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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 check digit is computed as follows: Drop the check digit from the number (if it's already present). This leaves the payload. Start with the payload digits. Moving from right to left, double every second digit, starting from the last digit. If doubling a digit results in a value > 9, subtract 9 from it (or sum its digits).
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.).
A cyclic redundancy check (CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital data. [1] [2] Blocks of data entering these systems get a short check value attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation is ...
The memo section of the check is optional, but it’s a good idea to fill it out because it can serve as a reminder of what the check is for. If you’re writing the check to pay for a haircut ...
Write the full, proper name of the person or company receiving the check on the “pay to” line. You’ll find it in the middle of the check, labeled “Pay to the order of.”
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The exception to this result is a bit pattern the same as that of the generator polynomial. All uneven bit errors are detected by generator polynomials with even number of terms. 2-bit errors in a (multiple) distance of the longest bitfilter of even parity to a generator polynomial are not detected; all others are detected.