3

Linear Codes

3.1    Linear Vector Spaces over Finite Fields

We will assume that the reader is familiar with the definition and properties of a vector (linear) space and finite field Fq, which is simply a subspace of the vector space Fqn. Linear codes C are vector spaces and their algebraic structures follow the rules of a linear space. Some examples of vector spaces over Fq are:

 (i)  For any q,C1=Fqn, and C2 = {0} = the zero vector (0, 0,…, 0) Fqn;

(ii)  For any q, C3 = { (λ,,λ):λFqn };

(iii)  For q = 2, C4 = {(0, 0, 0, 0,), (1, 0, 1, 0), (0, 1, 0, 1), (1, 1, 1, 1)};

(iv)  For q = 3, C5 = {(0, 0, 0), (0, 1, 2), (0, 2, 1)}.

From these examples, it is easy to see that for any q, C2 = {0} is a subspace of both C3 and C1Fqn, and C3 is a subspace ...

Get Algebraic and Stochastic Coding Theory now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.