Evaluation codes from smooth quadric surfaces and twisted Segre varieties

Authors:
Alain Couvreur;Iwan Duursma
Affiliations:
Institut de Mathématiques de Bordeaux, Université Bordeaux I, Talence Cedex, France 33405;Department of Mathematics, University of Illinois at Urbana---Champaign, Urbana, USA 61801-2975
Venue:
Designs, Codes and Cryptography
Year:
2013

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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 on these codes and using the BCH bound. The elliptic quadric is a twist of the surface P 1 脳 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.