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Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Groebner bases, to compute the length and the dimension of C_X*(d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that C_X^*(d) has the same basic parameters that C_Y(d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of C_X*(d). We show how to compute the vanishing ideals of X* and Y.
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We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus, with prescribed parameters. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.
Let K be a finite field. Let X* be a subset of the affine space Kn, which is parameterized by odd cycles. In this paper we give an explicit Grobner basis for the vanishing ideal, I(X*), of X*. We give an explicit formula for the regularity of I(X*) and finally if X* is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X* can be written as linear combination of Hilbert functions of degenerate torus.
Affine ind-varieties are infinite dimensional generalizations of algebraic varieties which appear naturally in many different contexts, in particular in the study of automorphism groups of affine spaces. In this article we introduce and develop the basic algebraic theory of topologically integrable derivations of complete topological rings. We establish a bijective algebro-geometric correspondence between additive group actions on affine ind-varieties and topologically integrable derivations of their coordinate pro-rings which extends the classical fruitful correspondence between additive group actions on affine varieties and locally nilpotent derivations of their coordinate rings.
The Cancellation Problem for Affine Spaces is settled affirmatively, that is, it is proved that : Let $ k $ be an algebraically closed field of characteristic zero and let $n, m in mathbb{N}$. If $R[Y_1,..., Y_m] cong_k k[X_1,..., X_{n+m}]$ as $k$-algebras, where $Y_1,..., Y_m, X_1,..., X_{n+m}$ are indeterminates, then $R cong_k k[X_1,..., X_n]$.
Let $mathbb{F}_{q}$ denote the finite field of order $q,$ let $m_1,m_2,cdots,m_{ell}$ be positive integers satisfying $gcd(m_i,q)=1$ for $1 leq i leq ell,$ and let $n=m_1+m_2+cdots+m_{ell}.$ Let $Lambda=(lambda_1,lambda_2,cdots,lambda_{ell})$ be fixed, where $lambda_1,lambda_2,cdots,lambda_{ell}$ are non-zero elements of $mathbb{F}_{q}.$ In this paper, we study the algebraic structure of $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ and their dual codes with respect to the standard inner product on $mathbb{F}_{q}^n.$ We provide necessary and sufficient conditions for the existence of a self-dual $Lambda$-multi-twisted code of length $n$ over $mathbb{F}_{q},$ and obtain enumeration formulae for all self-dual and self-orthogonal $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}.$ We also derive some sufficient conditions under which a $Lambda$-multi-twisted code is LCD. We determine the parity-check polynomial of all $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ from the generating sets of the codes. Besides this, we provide a trace description for all $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure.
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