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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]
For practical purposes, parity-check matrix of a binary Goppa code is usually converted to a more computer-friendly binary form by a trace construction, that converts the -by-matrix over () to a -by-binary matrix by writing polynomial coefficients of () elements on successive rows.
A self-dual code is one which is its own dual. This implies that n is even and dim C = n/2.If a self-dual code is such that each codeword's weight is a multiple of some constant >, then it is of one of the following four types: [1]
where H is the parity-check matrix of the Hamming code and is given by = []. The [[,,]] Steane code is the first in the family of ...
The ternary Golay code consists of 3 6 = 729 codewords. Its parity check matrix is [].Any two different codewords differ in at least 5 positions. Every ternary word of length 11 has a Hamming distance of at most 2 from exactly one codeword.
Indirect parity measurements coincide with the typical way we think of parity measurement as described above, by measuring an ancilla qubit to determine the parity of the input bits. Direct parity measurements differ from the previous type in that a common mode with the parities coupled to the qubits is measured, without the need for an ancilla ...
Thu Duc is a class-1 city of Ho Chi Minh City and the first city to use city-within-city model in Vietnam. [11] [12]Thủ Đức City has 34 wards: An Khánh, An Lợi Đông, An Phú, Bình Chiểu, Bình Thọ, Bình Trưng Đông, Bình Trưng Tây, Cát Lái, Hiệp Bình Chánh, Hiệp Bình Phước, Hiệp Phú, Linh Chiểu, Linh Đông, Linh Tây, Linh Trung, Linh Xuân, Long Bình ...
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} .