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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 ...
The following definition of the LRC follows from the description above: an [,,]-Locally Recoverable Code (LRC) of length is a code that produces an -symbol codeword from information symbols, and for any symbol of the codeword, there exist at most other symbols such that the value of the symbol can be recovered from them.
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 ...
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one [clarification needed] effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values ...
Linear interpolation on a data set (red points) consists of pieces of linear interpolants (blue lines). Linear interpolation on a set of data points (x 0, y 0), (x 1, y 1), ..., (x n, y n) is defined as piecewise linear, resulting from the concatenation of linear segment interpolants between each pair of data points.
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 ...
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.