Finding an asymptotically bad family of $q$-th power residue codes
P. Charters · Advances in Mathematics of Communications · 2009
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 totalcodeword length be large. In particular, for families of codesthis means that as the codeword length increases, we do notsignificantly decrease the amount of information transmitted; for eachfixed prime $q$ wecan construct a family of $q$-arycodes 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 binaryquadratic residue codes,named $q$-th power residue codes, where $q$ is a fixed odd prime, andfind an asymptotically bad subfamily of these codes. For each prime$l$ we willconstruct a $q$-th power residue code of length $p$ (determined by ourchoice of $l$ )and minimaldistance $d_p$, with the property that as $l$ approachesinfinity, $p$ also tends towards infinity, and $\lim_{lto \infty} \frac{d_p}{p} = 0$.