Linear Block Codes
... Introduction to linear codes A linear code over GF(q) [Galois Field] where q is a prime power is a subset of the vector space V(n, q) for some positive value of n. C is a subspace of V(n, q) iff (1) u v C for all u and v in C (2) a.u C for all u C, a GF (q) A binary code is linear iff the ...
... Introduction to linear codes A linear code over GF(q) [Galois Field] where q is a prime power is a subset of the vector space V(n, q) for some positive value of n. C is a subspace of V(n, q) iff (1) u v C for all u and v in C (2) a.u C for all u C, a GF (q) A binary code is linear iff the ...
VECTOR SPACES: FIRST EXAMPLES 1. Definition So far in the
... Example 4.2. Consider the subset U = {ax2 | a ∈ R} of P2 [x]. Show that U is a subspace, give a basis and find its dimension. Solution 1. We verify the three axioms. Throughout, keep in mind that to say that a polynomial p(x) is in U , is exactly to say that is of the form p(x) = ax2 + 0x + 0 for so ...
... Example 4.2. Consider the subset U = {ax2 | a ∈ R} of P2 [x]. Show that U is a subspace, give a basis and find its dimension. Solution 1. We verify the three axioms. Throughout, keep in mind that to say that a polynomial p(x) is in U , is exactly to say that is of the form p(x) = ax2 + 0x + 0 for so ...
Math 3101 Spring 2017 Homework 2 1. Let R be a unital ring and let
... a, b, c, d ∈ I c d is an ideal of Mn (R). (b) Let J be an ideal of Mn (R), and let E(J) denote the subset of R comprised of entries of matrices in J. Prove that E(J) is an ideal of R, and that J = Mn (E(J)). (Hint: Consider the elementary matrices eij for 1 ≤ i, j ≤ n.) Conclude that every ideal o ...
... a, b, c, d ∈ I c d is an ideal of Mn (R). (b) Let J be an ideal of Mn (R), and let E(J) denote the subset of R comprised of entries of matrices in J. Prove that E(J) is an ideal of R, and that J = Mn (E(J)). (Hint: Consider the elementary matrices eij for 1 ≤ i, j ≤ n.) Conclude that every ideal o ...
