As a [[linear subspace]] of <math>{\mathbb{F}_q}^n</math>, the entire code <math>''C</math>'' (which may be very large) may be represented as the [[span (linear algebra)|span]] of a minimal set of codewords (known as a [[basis (linear algebra)|basis]] in [[linear algebra]]). These basis codewords are often collated in the rows of a matrix known as a ''[[Generator matrix|generating matrix]]'' for the code <math>''C</math>''.
The subspace definition also gives rise to the important propertyguarantees that the minimum [[Hamming distance]] ''d'' between any given codeword ''c''<mathsub>c_00</mathsub> and the other codewords ''c'' ≠ ''c''<mathsub>c \neq c_00</mathsub> is constant. Since the difference ''c'' − ''c''<mathsub>c - c_00</mathsub> of two codewords in <math>''C</math>'' is also a codeword (iei.e., an [[element (mathematics)|element]] of the subspace <math>''C</math>''), and <math>''d''(''c'', c_0 c<sub>0</sub>) = ''d''(''c-c_0, '' − ''c''<sub>0)</mathsub>, 0), we see that
:<math>\min_{c \in C,\ c \neq c_0}d(c,c_0)=\min_{c \in C, c \neq c_0}d(c-c_0, 0)=\min_{c \in C, c \neq 0}d(c, 0).</math>
