Linear codes and weights

Let F be a finite field with q elements. A k dimensional subspace C of the vector space Fn of all n-tuples over F is called a linear code of length n and dimension k. Algebraically, C is just a k-dimensional vector space over F. However, as a particular subspace of Fn, C inherits some metric properties. Specifically, for every v E Fn, the weight of v, denoted by wt( v), is defined to be the number of non-zero entries in the vector v, and the distance between two vectors is the weight of their difference. The interplay between the algebraic structure of C and the metric structure induced by the weight function is central to coding theory. (Should we rename it "Finite Analysis"? ) The ubiquitous triangle inequality wt( v + w) :::; wt( v) + wt( w) does hold, but, as the following example shows, it is too weak to tell the whole story.