By virtue of the fact that the code isAs a [[linear subspace]] of <math>\mathbb{F}^n_2n_q</math>, the sumentire <math>c_1 + c_2</math> of two codewords incode <math>C</math> is also a codeword (ie an [[element (mathematics)|element]] of the subspace <math>C</math>). Thus the entire code (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 ''generating matrix'' for the code <math>C</math>.
The subspace definition also gives rise to the important property that the minimum [[Hamming distance]] between any given codeword <math>c_0</math> and the other codewords <math>c \neq c_0</math> is simplyconstant. Since the minimumdifference <math>c - c_0</math> of two codewords in <math>C</math> is also a codeword (ie an [[Hammingelement weight(mathematics)|element]] of allthe codewordssubspace <math>C</math>) and <math>d(c, c_0)=d(c-c_0, since:0)</math>
:<math>\min_{c \in C,\ c \neq c_0}d(c_1c, c_2c_0)=\min_{c \in C, c \neq c_0}d(c_1+c_2c-c_0, 0)=\min_{c \in C, c \neq 0}d(c, 0)</math>
