyle r_{out}} and relative distance δ o u t {\displaystyle \delta _{out}} satisfying r o u t ...
A code is considered "binary" if the codewords use symbols from the binary alphabet {,}. In particular, if all codewords have a fixed length n , then the binary code has length n . Equivalently, in this case the codewords can be considered elements of vector space F 2 n {\displaystyle \mathbb {F} _{2}^{n}} over the finite field F 2 ...