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*are:

_{q} (i) For any $q,{C}_{1}={F}_{q}^{n}$, and *C*_{2} = {0} = the zero vector (0, 0,…, 0) ${F}_{q}^{n}$;

(ii) For any *q, C _{3}* = $\left\{\left(\lambda ,\dots ,\lambda \right):\lambda \in {F}_{q}^{n}\right\}$;

(iii) For *q* = 2, *C*_{4} = {(0, 0, 0, 0,), (1, 0, 1, 0), (0, 1, 0, 1), (1, 1, 1, 1)};

(iv) For *q* = 3, *C*_{5} = {(0, 0, 0), (0, 1, 2), (0, 2, 1)}.

From these examples, it is easy to see that for any *q, C*_{2} = {0} is a subspace of both *C*_{3} and ${C}_{1}\in {F}_{q}^{n}$, and *C*_{3} is a subspace ...

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