The Ubiquity of Order Domains for the Construction of Error Control Codes

The order domains are a class of commutative rings introduced by H{\o}holdt, van Lint, and Pellikaan to simplify the theory of error control codes using ideas from algebraic geometry. The definition is largely motivated by the structures utilized in the Berlekamp-Massey-Sakata (BMS) decoding algorithm, with Feng-Rao majority voting for unknown syndromes, applied to one-point geometric Goppa codes constructed from curves. However, order domains are much more general, and O'Sullivan has shown that the BMS algorithm can be applied to decode all codes constructed from order domains by a suitable generalization of Goppa's procedure for curves. In this article we will first discuss the connection between order domains and valuations on function fields over a finite field. Under some mild conditions, we will see that a general projective variety over a finite field has projective models which can be used to construct order domains and Goppa-type codes for which the BMS algorithm is applicable. We will then give a slightly different interpretation of Geil and Pellikaan's extrinsic characterization of order domains via the theory of Gr\"obner bases, and show that their results are related to the existence of toric deformations of varieties. To illustrate the potential usefulness of these observations, we present a series of new explicit examples of order domains associated to varieties with many rational points over finite fields: Hermitian hypersurfaces, Grassmannians, and flag varieties.