Tangent Codes

We interpret the finite Zariski tangent spaces Ta(X,Fqm) to an affine variety X ⊂ Fq n as linear codes. If the degree ofX is relatively prime to the characteristic p = charFq, we show that the minimum distance of Ta(X,Fqm) stabilizes while varying a ∈ X(Fqm) := X ∩ Fnqm and m ∈ N. The article provides a pseudocode for obtaining the generic minimum distance of a tangent code and points, at which it is attained. On the other hand, any family of linear codes of fixed length, fixed dimension and arbitrary minimum distance is interpolated by the Fq-Zariski tangent bundle of an affine variety X ⊂ Fq n with explicit equations, whose degree is divisible by p = charFq. A basic problem in the theory of error correcting codes is the construction of linear codes with a priori given length, dimension and minimum distance. That is helpful for the presence of a unique decoding under appropriate upper bound on the number of the perturbed symbols. Except the construction of individual linear codes, one can investigate the stabilization or destabilization of the minimum distance within a family of linear codes of fixed length and dimension. Explicit families enable "to deform" a given linear code into one with the desired minimum distance or to produce a bunch of linear codes with one and a same parameters. In a similar vein, one looks for families of Hamming isometries of linear codes. In order to produce linear codes with given parameters or to construct their Hamming isometries, one makes use of structures, arising from other branches of mathematics. In the early 80’s, Goppa introduced the classical algebraic geometric codes. They consist of values of global holomorphic sections of line bundles over curves. Various geometric properties of the corresponding line bundles estimate the dimension and the minimum distance. Under the presence of extra assumptions, one is able to predict the exact values of these parameters. The present work introduces another application of algebraic geometry to the coding theory. It interprets the finite Zariski tangent spaces to an affine variety X as linear codes. If the degree of X is relatively prime to the characteristic p of the definition field of X, the generic minimum distance d of a finite Zariski tangent space to X is characterized by a global geometric property of X. We provide an effective procedure in a form of a pseudo-code, obtaining d and points a ∈ X, at which d is attained. On the other hand, an explicit construction of an affine variety X, whose This research is partially supported by Contract 101 / 19.04.2013 and Contract 015 / 9.04.2014 with the Scientific Foundation of the University of Sofia degree is divisible by p, illustrates the possibility for incorporating codes of arbitrary minimum distance within a single Zariski tangent bundle. Here is a synopsis of the paper. The first section characterizes the generic minimum distance of a tangent code Ta(X,Fqm) to an affine variety X ⊂ Fqn, defined over Fq, whose degree is relatively prime to p = charFq. More precisely, subsection 1.1 recalls the notion of a Zariski tangent space to an affine variety. The next subsection 1.2 translates the minimum distance d of a linear code in terms of its punctures at d−1 or d coordinates. This description becomes handy for the characterization of the generic minimum distance of a tangent code Ta(X,Fqm) to X, as far as the puncturings of Ta(X,Fqm) coincide with the differentials of the associated puncturings of X at a ∈ X. The same subsection 1.2 introduces the genericity index d− 1 of an irreducible affine varietyX ⊂ Fqn as the maximal non-negative integer, for which any n−d+1 coordinate functions xj1+I(X,Fq), . . . , xjn−d+1+I(X,Fq) ∈ Fq(X) contain a transcendence basis of the function field Fq(X) of X over the algebraic closure Fq of the definition field Fq of X. Let X/Fq ⊂ Fqn be an irreducible, (d−1)-generic affine variety, defined over Fq, whose degree is not divisible by p = charFq. Subsection 1.3 establishes that the generic finite tangent spaces to a complement of a hypersurface in X are of minimum distance d. This involves an analogue of the Implicit Function Theorem, deriving a lower bound on the genericity index of X from a lower bound on the generic minimum distance of a tangent code. A sort of an inverse statement of the Implicit Function Theorem provides a lower bound on the generic minimum distance of a finite Zariski tangent space to X, which follows from a lower bound on the genericity index of X. The second section is devoted to an algorithm for obtaining the generic minimum distance of a tangent code and points, at which it is attained. Its first subsection recalls the Hilbert polynomial of an affine variety and an algorithm for its computing. It sketches one of the well known explicit procedures for decomposition of an affine variety into a union of irreducible components. Subsection 2.2 proposes an easy way for computing the genericity index d − 1 of an irreducible affine variety X ⊂ Fqn, out of the set of all the coordinate transcendence bases of the function field Fq(X) of X over Fq. The third subsection provides an algorithm for obtaining the discriminant locus of all the puncturings of X at d−1 coordinates and explicit points from its complement. Subsection 2.4 synthesizes the algorithms from the previous three subsections in a pseudo-code. The third section illustrates the stabilization, respectively, the destabilization of the minimum distance within a Zariski tangent bundle to an affine variety X ⊂ Fqn, whose degree is not divisible, respectively, is divisible by the characteristic p = charFq. This is done in the corresponding subsections by explicit constructions of the desired affine varieties X by their defining equations. The last, fourth section relates the linear Hamming isometries of the tangent codes Ta(X,Fqm) to an affine variety X ⊂ Fqn with the differentials of appropriate morphisms of X. Subsection 4.1 provides a pattern for construction of a morphism ψ : Fq n → Fqn, whose differentials restrict to linear Hamming isometries (dψ)a : Ta(X,Fqm) → Tψ(a)(ψ(X),Fqm) on the generic Zariski tangent spaces to the affine varieties X ⊂ Fqn, which are not contained in an explicitly given hypersurface V (ψo) ⊂ Fqn, depending on ψ. Subsection 4.2 realizes arbitrary families of linear Hamming isometries Fq → Fq by differentials of appropriate explicit morphisms Fq n → Fqn. It observes also that the Frobenius automorphism Φq of