Suppose that we have three arbitrary vectors, **u**, **v**, and **w**, all of dimension . We wish to show that (**u**,**w**)(**u**,**v**)(**v**,**w**).

Suppose there are positions in which **u** and **w** differ. Thus (**u**,**w**). If represents identical elements and represents differing elements between **u** and **w**, a graphical representation might look like the following:

Out of the positions where **u** and **v** differ, there are positions in which **v**'s elements are identical to those of **u**, but not identical to those of **w**. And likewise, out of these positions, there are positions in which **v**'s elements are identical to those of **w**, but not identical to those of **u**. And in the positions in which **u** matches **w**, there are positions in which the elements of **v** do not match either **u** or **w**.

We can now see that (**v**,**w**) and (**u**,**v**).

Adding (**u**,**v**) and (**v**,**w**) we have

(**u**,**v**)(**v**,**w**)
(**u**,**w**).

This is what we wished to show. Therefore, the Hamming distance is a metric.
Let **v** **V** and let . Then the *Hamming sphere of radius r and centre v* is defined by
S(**v**):={**x** **V** :
.

Let
. A subset of
is called a *t-error correcting code* if any two distinct elements **v**,**w** **C** satisfy
.

If **C** is a -error correcting code, we define its *minimum distance* to be (**C**):= min{(**c**,**c'**) : **c**,**c'** **C**,**c** **c'**}. Note that by Definition 1.4, the minimum distance for a **C** must be greater than or equal to .

If **C** is a -error correcting code and **v** **C** such that
, then **v** is called a *codeword of C*.

Let **C** be a -error correcting code. Then for each **v** **V** there exists at most one codeword **c** **C** such that
.