Algebraic geometric codes over rings
Let \(A\) denote a local Artinian ring with finite residue field. Let \(X\) be a smooth curve of genus \(g\) over \(\text{Spec }A\). Given a line bundle \({\mathcal L}\) on \(X\), a set \(\mathcal Z=\{Z_1,\dots,Z_n\}\) of disjoint \(A\)-points on \(X\), and chosen isomorphisms \(\gamma_i: \Gamma(Z_i,{\mathcal L}|_{Z_i})\rightarrow A\), the author defines the algebraic geometric code \(C_A(X,\mathcal Z,{\mathcal L},\gamma)\) by essentially evaluating sections of \({\mathcal L}\) at the \(Z_i\). It is shown that this code is a free \(A\)-module of rank \(n\). The author then establishes that these codes have analogous properties to algebraic geometric (Goppa) codes defined using curves over a finite field. Specifically: she proves a version of the Riemann-Roch Theorem for curves over Artin local rings in order to show that the dimension of this code is deg \({\mathcal L} +1-g\) if \(2g-2<\text{deg }{\mathcal L}<n\); shows that the minimum distance \(d\) of this code satisfies \(d\geq n-\text{deg }{\mathcal L}\); and, under the further assumption that \(A\) is Gorenstein, shows that the dual of this code is also an algebraic geometric code over \(A\). The author has used this theory in [IEEE Trans. Inf. Theory 43, 1588-1593 (1997; Zbl 0904.94027)] to show that the Nordström-Robinson code is the image under the Gray map of an algebraic geometric code over \({\mathcal Z}/4\).
scientific article; zbMATH DE number 1384135