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A generator matrix for a linear [,,]-code has format , where n is the length of a codeword, k is the number of information bits (the dimension of C as a vector subspace), d is the minimum distance of the code, and q is size of the finite field, that is, the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc.).
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 [1] would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. [2]
It can be shown that if is a pseudo-random number generator for the uniform distribution on (,) and if is the CDF of some given probability distribution , then is a pseudo-random number generator for , where : (,) is the percentile of , i.e. ():= {: ()}. Intuitively, an arbitrary distribution can be simulated from a simulation of the standard ...
As with other codes, the maximum likelihood decoding of an LDPC code on the binary symmetric channel is an NP-complete problem, [24] shown by reduction from 3-dimensional matching. So assuming P != NP, which is widely believed, then performing optimal decoding for an arbitrary code of any useful size is not practical.
MATLAB (an abbreviation of "MATrix LABoratory" [18]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
The size of a code is the number of codewords and equals q k. ... Example: The linear block code with the following generator matrix is a [,,] ...
The second row is the same generator with a seed of 3, which produces a cycle of length 2. Using a = 4 and c = 1 (bottom row) gives a cycle length of 9 with any seed in [0, 8]. A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation.
The generator polynomial of the BCH code is defined as the least common multiple = ((), …, + ()). Note: if n = q m − 1 {\displaystyle n=q^{m}-1} as in the simplified definition, then g c d ( n , q ) {\displaystyle {\rm {gcd}}(n,q)} is 1, and the order of q {\displaystyle q} modulo n {\displaystyle n} is m . {\displaystyle m.}