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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 repeated ...
The cyclic redundancy check (CRC) is a check of the remainder after division in the ring of polynomials over GF (2) (the finite field of 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 ...
Finally the intermediate remainder can be reduced modulo the standard polynomial in a second shift register to yield the CRC-32 remainder. [12] If 3- or 4-input XOR gates are permitted, shorter intermediate polynomials of degree 71 or 53, respectively, can be used.
The Chebyshev polynomials form a complete orthogonal system. The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives.
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the ...
Primitive polynomial (field theory) In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF (pm). This means that a polynomial F(X) of degree m with coefficients in GF (p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF (pm) such that ...
Reciprocal polynomials have several connections with their original polynomials, including: is not 0. p(x) = xnp∗(x−1). [2] α is a root of a polynomial p if and only if α−1 is a root of p∗. [4] If p(x) ≠ x then p is irreducible if and only if p∗ is irreducible. [5] p is primitive if and only if p∗ is primitive.
Here the function is and therefore the three real roots are 2, −1 and −4. In algebra, a cubic equation in one variable is an equation of the form in which a is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic ...