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General linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
Let k elements of Fqn be given. You want to find the dimension of the code they generate. Write your tuples as rows of a (k, n)-matrix G in some order.Use row operations to bring the matrix in row-echelon form. Call the resulting matrix G′.Row operations will not change the row space (the code). The dimension of the code is the number of nonzero rows of G′. These nonzero rows form a basis of the code.
Matrices
Published in Lina Oliveira, Linear Algebra, 2022
Matrices are crucial in the book, and several fundamental notions related with matrices were established here and will be relied upon in the remainder of the book, notably, the rank and the inverse of a matrix. To be determined, both rank and inverse lean on two methods known as Gaussian elimination and Gauss–Jordan elimination. These methods aim at finding, respectively, a row echelon form and the (unique) reduced row echelon form of a matrix. Gaussian and Gauss–Jordan eliminations will be used extensively throughout the book and matrices are mostly what this book is about.
Fundamentals of Systems of Differential Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
Both (5.18) and (5.19) are upper triangular matrices, the only difference being that (5.19) has the number one as the first nonzero entry of each row. A matrix of this form is said to be in row-echelon form. In the event that some rows would end up with all zeros, row-echelon form requires that we move these rows to the end and keep the leading entry in this staircase layout. This should be believable because it is equivalent to switching the physical location of two equations and why should this change the solution? (It shouldn’t.)
Linear Algebra on Parallel Structures Using Wiedemann Algorithm to Solve Discrete Logarithm Problem
Published in IETE Journal of Research, 2022
K S Spoorthi, R. Padmavathy, S K Pal, S Ravi Chandra
Gaussian Elimination is the oldest available algorithm to solve the system of linear equations. Gaussian elimination reduces the matrices to row echelon form, then solves the system of linear equations. For a matrix M of size , the complexity of the method can be given by or . The complexity is too high for large matrices. It does not utilize the sparsity of the matrix. Hence FFS does not use Gaussian Elimination.
Feasible rounding approaches for equality constrained mixed-integer optimization problems
Published in Optimization, 2023
Christoph Neumann, Oliver Stein
For its preparation recall that a matrix F is said to possess row echelon form (REF) if it has the shape resulting from Gaussian elimination. Moreover, it possesses reduced row echelon form (RREF) if it has the shape resulting from Gauss–Jordan elimination, that is, it has REF, the leading entry in each non-zero row is a 1, and each column containing such a leading 1 exhibits zeros in all its other entries.