Projective toric codes

2021 ◽  
pp. 1-26
Author(s):  
Jade Nardi
Keyword(s):  
Error Correcting Code ◽  
Convex Polytope ◽  
Global Section ◽  
Explicit Construction ◽  
Geometric Error ◽  
Integral Points ◽  
Toric Codes ◽  
Algorithmic Technique ◽  
The One ◽  
Integral Convex Polytope

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.

scholarly journals Multiple Observations for Secret-Key Binding with SRAM PUFs

2021 ◽  
Author(s):  
Lieneke Kusters ◽  
Frans M.J. Willems
Keyword(s):  
Error Correcting Code ◽  
Ldpc Code ◽  
Soft Decision ◽  
Secret Key ◽  
Multiple Observations ◽  
Reconstruction Performance ◽  
New Strategy ◽  
The One ◽  
Helper Data ◽  
Data Scheme

We present a new Multiple-Observations (MO) helper data scheme for secret-key binding to an SRAM PUF. This MO scheme binds a single key to multiple enrollment observations of the SRAM PUF. Performance is improved in comparison to classic schemes which generate helper data based on a single enrollment observation. The performance increase can be explained by the fact that the reliabilities of the different SRAM cells are modeled (implicitly) in the helper data. We prove that the scheme achieves secret-key capacity for any number of enrollment observations, and, therefore it is optimal. We evaluate performance of the scheme using Monte Carlo simulations, where an off-the-shelf LDPC code is used to implement the linear error-correcting code. Another scheme that models the reliabilities of the SRAM cells is the so-called Soft-Decision (SD) helper data scheme. The SD scheme considers the one-probabilities of the SRAM cells as an input, which in practice are not observable. We present a new strategy for the SD scheme that considers the binary SRAM-PUF observations as an input instead and show that the new strategy is optimal and achieves the same reconstruction performance as the MO scheme. Finally, we present a variation on the MO helper data scheme that updates the helper data sequentially after each successful reconstruction of the key. As a result, the error-correcting performance of the scheme is improved over time.

Quickest Change Point Detection in Shape Inspection of Additively Manufactured Parts Under a Multi-Resolution Framework

2020 ◽  
Author(s):  
Yu Jin ◽  
Haitao Liao ◽  
Harry Pierson
Keyword(s):  
Point Cloud ◽  
Geometric Error ◽  
Change Point Detection ◽  
Layer By Layer ◽  
Shape Inspection ◽  
The One ◽  
Alignment Errors ◽  
Point Detection

Abstract In-situ layer-by-layer inspection is essential to achieving the full capability and advantages of additive manufacturing in producing complex geometries. The shape of each inspected layer can be described by a 2D point cloud obtained by slicing a thin layer of 3D point cloud acquired from 3D scanning. In practice, a scanned shape must be aligned with the corresponding base-truth CAD model before evaluating its geometric accuracy. Indeed, the observed geometric error is attributed to systematic, random, and alignment errors, where the systematic error is the one that triggers an alarm of system anomalies. In this work, a quickest change detection (QCD) algorithm is applied under a multi-resolution alignment and inspection framework 1) to differentiate errors from different error sources, and 2) to identify the layer where the earliest systematic deviation distribution changes during the printing process. Numerical experiments and a case study on a human heart are conducted to illustrate the performance of the proposed method in detecting layer-wise geometric error.

Mathematical Symbols: Insight through Invention

1985 ◽  
Vol 32 (6) ◽  
pp. 20-22
Author(s):  
Daiyo Sawada
Keyword(s):  
Mathematics Education ◽  
Mathematical Thinking ◽  
The Other ◽  
Mathematical Symbols ◽  
Algorithmic Technique ◽  
The One ◽  
New Math ◽  
Basic Movement ◽  
Do So

Over the years, like a pendulum. the emphasis in mathematics education has swung from a focus on concepts and understanding (e.g., the new-math movement) on the one hand to skill with facts and algorithms (e.g., the back-to-basic movement) on the other. Currently. children can adequately perform algorithms, but they may do so with little understanding of the underlying concepts (Resnick 1982, 136–55). In part, the difficulty lies in students having lo t sight of the role of symbols in mathematical thinking. The development of approache. that help children integrate the insight of symbolic understanding with the power of algorithmic technique should be of value. Accordingly, the intent of this article is to suggest how children can be guided to see and personally feel the power and simplicity that thinking with and about mathematical symbols can bring to their algorithmic competence. Although, for the sake of concretene and pecificity, attention hall be confined to computation, stress shall be placed on an approach that the reader may find generalize to other areas.

scholarly journals Multiple Observations for Secret-Key Binding with SRAM PUFs

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 590
Author(s):  
Lieneke Kusters ◽  
Frans M. J. Willems
Keyword(s):  
Error Correcting Code ◽  
Ldpc Code ◽  
Soft Decision ◽  
Secret Key ◽  
Multiple Observations ◽  
Reconstruction Performance ◽  
New Strategy ◽  
The One ◽  
Helper Data ◽  
Data Scheme

We present a new Multiple-Observations (MO) helper data scheme for secret-key binding to an SRAM-PUF. This MO scheme binds a single key to multiple enrollment observations of the SRAM-PUF. Performance is improved in comparison to classic schemes which generate helper data based on a single enrollment observation. The performance increase can be explained by the fact that the reliabilities of the different SRAM cells are modeled (implicitly) in the helper data. We prove that the scheme achieves secret-key capacity for any number of enrollment observations, and, therefore, it is optimal. We evaluate performance of the scheme using Monte Carlo simulations, where an off-the-shelf LDPC code is used to implement the linear error-correcting code. Another scheme that models the reliabilities of the SRAM cells is the so-called Soft-Decision (SD) helper data scheme. The SD scheme considers the one-probabilities of the SRAM cells as an input, which in practice are not observable. We present a new strategy for the SD scheme that considers the binary SRAM-PUF observations as an input instead and show that the new strategy is optimal and achieves the same reconstruction performance as the MO scheme. Finally, we present a variation on the MO helper data scheme that updates the helper data sequentially after each successful reconstruction of the key. As a result, the error-correcting performance of the scheme is improved over time.

scholarly journals The $h$-Vector of a Gorenstein Toric Ring of a Compressed Polytope

10.37236/1891 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Hidefumi Ohsugi ◽  
Takayuki Hibi
Keyword(s):  
Convex Polytope ◽  
Integral Convex Polytope

A compressed polytope is an integral convex polytope all of whose pulling triangulations are unimodular. A $(q - 1)$-simplex $\Sigma$ each of whose vertices is a vertex of a convex polytope ${\cal P}$ is said to be a special simplex in ${\cal P}$ if each facet of ${\cal P}$ contains exactly $q - 1$ of the vertices of $\Sigma$. It will be proved that there is a special simplex in a compressed polytope ${\cal P}$ if (and only if) its toric ring $K[{\cal P}]$ is Gorenstein. In consequence it follows that the $h$-vector of a Gorenstein toric ring $K[{\cal P}]$ is unimodal if ${\cal P}$ is compressed.

scholarly journals Higher Spin Alternating Sign Matrices

10.37236/1001 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Roger E. Behrend ◽  
Vincent A. Knight
Keyword(s):  
Convex Polytope ◽  
Ehrhart Polynomial ◽  
Alternating Sign Matrices ◽  
Integer Points ◽  
Wall Boundary Conditions ◽  
Alternating Sign Matrix ◽  
Higher Spin ◽  
Statistical Mechanical ◽  
Sign Matrix ◽  
Integral Convex Polytope

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer $r$, all complete row and column sums are $r$, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin $r/2$ statistical mechanical vertex models with domain-wall boundary conditions. The case $r=1$ gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size $n$ are the integer points of the $r$-th dilate of an integral convex polytope of dimension $(n{-}1)^2$ whose vertices are the standard alternating sign matrices of size $n$. It then follows that, for fixed $n$, these matrices are enumerated by an Ehrhart polynomial in $r$.

scholarly journals Non-Normal Very Ample Polytopes and their Holes

10.37236/3656 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Akihiro Higashitani
Keyword(s):  
Convex Polytope ◽  
Integral Convex Polytope

In this paper, we show that for given integers $h$ and $d$ with $h \geq 1$ and $d \geq 3$, there exists a non-normal very ample integral convex polytope of dimension $d$ which has exactly $h$ holes.

scholarly journals Classification of Ehrhart polynomials of integral simplices

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Akihiro Higashitani
Keyword(s):  
Nous Avons ◽  
Normal Forms ◽  
Convex Polytope ◽  
Ehrhart Polynomials ◽  
International Audience ◽  
Integral Convex Polytope

International audience Let $δ (\mathcal{P} )=(δ _0,δ _1,\ldots,δ _d)$ be the $δ$ -vector of an integral convex polytope $\mathcal{P}$ of dimension $d$. First, by using two well-known inequalities on $δ$ -vectors, we classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i ≤3$. Moreover, by means of Hermite normal forms of square matrices, we also classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i = 4$. In addition, for $\sum_{i=0}^d δ _i ≥5$, we characterize the $δ$ -vectors of integral simplices when $\sum_{i=0}^d δ _i$ is prime. Soit $δ (\mathcal{P} )=(δ _0,δ _1,\ldots,δ _d)$ le $δ$ -vecteur d'un polytope intégrante de dimension d. Tout d'abord, en utilisant deux bien connus des inégalités sur $δ$ -vecteurs, nous classons les $δ$ -vecteurs possibles avec $\sum_{i=0}^d δ _i ≤3$ En outre, par le biais de Hermite formes normales, nous avons également classer les $δ$ -vecteurs avec $\sum_{i=0}^d δ _i = 4$. De plus, pour $\sum_{i=0}^d δ _i ≥5$, nous caractérisons les $δ$-vecteurs des simplex inégalités lorsque $\sum_{i=0}^d δ _i$ est premier.

scholarly journals PREPOTENTIALS, BILINEAR FORMS ON PERIODS AND ENHANCED GAUGE SYMMETRIES IN TYPE II STRINGS

1999 ◽  
Vol 14 (08) ◽  
pp. 1177-1203 ◽  
Author(s):  
TAKAHIRO MASUDA ◽  
HISAO SUZUKI
Keyword(s):  
Gauge Symmetry ◽  
Bilinear Form ◽  
Explicit Construction ◽  
Bilinear Forms ◽  
Type Ii ◽  
Yang Mills ◽  
Gauge Symmetries ◽  
The One ◽  
Flat Coordinates

We construct a bilinear form on the periods of Calabi–Yau spaces. This is used to obtain the prepotentials around conifold singularities in type II strings compactified on Calabi–Yau space. Explicit construction of the bilinear forms is achieved for the one-modulus models as well as two-modulus models with K3 fibrations where the enhanced gauge symmetry is known to be observed at the conifold locus. We also show how these bilinear forms are related with the existence of flat coordinates. We list the resulting prepotentials in two-modulus models around the conifold locus, which contains α′ corrections of 4D N=2 SUSY SU(2) Yang–Mills theory as the stringy effect.

scholarly journals DUAL POLYHEDRA, MIRROR SYMMETRY AND GINZBURG-LANDAU ORBIFOLDS

1996 ◽  
Vol 11 (05) ◽  
pp. 389-396 ◽  
Author(s):  
HITOSHI SATO
Keyword(s):  
Mirror Symmetry ◽  
Toric Geometry ◽  
Integral Points ◽  
Ginzburg Landau ◽  
Geometrical Features ◽  
Blowing Up ◽  
Typical Type ◽  
One To One ◽  
The One ◽  
Mirror Map

New geometrical features of the Ginzburg-Landau orbifolds are presented, for models with a typical type of superpotential. We show the one-to-one correspondence between some of the (a, c) states with U(1) charges (−1, 1) and the integral points on the dual polyhedra, which are useful tools for the construction of mirror manifolds. Relying on toric geometry, these states are shown to correspond to the (1, 1) forms coming from blowing-up processes. In terms of the above identification, it can be checked that the monomial-divisor mirror map for Ginzburg-Landau orbifolds, proposed by the author, is equivalent to that mirror map for Calabi-Yau manifolds obtained by the mathematicians.

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