Evaluation Codes from smooth Quadric Surfaces and Twisted Segre Varieties

TL;DR

This paper determines the parameters of evaluation codes on smooth quadric surfaces, revealing BCH structures in codes on elliptic quadrics and their twists, thus advancing algebraic coding theory on algebraic surfaces.

Contribution

It provides explicit parameters for evaluation codes on smooth quadrics and uncovers BCH structures in codes on elliptic quadrics and their twists.

Findings

Parameters of codes on hyperbolic quadrics derived from product codes.

Identification of BCH structures in codes on elliptic quadrics.

Extension of BCH structure detection to twists of Segre embeddings.

Abstract

We give the parameters of any evaluation code on a smooth quadric surface. For hyperbolic quadrics the approach uses elementary results on product codes and the parameters of codes on elliptic quadrics are obtained by detecting a BCH structure of these codes and using the BCH bound. The elliptic quadric is a twist of the surface P^1 x P^1 and we detect a similar BCH structure on twists of the Segre embedding of a product of any d copies of the projective line.