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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]
CLHEP (short for A Class Library for High Energy Physics) is a C++ library that provides utility classes for general numerical programming, vector arithmetic, geometry, pseudorandom number generation, and linear algebra, specifically targeted for high energy physics simulation and analysis software. [1]
An additional rule holds for combinations of j 1, j 2, and j 3 that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol: (+ +) = This identity also holds if the sign of any j i is reversed, or if any of them are substituted with an m i instead.
This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics. The overall sign of the coefficients for each set of constant j 1 {\displaystyle j_{1}} , j 2 {\displaystyle j_{2}} , j {\displaystyle j} is arbitrary to some degree and has been fixed according to the Condon–Shortley and Wigner sign ...
It is apparent that the 3-j symbol treats all three angular momenta involved in the addition problem on an equal footing and is therefore more symmetrical than the CG coefficient. Since the state | 0 0 {\displaystyle |0\,0\rangle } is unchanged by rotation, one also says that the contraction of the product of three rotational states with a 3- j ...
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. . Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are kn
newton per coulomb (N⋅C −1), or equivalently, volt per meter (V⋅m −1) energy: joule (J) Young's modulus: pascal (Pa) or newton per square meter (N/m 2) eccentricity: unitless Euler's number (2.71828, base of the natural logarithm) unitless electron: unitless elementary charge: coulomb (C) force
For any positive integer i, let m i (x) be the minimal polynomial with coefficients in GF(q) of α i. The generator polynomial of the BCH code is defined as the least common multiple g(x) = lcm(m 1 (x),…,m d − 1 (x)). It can be seen that g(x) is a polynomial with coefficients in GF(q) and divides x n − 1.