The dual minimum distance of arbitrary dimensional algebraic--geometric codes

源自极大曲线的自正交码与自对偶码

Self-orthogonal and self-dual codes from maximal curves

本文利用有限域代数结构和代数曲线几何性质描述微分，构造了参数 $[n, k, d]_{q^2}$ 且 $k+d$ 接近 $n$ 的自正交与自对偶码，以及大最小距离的量子码。

On the minimum distance and the minimum weight of Goppa codes from a quotient of the Hermitian curve

In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form $y^q+y=x^m$, $q$ being a prime power and $m$ a positive integer which divides $q+1$. The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual codes through a geometric characterization of the supports, and provide their number. Finally, we apply our results to describe Goppa codes of classical interest on such curves.

Differential approach for the study of duals of algebraic-geometric codes on surfaces

The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this geometrical description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code on a curve is a differential code. We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples of codes on surfaces, and in particular surfaces with Picard number 1 like elliptic quadric…

The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes

A flag of codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}_q^n$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q^*)^n$ such that the code $C_i$ is {\bf x}-isometric to the dual code $C_{s-i}^\perp$ for all $i=0,\ldots, s$. For $P$ and $Q$ rational places in a function field ${\mathcal F}$, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes $$C_\mathcal L(D, a_0P+bQ)\subsetneq C_\mathcal L(D, a_1P+bQ)\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+bQ),$$ …

双曲面上的代数几何码的对偶

Dual of Algebraic Geometry codes from Hirzebruch surfaces

本文研究了Hirzebruch面上参数为aS_e + bF_e的代数几何码C_e(a,b)的对偶形式，并给出了其最小距离下界，同时探讨了码的截断及量子码构造。

Self-dual projective algebraic varieties associated with symmetric spaces

We discover a class of projective self-dual algebraic varieties. Namely, we consider actions of isotropy groups of complex symmetric spaces on the projectivized nilpotent varieties of isotropy modules. For them, we classify all orbit closures $X$ such that $X=\check X$ where $\check X$ is the projective dual of $X$. We give algebraic criteria of projective self-duality for the considered orbit closures.

Bounding minimum distances of cyclic codes using algebraic geometry

There are many results on the minimum distance of a cyclic code of the form that if a certain set T is a subset of the defining set of the code, then the minimum distance of the code is greater than some integer t. This includes the BCH, Hartmann-Tzeng, Roos, and shift bounds and generalizations of these. In this paper we define certain projective varieties V(T,t) whose properties determine whether, if T is in the defining set, the code has minimum distance exceeding t. Thus our attention shifts to the study of these varieties. By investigating them using class field theory and arithmetical ge…

Algebraic Boundaries of Convex Semi-algebraic Sets

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalize the correspondence of facets of a polytope to the vertices of the dual polytope to general semi-algebraic convex bodies. In the general setup, exceptional families of extreme points might exist and we characterize them semi-algebraically. We also give an algorithm to compute a complete list of exceptional families, given the algebraic boundary of the dual convex set.

Geometric Langlands duality and representations of algebraic groups over commutative rings

In this paper we give a geometric version of the Satake isomorphism. Given a connected complex reductive algebraic group, we show that the category of representations of its Langlands dual is naturally equivalent to a certain category of perverse sheaves on the complex affine Grassmannian. We can work with perverse sheaves with coefficients in an arbitrary commutative ring and then we recover the representation theory of the split form of the dual group over the commutative ring.

交换代数群的局部对偶定理

Local duality theorems for commutative algebraic groups

本研究构建任意域上含交换代数群对偶对象的k-1-motives范畴，建立任意特征局部域下关联群fppf上同调与对偶k-1-motive超上同调的Pontryagin对偶定理，还得到任意k-1-motive二阶上同调的对偶定理，可推广Lichtenbaum-van Hamel对偶。

Minimum distance functions of graded ideals and Reed-Muller-type codes

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives an algebraic formulation of the minimum distance of a projective Reed-Muller-type code in terms of the algebraic invariants and structure of the underlying vanishing ideal. Then we give a method, based on Groebner bases and Hilbert functions, to find lower bounds for the minimum distance of certain Reed-Muller-type codes. Finally we show explicit upper bound…

Projective duals to algebraic and tropical hypersurfaces

We study a tropical analogue of the projective dual variety of a hypersurface. When $X$ is a curve in $\mathbb{P}^2$ or a surface in $\mathbb{P}^3$, we provide an explicit description of $\text{Trop}(X^*)$ in terms of $\text{Trop}(X)$, as long as $\text{Trop}(X)$ is smooth and satisfies a mild genericity condition. As a consequence, when $X$ is a curve we describe the transformation of Newton polygons under projective duality, and recover classical formulas for the degree of a dual plane curve. For higher dimensional hypersurfaces $X$, we give a partial description of $\text{Trop}(X^*)$.

Quasi-period collapse for duals to Fano polygons: an explanation arising from algebraic geometry

The Ehrhart quasi-polynomial of a rational polytope $P$ is a fundamental invariant counting lattice points in integer dilates of $P$. The quasi-period of this quasi-polynomial divides the denominator of $P$ but is not always equal to it: this situation is called quasi-period collapse. Polytopes experiencing quasi-period collapse appear widely across algebra and geometry, and yet the phenomenon remains largely mysterious. Using techniques from algebraic geometry - specifically the $\mathbb{Q}$-Gorenstein deformation theory of orbifold del Pezzo surfaces - we explain quasi-period collapse for ra…

Homological Projective Duality

We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties $X$ and $Y$ in dual projective spaces are Homologically Projectively Dual, then we prove that the orthogonal linear sections of $X$ and $Y$ admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate Homological Projective Duality for projectivizations of vec…

Eliminating dual spaces

Macaulay dual spaces provide a local description of an affine scheme and give rise to computational machinery that is compatible with the methods of numerical algebraic geometry. We introduce eliminating dual spaces, use them for computing dual spaces of quotient ideals, and develop an algorithm for detection of embedded points on an algebraic curve.

The dual geometry of Hermitian two-point codes

In this paper we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through an explicit geometric characterization of their supports. In particular, we show that they appear as sets of collinear points in many cases.

对称星变换及其对偶微分算子的代数几何

Symmetric star transforms and the algebraic geometry of their dual differential operators

本文研究星变换的对偶微分算子代数几何性质，证明对称分支向量下算子象征属于不变多项式环，退化对称星变换对应初等对称多项式零点集的线性子空间，二维不可逆星变换对应Cayley结点三次曲面的实直线。

Generalized Albanese and its dual

We consider categories of rational maps to algebraic groups and study existence and construction of universal objects for such categories, using the duality theory of Laumon 1-motives. In particular, we obtain functorial descriptions of the Albanese of a singular projective variety X over an algebraically closed field of characteristic zero, which is the universal regular quotient of a relative Chow group of points on X, and of its dual.

Tannaka Duality for Geometric Stacks

We show that, under appropriate hypothesis, the groupoid of maps from S to an an algebraic stack X can be identified with a category of tensor functors from coherent sheaves on X to coherent sheaves on S. As an application, we show that if S is a proper variety over the field of complex numbers, then every ``analytic'' map from S to X is ``algebraic''.

凸半代数集代数边界的惠特尼分层

Whitney Stratification of Algebraic Boundaries of Convex Semi-algebraic Sets

本文将凸半代数集代数边界的迭代奇异轨迹分层细化为 Whitney (a) 分层，并基于 Teissier 判据提出利用余正规空间和素分解的计算算法，为对偶凸体代数边界的不可约分量提供候选簇列表。

The dual Kaehler cone of compact Kaehler threefolds

We consider a compact Kaehler manifold whose dual Kaehler cone contains a rational interior point. The general problem we have in mind is how far the manifold is from being projective; i.e. we ask for the algebraic dimension. We prove e.g. that if the interior point is represented by an effective curve in dimension 3, then the manifold has algebraic dimension at least 2 unless the variety is a so-called simple non-Kummer 3-fold. We also investigate 3-folds containing curves with ample normal bundle. It seems likely that examples with algebraic dimension 2 really exist.

On symbolic group varieties and dual surjunctivity

Let $G$ be a group. Let $X$ be an algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of rational points of $X$. We study algebraic group cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M \to X$ where $M$ is a finite memory. When $G$ is sofic and $K$ is uncountable, we show that if $τ$ is post-surjective then it is weakly pre-injective. Our result extends the dual version of Gottschalk's Conjecture for finite alphabets proposed by Capobianco, Kari, and Taati. When $G$ is amenable, we prove that …

Tropical Effective Primary and Dual Nullstellensätze

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz and moreover we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomial…

关于一般内积下秩一近似问题的最小代数复杂度

On the minimal algebraic complexity of the rank-one approximation problem for general inner products

本文研究了在一般内积下秩一近似问题的最小代数复杂度，通过验证弗罗贝尼乌斯内积为ED度数的局部最小值并探讨其全局最小性，结合奇点理论、莫尔斯理论和代数几何工具进行分析。