Some Optimal Codes from Algebraic Geometry and Their Covering Radii

Abstract

We show that many Goppa codes from algebraic geometry are optimal. Many of these codes attain the Griesmer bound and provide first examples of Griesmer codes which have relatively large dimension compared with the minimum distance. We give a lower bound on the covering radius of algebraic geometric codes in terms of the number of rational points and the genus of the underlying curve. We use this lower bound and some upper bounds on the covering radius proved elsewhere to determine the covering radii of many optimal codes mentioned above exactly. We use our results to give for the first time many non-trivial examples of non-binary normal codes. We also point out connections between our results and many geometric structures such as saturation configurations, t-independent sets, min-hyper (max-hyper) etc. Finally, we show that Goppa codes have at least the potential of being among the best covering and packing codes discovered so far.