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Chapter 5 studies cyclic codes and Chapter 6 studies a special case of cyclic codes, the quadratic residue codes. Chapter 7 returns to BCH codes. [1] [6] After these discussions of specific codes, the next chapter concerns enumerator polynomials, including the MacWilliams identities, Pless's own power moment identities, and the Gleason ...
For example, given a linear map T : V → W, the image T(V) of V, and the inverse image T −1 (0) of 0 (called kernel or null space), are linear subspaces of W and V, respectively. Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums
For example, p 2 provides an even parity for bits 2, 3, 6, and 7. It also details which transmitted bit is covered by which parity bit by reading the column. For example, d 1 is covered by p 1 and p 2 but not p 3 This table will have a striking resemblance to the parity-check matrix (H) in the next section.
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis.
Download QR code; Print/export ... This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and ...
For many problems in applied linear algebra, it is useful to adopt the perspective of a matrix as being a concatenation of column vectors. For example, when solving the linear system =, rather than understanding x as the product of with b, it is helpful to think of x as the vector of coefficients in the linear expansion of b in the basis formed by the columns of A.
Linear block codes are frequently denoted as [n, k, d] codes, where d refers to the code's minimum Hamming distance between any two code words. (The [n, k, d] notation should not be confused with the (n, M, d) notation used to denote a non-linear code of length n, size M (i.e., having M code words), and minimum Hamming distance d.)
aaa is a one- to three-letter code describing the actual algorithm implemented in the subroutine, e.g. SV denotes a subroutine to solve linear system, while R denotes a rank-1 update. For example, the subroutine to solve a linear system with a general (non-structured) matrix using real double-precision arithmetic is called DGESV . [ 2 ] : "
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