Orange County Community College
... with proof, I will not schedule a make-up exam and a zero grade will be recorded. In general, make-up exams may be of greater difficulty than regular exams because of the additional study time you have had. (The exams are usually announced one week in advance.) Homework Quizzes and Weekly Quizzes wi ...
... with proof, I will not schedule a make-up exam and a zero grade will be recorded. In general, make-up exams may be of greater difficulty than regular exams because of the additional study time you have had. (The exams are usually announced one week in advance.) Homework Quizzes and Weekly Quizzes wi ...
Galois Field Computations A Galois field is an algebraic field that
... The definitions imply that a primitive element is a root of a corresponding primitive polynomial. This section describes how to create a Galois array, which is a MATLAB expression that represents elements of a Galois field. This section also describes how MATLAB interprets the numbers that you use i ...
... The definitions imply that a primitive element is a root of a corresponding primitive polynomial. This section describes how to create a Galois array, which is a MATLAB expression that represents elements of a Galois field. This section also describes how MATLAB interprets the numbers that you use i ...
Paul Mitchener's notes
... Proposition 20 Let p(x) ∈ F [x] be irreducible, with degree n. Let q1 (x) and q2 (x) be non-zero polynomials of degree less than n. Then p(x) is not a factor of the product q1 (x)q2 (x). Proof: Suppose we can find non-zero polynomials q1 (x) and q2 (x) of degree less than n such that p(x) is a facto ...
... Proposition 20 Let p(x) ∈ F [x] be irreducible, with degree n. Let q1 (x) and q2 (x) be non-zero polynomials of degree less than n. Then p(x) is not a factor of the product q1 (x)q2 (x). Proof: Suppose we can find non-zero polynomials q1 (x) and q2 (x) of degree less than n such that p(x) is a facto ...
A Complete Characterization of Irreducible Cyclic Orbit - HAL
... The set of all k-dimensional subspaces of a vector space V is often referred to as the Grassmann variety (or simply Grassmannian) and denoted by G(k, V ). Constant dimension codes are subsets of G(k, Fnq ), where Fq is some finite field. The general linear group GL(V ) consisting of all invertible t ...
... The set of all k-dimensional subspaces of a vector space V is often referred to as the Grassmann variety (or simply Grassmannian) and denoted by G(k, V ). Constant dimension codes are subsets of G(k, Fnq ), where Fq is some finite field. The general linear group GL(V ) consisting of all invertible t ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.