TABLE OF CONTENTS A linear code of length n over the finite field F q is simply a subspace of the vector space 
. Since linear codes are vector spaces, their algebraic structures often make them easier to describe and use than nonlinear codes. In most of this book, we focus our attention on linear codes over finite fields.
We recall some definitions and facts about vector spaces over finite fields. While the proofs of most of the facts stated in this section are omitted, it should be noted that many of them are practically identical to those in the case of vector spaces over R or C.
Let F q be the finite field of order q. A nonempty set V, together with some (vector) addition + and scalar multiplication by elements of F q, is a vector space (or linear space) over F q if it satisfies all of the following conditions. For all u , v , w ? V and for all ?, ? ? F q:
u + v ? V ;
( u + v) + w = u + ( v + w);
there is an element 0 ? V with the property 0 + v = v = v + 0 for all v ? V ;
for each u ? V there...
