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Error Control Coding
Published in Jerry D. Gibson, Mobile Communications Handbook, 2017
Let GF(q) denote a field with q elements; then it is possible to construct the field GF(pm) for any prime number p and any positive integer m, and it is impossible to construct a field with q elements if q is not a power of a prime. To construct the field GF(p) for prime p, the field consists of the integers {0,1,2, …, p − 1} and both addition and multiplication are performed modulo p. To construct the field GF(pm) for m > 1 requires more subtlety; the field elements of GF(pm) are the polynomials of degree at most m − 1, with coefficients drawn from GF(p). There are pm such polynomials, and if addition and multiplication are defined appropriately, then the resulting structure is a field. Here, “appropriately” means that addition is performed in the “usual” polynomial fashion, component-wise modulo p, while multiplication is performed in polynomial fashion, but the result is reduced modulo π(x), where π(x) is an irreducible polynomial with coefficients in GF(p) of degree exactly m. An irreducible polynomial is one that cannot be factored; it is analogous among polynomials to a prime number among integers.
The last chapter
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
In Section 3.2 we constructed finite fields as extensions of prime fields Fp, using irreducible polynomials. There is a similar mechanism producing extension rings of the rings Zn = ℤ/nℤ. These finite rings are known as Galois rings and were first studied by W. Krull [128]. In coding theory the case when the underlying ring is Z4 has proved particularly fruitful. We concentrate on this case. The corresponding Galois rings are GR(m, Z4).
Finite Field Arithmetic Architecture
Published in Keshab K. Parhi, Takao Nishitani, Digital Signal Processing for Multimedia Systems, 2018
The concept of irreducible polynomial is analogous to prime numbers and implies that there are no factors for a given polynomial. Considering the fact that an irreducible polynomial of degree m over GF(2) is a factor of the polynomial x2m–1 + 1, it is, therefore, possible to obtain the irreducible polynomial of degree m by considering all the factors of the polynomial x2m–1 + 1.
A Framework for Filtering Step of Number Field Sieve and Function Field Sieve
Published in IETE Journal of Research, 2023
Rahul Janga, R. Padmavathy, S. K. Pal, S. Ravichandra
The relations for were generated using the CADO-NFS with the parameters mentioned in Table 1. The parameters given in the table are needed to generate the relations using the CADO-NFS tool. The factor base bound and sieve range are two parameters needed for the sieving phase of FFS as mentioned in the previous section. qstart and qend are starting and ending special−q sieving range and lpb is the factor base bound. The lpb represents the highest degree of the irreducible polynomial. Three relation sets namely A, B and C are generated and given as inputs to the filtering module. The matrix reduction percentage for each phase has been reported in Table 2. A massive reduction of above 90% has been recorded for every set of relations.
Darboux theory of integrability for polynomial vector fields on 𝕊n
Published in Dynamical Systems, 2018
Let be a C1 map. A hypersurface Ω = {(x1,… ,xn + 1) is said to be regular if the gradient ∇G of G is not equal to zero on . Of course, if Ω is regular, then it is smooth. We say that Ω is algebraic if G is an irreducible polynomial. If the degree of the polynomial G is d, then we say that Ω is algebraic of degreed. A polynomial vector fieldon the regular hypersurface Ω (or simply a polynomial vector field on Ω) is a polynomial vector field in satisfying where the dot denotes the inner product of two vectors in . If the polynomial vector field in has degree m, then we say that the vector field on Ω is of degree m.
Block conjugacy of irreducible toral automorphisms
Published in Dynamical Systems, 2019
Lennard F. Bakker, Pedro Martins Rodrigues
We fix an irreducible polynomial of degree n. This determines an algebraic number field . The class of t, which we denote from now on by β, is a simple root of f in K and we identify K with the isomorphic field . The notation denotes the ring of all algebraic integers of K. An automorphism with characteristic polynomial acts on and so there exists a right eigenvector associated to β, namely, , where the entries of u determine a (fractional) ideal of . The action of A, on the right, on , is thus identified with multiplication by β in I, namely, if for some , then . That the entries of u are a basis of the free -module I has the obvious but important consequence that if then m=0, and so, if (an integer matrix) and Mu=0 then M=0.