Algebraic and Stochastic Coding Theory

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 F_q, which is simply a subspace of the vector space $F_q^n$. Linear codes C are vector spaces and their algebraic structures follow the rules of a linear space. Some examples of vector spaces over F_q are:

 (i)  For any $q,C_1=F_q^n$, and C₂ = {0} = the zero vector (0, 0,…, 0) $F_q^n$;

(ii)  For any q, C₃ = $\left\{\left(\lambda ,\dots ,\lambda \right):\lambda \in F_q^n\right\}$;

(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 $C_1\in F_q^n$, and C₃ is a subspace ...