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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.
Axiomatic characterizations of (𝔾, 𝕆)-fuzzy rough approximation operators via overlap and grouping functions on a complete lattice
Published in International Journal of General Systems, 2023
Yan Sun, Bin Pang, Ju-Sheng Mi
If a continuous t-norm1 is positive, then it is an overlap function.Given a frame L with a meet-irreducible element , a binary function : defined by is an overlap function.Given a chain L, for each , the binary function : defined by is an overlap function.
Geometry of symplectic partially hyperbolic automorphisms on 4-torus
Published in Dynamical Systems, 2020
Let be some reciprocal polynomial with integer coefficients that is irreducible over the field and suppose it to have two complex conjugate roots on the unit circle and two real eigenvalues outside of the unit circle. The companion matrix of this polynomial has the form This matrix is not symplectic with respect to the standard symplectic 2-form , . Let us show, however, A to be symplectic with respect to a non-standard symplectic structure on defined as with a skew-symmetric integer non-degenerate matrix J. We consider the identity as the algebraic system for the entries of the matrix J. A solution to this matrix equation gives Thus, the matrix A becomes symplectic with respect to this non-standard symplectic structure. This will be used later on.
Block conjugacy of irreducible toral automorphisms
Published in Dynamical Systems, 2019
Lennard F. Bakker, Pedro Martins Rodrigues
For an integer , two matrices A and B with the same irreducible characteristic polynomial are k-block conjugate if there exist matrices such that , and .