In coding theory, we are often interested in finding codes with a ''good''
relative minimum distance.
To create a code that transmits information efficiently,
we would like to see the ratio of bits containing information to total
codeword length be large. In particular, for families of codes
this means that as the codeword length increases, we do not
significantly decrease the amount of information transmitted; for each
fixed prime $q$ we
can construct a family of $q$-ary
codes with lengths $p$ and minimal distances $d_p$
with the property that as the length of these codes approaches infinity,
the ratio of their minimal distance to their total length tends towards
$\epsilon > 0$.
In this paper, we examine families of generalized binary
quadratic residue codes,
named $q$-th power residue codes, where $q$ is a fixed odd prime, and
find an asymptotically bad subfamily of these codes. For each prime
$l$ we will
construct a $q$-th power residue code of length $p$ (determined by our
choice of $l$ )
and minimal
distance $d_p$, with the property that as $l$ approaches
infinity, $p$ also tends towards infinity, and $\lim_{l
to \infty} \frac{d_p}{p} = 0$.