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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]
The first such family is the Am2900 series of integrated circuits (ICs) created in 1975 designed in bit-slice topology so they could be used as modular components each representing a different aspect of a computer control unit (CCU).
9-bit parity generator/checker, bus driver parity I/O port 14 SN74AS286: 74x287 1 1024-bit PROM (256x4) three-state 16 SN74S287: 74x288 1 256-bit PROM (32x8) three-state 16 SN74S288: 74x289 1 64-bit RAM (16x4), 4 data inputs, 4 inverted data outputs open-collector 16 SN74S289: 74x290 1 decade counter (separate divide-by-2 and divide-by-5 ...
The simplest error-detection system, the parity bit, is in fact a 1-bit CRC: it uses the generator polynomial x + 1 (two terms), [5] and has the name CRC-1. Application [ edit ]
The parity bit may be used within another constituent code. In an example using the DVB-S2 rate 2/3 code the encoded block size is 64800 symbols (N=64800) with 43200 data bits (K=43200) and 21600 parity bits (M=21600). Each constituent code (check node) encodes 16 data bits except for the first parity bit which encodes 8 data bits.
Download QR code; Print/export ... 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 ...
There exists a few papers that systematically compare various model checkers on a common case study. The comparison usually discusses the modelling tradeoffs faced when using the input languages of each model checker, as well as the comparison of performances of the tools when verifying correctness properties. One can mention:
The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code, also known as a Simplex code. The parity-check matrix has the property that any two columns are pairwise linearly independent.