A linear code C of length n over F _q is a subspace of .

The following are linear codes:

1. C = {( ?, ?, , ?): ? ? F _q}. This code is often called a repetition code (refer also to Example 1.0.3).

2. ( q = 2) C = {000 , 001 , 010 , 011}.

3. ( q = 3) C = {0000 , 1100 , 2200 , 0001 , 0002 , 1101 , 1102 , 2201 , 2202}.

4. ( q = 2) C = {000 , 001 , 010 , 011 , 100 , 101 , 110 , 111}.

Let C be a linear code in .

(i) The dual code of C is C ^?, the orthogonal complement of the subspace C of .

(ii) The dimension of the linear code C is the dimension of C as a vector space over F _q, i.e., dim( C).

Let C be a linear code of length n over F _q . Then,

1. C = q ^{dim( C )} , i.e., dim( C) = log _q