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Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
linear medium linear approximation any technique used for the purpose of analysis and design of nonlinear systems. For example, one way of analyzing the stability of a system described by nonlinear differential equations is to linearize the equations around the equilibrium point of interest and check the location of the eigenvalues of the linear system approximation. linear block code a block coding scheme for which the mapping can be described by a linear transformation of the message block. The transformation matrix is referred to as the generator matrix. linear code a forward error control code or line code whose code words form a vector space. Equivalently, a code where the element-wise finite field addition of any two code words forms another code word. linear constant-coefficient equation a general N th-order linear constant-coefficient differential equation is of the form
Lattice-Based Coding to Enhance Error Performance of the Hidden Direct Sequence Spread Spectrum
Published in IETE Journal of Research, 2021
Nader Sanandaji, Abolfazl Falahati
According to (2) for a random linear code, the expected number of non-zero codewords of a random linear block code intersecting with a modulo-p set s in depends only on the cardinality of the set s. Lifting the modulo-p linear block code to by “construction A” results in an RCA lattice as (3) with the period of . Hence, the expected number of non-zero points in RCA lattice overlapping with a convex set in (such as an n-dimensional ball), relies only on the volume of the set. Thus, balls of the same radius but different centres have the same expected number of intersecting points with the lattice which is the “construction A” of a random linear block code. Now, referring to (30), the term should be independent of the ball centre . Thus, it can be replaced by .
Fuzzy Linear Codes
Published in Fuzzy Information and Engineering, 2018
S. Atamewoue Tsafack, S. Ndjeya, L. Strüngmann, C. Lele
Assume that and , that means that k = 2 and n = 3. Let be a linear code over F, in the classical case, when we send a codeword through a communication channel, the signal receive can be read as and modulate to . Thus to know if belong to the code C, we use syndrome calculation [9]. Since the modulation have gave a wrong word, we can consider that have more information than , in the sense that we can estimate a level to which a word 0 is modulate to 1, and a word 1 is modulate to 0. Therefore it is possible to use the idea of fuzzy logic to recover the transmit codeword.
A Quantum Resistant Chameleon Hashing and Signature Scheme
Published in IETE Journal of Research, 2022
A linear code C is a k-dimensional linear subspace of and all the elements of C are called codewords. The weight of an element x in is the number of nonzero coordinates of x . The minimum distanced of a linear code is the minimum weight of any nonzero element of the code. A code is specified either by a generator matrix G or by a parity check matrix H. A generator matrix of a linear code C is a matrix G whose row vectors form a basis for the vector subspace C, i.e.,. A parity-check matrixH of a linear code C is an matrix whose row vectors form a basis for the orthogonal complement of the vector subspace C. Hence . The syndrome of a word by H is a vector defined by . If the error correcting capability of the code is then for any syndrome s, there is at most one word of weight , such that . If there exists a word x corresponding to a vector , then s is said to be t-decodable (or simply decodable) in the code defined by H. Given a random syndrome s, finding whether s is t-decodable in the code defined by a random matrix H has been proved to be NP-complete [16]. Decoding can be easily performed for more structured codes.