Tangent Codes

TL;DR

This paper explores the properties of finite Zariski tangent spaces to affine varieties, viewing them as linear codes, and investigates their geometric characteristics, decoding capabilities, and constructions of special tangent codes.

Contribution

It introduces a geometric framework for tangent codes, analyzes their minimum distance behavior, and provides explicit constructions of varieties with near MDS, cyclic, or Hamming tangent codes.

Findings

Tangent codes admit simultaneous decoding.

Duals of tangent codes are gradients of polynomials in the ideal of X.

Constructed affine varieties with near MDS, cyclic, or Hamming 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.