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February  2009, 3(1): 53-58. doi: 10.3934/amc.2009.3.53

Finding an asymptotically bad family of $q$-th power residue codes

P. Charters 1,

1. 

The University of Texas at Austin, Mathematics Department, C1200, 1 University Station, Austin, TX, 78712, United States

Received  October 2008 Revised  January 2009 Published  January 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 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 $\lim_{l to \infty} \frac{d_p}{p} = 0$.
Keywords: Chebotarev's density theorem., Coding theory, quadratic residue codes.
Mathematics Subject Classification: Primary: 11T71, 11H71; Secondary: 11R3.
Citation: P. Charters. Finding an asymptotically bad family of $q$-th power residue codes. Advances in Mathematics of Communications, 2009, 3 (1) : 53-58. doi: 10.3934/amc.2009.3.53
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