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Dual Of BCH Is An Independent Source
A certain family of BCH codes have a particularly useful property, which is that treated as linear operators, their dual operators turns their input into an \ell-wise independent source. That is, the set of vectors from the input vector space are mapped to an \ell-wise independent source. The proof of this fact below as the following Lemma and Corollary is useful in derandomizing the algorithm for a 1-2^-approximation to MAXEkSAT MAXEkSAT is a problem in computational complexity theory that is a maximization version of the Boolean satisfiability problem 3SAT. In MAXEkSAT, each clause has exactly ''k'' literals, each with distinct variables, and is in conjunctive normal form .... Lemma Let C\subseteq F_2^n be a linear code such that C^\perp has distance greater than \ell +1. Then C is an \ell-wise independent source. Proof of lemma It is sufficient to show that given any k \times l matrix ''M'', where ''k'' is greater than or equal to ''l'', such that the rank of ''M'' is ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BCH Code
In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called ''Galois field''). BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem, and independently in 1960 by Raj Chandra Bose and D.K. Ray-Chaudhuri. The name ''Bose–Chaudhuri–Hocquenghem'' (and the acronym ''BCH'') arises from the initials of the inventors' surnames (mistakenly, in the case of Ray-Chaudhuri). One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an algebraic method known as syndrome decoding. This simplifies the design of the decoder for these codes, using small low-pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Basis
In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with the same index set ''I'' such that ''B'' and ''B''∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span ''V''∗. If it does span ''V''∗, then ''B''∗ is called the dual basis or reciprocal basis for the basis ''B''. Denoting the indexed vector sets as B = \_ and B^ = \_, being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes are equal, and equal to 0 otherwise. Symbolically, evaluating a dual vector in ''V''∗ on a vector in the original space ''V'': : v^i\cdot v_j = \delta^i_j = \begin 1 & \text i = j\\ 0 & \text i \ne j\text \end where \delta^i_j is the Kronecker delta symbol. Introduction To perform operations with a vector, we must ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairwise Independence
In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite variance are uncorrelated. A pair of random variables ''X'' and ''Y'' are independent if and only if the random vector (''X'', ''Y'') with joint cumulative distribution function (CDF) F_(x,y) satisfies :F_(x,y) = F_X(x) F_Y(y), or equivalently, their joint density f_(x,y) satisfies :f_(x,y) = f_X(x) f_Y(y). That is, the joint distribution is equal to the product of the marginal distributions. Unless it is not clear in context, in practice the modifier "mutual" is usually dropped so that independence means mutual independence. A statement such as " ''X'', ''Y'', ''Z'' are independent random variables" means that ''X'', ''Y'', ''Z'' are mutua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MAXEkSAT
MAXEkSAT is a problem in computational complexity theory that is a maximization version of the Boolean satisfiability problem 3SAT. In MAXEkSAT, each clause has exactly ''k'' literals, each with distinct variables, and is in conjunctive normal form. These are called k-CNF formulas. The problem is to determine the maximum number of clauses that can be satisfied by a truth assignment to the variables in the clauses. We say that an algorithm ''A'' provides an ''α''- approximation to MAXEkSAT if, for some fixed positive ''α'' less than or equal to 1, and every kCNF formula ''φ'', ''A'' can find a truth assignment to the variables of ''φ'' that will satisfy at least an ''α''-fraction of the maximum number of satisfiable clauses of ''φ''. Because the NP-hard ''k''-SAT problem (for ''k'' ≥ 3) is equivalent to determining if the corresponding MAXEkSAT instance has a value equal to the number of clauses, MAXEkSAT must also be NP-hard, meaning that there is no polynomial time algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
