Algebraic Geometry Curves and Elkies Codes

Error control codes are widely used to increase the reliability of transmis-sion of information over various forms of communications channels. Algebraic geometry (AG) codes were introduced by V. Goppa in the 1980’s. The code-words in an AG code are obtained by evaluating functions at he points of an algebraic curve over a finite field; Goppa’s construction produces some of the best codes currently known. N. Elkies has recently proposed a generalization of this construction that produces non-linear codes with even better parameters. Their non-linearity is a serious drawback for their use in applications. Our re-search involves analyzing Elkies codes on the line and on Hermitian curves over finite fields. We show that Elkies codes contain relatively large linear subcodes, and examine the codeword recognition problem using properties of these linear subcodes.