In a single-variable polynomial ring
whose coefficients are
elements of the finite field
of size 2, an irreductible polynomial
of degree
is said to be primitive it it divides
for
and it does not divide
for any
with
.
We are asked to show that the minimum Hamming code is 3. What is the link
with the the above ring? We know that we can represent cyclic codes thank
to polynomials.
will in fact be the set of our code words. It precisely
contains
distinct polynomials. Because we want to achieve a code word
length of 7, we need a generator polynomial
and a message polynomial
of the right length. Our cyclic code generator polynomial is
constructed from the factors of
. As
,
:
There, we notice that the minimum Hamming weight of the nonzero code words is 3, and therefore, this code has a minimum Hamming distance of 3.