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Computation of a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message string, with a fixed number of zeroes appended, by the "generator polynomial" string except that exclusive or operations replace subtractions.
LDPC codes functionally are defined by a sparse parity-check matrix. This sparse matrix is often randomly generated, subject to the sparsity constraints—LDPC code construction is discussed later. These codes were first designed by Robert Gallager in 1960. [5] Below is a graph fragment of an example LDPC code using Forney's factor graph notation.
Binary-coded decimal (BCD) is a binary encoded representation of integer values that uses a 4-bit nibble to encode decimal digits. Four binary bits can encode up to 16 distinct values; but, in BCD-encoded numbers, only ten values in each nibble are legal, and encode the decimal digits zero, through nine. The remaining six values are illegal and ...
When counting in binary, the digit sum modulo 2 is the Thue–Morse sequence. To compute the nth element t n, write the number n in binary. If the number of ones in this binary expansion is odd then t n = 1, if even then t n = 0. [2] That is, t n is the even parity bit for n.
A pseudorandom binary sequence (PRBS), pseudorandom binary code or pseudorandom bitstream is a binary sequence that, while generated with a deterministic algorithm, is difficult to predict [ 1] and exhibits statistical behavior similar to a truly random sequence. PRBS generators are used in telecommunication, such as in analog-to-information ...
In computing, a linear-feedback shift register ( LFSR) is a shift register whose input bit is a linear function of its previous state. The most commonly used linear function of single bits is exclusive-or (XOR). Thus, an LFSR is most often a shift register whose input bit is driven by the XOR of some bits of the overall shift register value.
Rule 30 has also been used as a random number generator in Mathematica, and has also been proposed as a possible stream cipher for use in cryptography. Rule 30 is so named because 30 is the smallest Wolfram code which describes its rule set (as described below). The mirror image, complement, and mirror complement of Rule 30 have Wolfram codes ...
The cyclic redundancy check (CRC) is based on division in the ring of polynomials over the finite field GF (2) (the integers modulo 2 ), that is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around. Any string of bits can be interpreted as the coefficients of a message polynomial of this ...