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This is a consequence of the fact that a parity check matrix of is a generator matrix of the dual code. G is a matrix, while H is a () matrix. Equivalent codes. Codes C 1 and C 2 are equivalent (denoted C 1 ~ C 2) if one code can be obtained from the other via the following two transformations: arbitrarily permute the components, and
Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥. This means that a codeword c is in C if and only if the matrix-vector product Hc ⊤ = 0 (some authors would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations.
LDPC codes functionally are defined by a sparse parity-check matrix. This sparse matrix is often randomly generated, subject to the sparsity constraints—LDPC code construction is discussed later. These codes were first designed by Robert Gallager in 1960. Below is a graph fragment of an example LDPC code using Forney's factor graph notation.
The Gilbert–Varshamov bound for linear codes asserts the existence of q -ary linear codes for any relative minimum distance less than the given bound that simultaneously have high rate. The existence proof uses the probabilistic method, and thus is not constructive. The Gilbert–Varshamov bound is the best known in terms of relative distance ...
In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form . The dimension of C and its dual always add up to the length n : A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code.
In addition, management's free-cash-flow (FCF) target of $10 billion in 2025 to 2026 means the stock trades on just 11.1 times FCF in 2026 -- a startlingly cheap valuation for a stock with such a ...
Ahead of Friday's June jobs report, data from the CME Group showed investors pricing in a 75% chance the Fed would cut rates in September. Those odds were little changed after the release. Writing ...
Cheating in video games involves a video game player using various methods to create an advantage beyond normal gameplay, usually in order to make the game easier.Cheats may be activated from within the game itself (a cheat code implemented by the original game developers), or created by third-party software (a game trainer or debugger) or hardware (a cheat cartridge).