Algebraic geometric codes over rings

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980's. Recently, there has been an increased interest in the study of linear codes over finite rings. In this thesis, we combine these two approaches to coding theory by introducing and studying algebraic geometric codes over rings.

We define algebraic geometric codes over any local Artinian ring A and compute their parameters. Under the additional hypothesis that A is a Gorenstein ring, we show that the class of codes we have defined is closed under duals. We show that the coordinatewise projection of an algebraic geometric code defined over A is an algebraic geometric code defined over the residue field of A. As an example of our construction, we show that the linear $\doubz$/4-code which Hammons, et al., project nonlinearly to obtain the Nordstrom-Robinson code is an algebraic geometric code. In the case where A is either $\doubz$/q or a Galois ring, we find an expression for the minimum Euclidean weights of (trace codes of) certain algebraic geometric codes over A in terms of an exponential sum.

U of I Only

ETDs are only available to UIUC Users without author permission