Tangent Codes

Abstract

The present article studies the finite Zariski tangent spaces to an affine variety X as linear codes, in order to characterize their typical or exceptional properties by global geometric conditions on X. The discussion concerns the generic minimum distance of a tangent code to X, its lower semi-continuity under a deformation of X, as well as the existence of Zariski tangent spaces to X with exceptional minimum distance. Tangent codes are shown to admit simultaneous decoding. The duals of the tangent codes to X are realized by gradients of polynomials from the ideal of X. We provide constructions of affine varieties with near MDS, cyclic or Hamming tangent codes. Puncturing, shortening and extending finite Zariski tangent spaces are related to the corresponding operations on affine varieties. The (u|u + v) construction of tangent codes is associated with a fibered product of varieties. Explicit constructions realize linear Hamming isometries as differentials of morphisms of affine varieties.