Extending work of M. Zarzar, we evaluate the potential of Goppatype evaluation codes constructed ... more Extending work of M. Zarzar, we evaluate the potential of Goppatype evaluation codes constructed from linear systems on projective algebraic surfaces with small Picard number. Putting this condition on the Picard number provides some control over the numbers of irreducible components of curves on the surface and hence over the minimum distance of the codes. We find that such surfaces do not automatically produce good codes; the sectional genus of the surface also has a major influence. Using that additional invariant, we derive bounds on the minimum distance under the assumption that the hyperplane section class generates the Néron-Severi group. We also give several examples of codes from such surfaces with minimum distance better than the best known bounds in Grassl's tables.
Advances in Biochemical Engineering/Biotechnology, 2018
After initial publication of the book, various errors were identified that needed correction. The... more After initial publication of the book, various errors were identified that needed correction. The following corrections have been updated within the current version, along with all known typographical errors.
If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, a... more If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, and K on ΓA cutting those sides in the same ratio: AH : HB = BΘ : ΘΓ = ΓK : KA, then Pappus of Alexandria proved that the triangles ABΓ and HΘK have the same centroid (center of mass). We present two proofs of this result: an English translation of Pappus's original synthetic proof and a modern algebraic proof making use of Cartesian coordinates and vector concepts. Comparing the two methods, we can see that while the algebraic proof gets to the heart of the matter more efficiently, the synthetic proof does a better job of revealing hidden aspects of the geometric configuration. Moreover, as Pappus presents it, the synthetic proof provides a real element of surprise and a sense of discovering unexpected connections. We conclude with some general observations about synthetic versus algebraic techniques in geometry and in the teaching and learning of mathematics.
Preface to the Second Edition The first edition of this book was published 5 years ago. When we h... more Preface to the Second Edition The first edition of this book was published 5 years ago. When we have been asked to prepare another edition, we decided not only to correct typographical errors, update the references, and improve some of the proofs but also to add new material, some appearing in printed form for the first time. The major changes in this edition are the following: (1) A new section about non-commutative Gröbner basis is added to chapter one, written mainly by Viktor Levandovskyy. (2) Two new sections about characteristic sets and triangular sets together with the corresponding decomposition-algorithm are added to chapter four. (3) There is a new appendix about polynomial factorization containing univariate factorization over F p and Q and algebraic extensions, as well as multivariate factorization over these fields and over the algebraic closure of Q. (4) The system Singular has improved quite a lot. A new CD is included, containing the version 3-0-3 with all examples of the book and several new Singular-libraries. (5) The appendix concerning Singular is rewritten corresponding to the version 3-0-3. In particular, more examples on how to write libraries and about the communication with other systems are given. We should like to thank many readers for helpful comments and finding typographical errors in the first edition. We thank the Singular Team for the support in producing the new CD. Special thanks to
In this paper we classify the singular curves whose theta divisors in their generalized Jacobians... more In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.
This essay describes the author\u27s recent encounter with two well-known passages in Plutarch th... more This essay describes the author\u27s recent encounter with two well-known passages in Plutarch that touch on a crucial episode in the history of the Greek mathematics of the fourth century BCE involving various approaches to the problem of the duplication of the cube. One theme will be the way key sources for understanding the history of our subject sometimes come from texts that have much wider cultural contexts and resonances. Sensitivity to the history, to the mathematics, and to the language is necessary to tease out the meaning of such texts. However, in the past, historians of mathematics often interpreted these sources using the mathematics of their own times. Their sometimes anachronistic accounts have often been presented in the mainstream histories of mathematics to which mathematicians who do not read Greek must turn to learn about that history. With the original sources, the tidy and inevitable picture of the development of mathematics disappears and we are left with a m...
We study a sort of analog of the key equation for decoding Reed-Solomon and BCH codes and identif... more We study a sort of analog of the key equation for decoding Reed-Solomon and BCH codes and identify a key equation for all codes from order domains which have finitely-generated value semigroups (the field of fractions of the order domain may have arbitrary transcendence degree, however). We provide a natural interpretation of the construction using the theory of Macaulay's inverse systems and duality. O'Sullivan's generalized Berlekamp-Massey-Sakata (BMS) decoding algorithm applies to the duals of suitable evaluation codes from these order domains. When the BMS algorithm does apply, we will show how it can be understood as a process for constructing a collection of solutions of our key equation.
This chapter will study systematic methods for eliminating variables from systems of polynomial e... more This chapter will study systematic methods for eliminating variables from systems of polynomial equations. The basic strategy of elimination theory will be given in two main theorems: the Elimination Theorem and the Extension Theorem. We will prove these results using Groebner bases and the classic theory of resultants. The geometric interpretation of elimination will also be explored when we discuss the Closure Theorem. Of the many applications of elimination theory, we will treat two in detail: the implicitization problem and the envelope of a family of curves.
ABSTRACT This appendix will discuss several computer algebra systems that can be used in conjunct... more ABSTRACT This appendix will discuss several computer algebra systems that can be used in conjunction with the text. We will describe AXIOM, Maple, Mathematica and REDUCE in some detail, and then mention some other systems. These are all amazingly powerful programs, and our brief discussion will not do justice to their true capability.
A canonically-embedded curve of genus $g$ is a pure 1-dimensional, non-degenerate subscheme $C$ o... more A canonically-embedded curve of genus $g$ is a pure 1-dimensional, non-degenerate subscheme $C$ of ${\bf P}^{g-1}$ over an algebraically closed field $k$, for which ${\cal O}_C(1) \cong \omega_C$, (the dualizing sheaf)$ and $h^0(C, {\cal O}_C) = 1$, $h^0(C, \omega_C) = g$. The singularities of $C$ (if any) are Gorenstein, and $C$ is connected of degree $2g-2$ and arithmetic genus $g$. In a recent paper, Schreyer has proved that Petri's normalization of the homogeneous ideal $I(C)$ of a smooth canonically-embedded curve can be also carried out for singular curves, provided that the curve has a simple $(g-2)$-secant (a linear ${\bf P}^{g-3}$ intersecting $C$ transversely at exactly $g-2$ (smooth) points). We use the Petri normalization to study the Hilbert scheme of curves of degree $2g-2$ and arithmetic genus $g$ in ${\bf P}^{g-1}$ in the low-genus cases $g = 5,6$. The main results are that the Hilbert points of all curves for which Petri's approach applies lie on one irred...
The recent extensive work on different approaches to the Schottky problem has produced marked pro... more The recent extensive work on different approaches to the Schottky problem has produced marked progress on several fronts. At the same time, it has become apparent that there exist very close connections between the various characterizations of Jacobian varieties described in Mumford's classic lectures {\it Curves and Their Jacobians\/}. Until now, the approach via double translation manifolds has seemed to be quite different from other approaches to the Schottky problem. The purpose of this paper is to bring this last approach ``into the fold'' as it were, and to show precisely how it relates to characterizations of Jacobians based on trisecants and flexes of the Kummer variety, and the K.P. equation.
Let X be an irreducible rational nodal curve of arithmetic genus g > 2, and let S* be a non-speci... more Let X be an irreducible rational nodal curve of arithmetic genus g > 2, and let S* be a non-special, effective invertible sheaf on X. Let W(= S ?) denote the set of smooth Weierstrass points of 3? and all its positive tensor powers on I. In this paper, we study the distribution of W{3?) on X. In particular, we will show that is not dense on X, generalizing an example of R. F. Lax.
Any linear code with a nontrivial automorphism has the structure of a module over a polynomial ri... more Any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring. The theory of Griihner bases for modules gives a compact description and implementation of a systematic encoder. We present examples of algebraic-geometric Goppa codes that can be encoded by these methods, including the one-point Hermitian codes. By row operations and, if necessary, column interchanges (i.e., relabeling the points of D), we can take M to the form M' = [Ik (B] (1) where I, is a k x k identity matrix, and B is some Ic x (n-k) matrix. Multiplying the row vector w E Ft on the right by Manuscript
We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf ... more We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf 𝓛 on an irreducible rational curve X with g ≧ 2 ordinary cusps. Using an idea from B. Olsen's study of the analogous question on smooth curves, and an explicit formula for the "theta function" of a cuspidal rational curve, we show that W(𝓛) is never dense on X (in contrast to the case of smooth curves of genus g ≧ 2).
Let X,,, denote the Hermitian curve xm+' = ym + y over the field F,,,z. Let Q be the single point... more Let X,,, denote the Hermitian curve xm+' = ym + y over the field F,,,z. Let Q be the single point at infinity, and let D be the sum of the other rn3 points of X,,, rational over F,,,z, each with multiplicity 1. X,,, has a cyclic group of automorphisms of order m2-1, which induces automorphisms of each of the the one-point algebraic geometric Goppa codes &(D,uQ) and their duals. As a result, these codes have the structure of modules over the ring F,[t], and this structure can be used to good effect in both encoding and decoding. In this paper we examine the algebraic structure of these modules by means of the theory of Groebner bases. We introduce a roof diagram for each of these codes (analogous to the set of roots for a cyclic code of length q-1 over Fq), and show how the root diagram may be determined combinatorially from a. We also give a specialized algorithm for computing Groebner bases, adapted to these particular modules. This algorithm has a much lower complexity than general Groebner basis algorithms, and has been successfully implemented in the Maple computer algebra system. This permits the computation of Groebner bases and the construction of compact systematic encoders for some quite large codes (e.g. codes such as C~(D,4010Q) on the curve X16, with parameters n = 4096, k = 3891). @ 1997 Elsevier Science B.V.
John B. Little is the translator. This is a Latin to English translation of Geometria Practica by... more John B. Little is the translator. This is a Latin to English translation of Geometria Practica by Chrisopher Clavius, S.J. (1538-1612), the preeminent Jesuit mathematician and mathematical astronomer of his time. The first edition of Geometria Practica appeared in 1604. This translation is of the second edition from 1606, produced by the printshop of Johann Albin in Mainz. In preparing this translation we have made use of the electronic version of the 1606 edition of the Geometria Practica maintained by the Bayerische StaatsBibliothek. In particular, all of the figures have been copied from the scanned images here. The typesetting was done with the LaTeX system. In an attempt to duplicate the organization of the original book as much as possible, the marginal references and labels as in Clavius\u27s original have been included. References in the form Book X, Prop. Y are references to Clavius\u27s own edition of Euclid\u27s Elements. This was very influential and a standard text in J...
We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and compr... more We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and comprehensive textbook of practical geometry, whose first edition appeared in 1604. Our focus is on four particular sections from Books IV and VI where Clavius has either used his sources in an interesting way or where he has been uncharacteristically reticent about them. These include the treatments of Heron’s Formula, Archimedes’ Measurement of the Circle, four methods for constructing two mean proportionals between two lines, and finally an algorithm for computing nth roots of numbers.
Toric codes are a class of m-dimensional cyclic codes introduced recently by J. Hansen in [7], [8... more Toric codes are a class of m-dimensional cyclic codes introduced recently by J. Hansen in [7], [8], and studied in [9], [5], [10]. They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope P ⊆ R m. As such, they are in a sense a natural extension of Reed-Solomon codes. Several articles cited above use intersection theory on toric surfaces to derive bounds on the minimum distance of some toric codes with m = 2. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all m ≥ 2. Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from rectangular polytopes and simplices. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope P 1 to a second polytope P 2 , then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension.
Extending work of M. Zarzar, we evaluate the potential of Goppatype evaluation codes constructed ... more Extending work of M. Zarzar, we evaluate the potential of Goppatype evaluation codes constructed from linear systems on projective algebraic surfaces with small Picard number. Putting this condition on the Picard number provides some control over the numbers of irreducible components of curves on the surface and hence over the minimum distance of the codes. We find that such surfaces do not automatically produce good codes; the sectional genus of the surface also has a major influence. Using that additional invariant, we derive bounds on the minimum distance under the assumption that the hyperplane section class generates the Néron-Severi group. We also give several examples of codes from such surfaces with minimum distance better than the best known bounds in Grassl's tables.
Advances in Biochemical Engineering/Biotechnology, 2018
After initial publication of the book, various errors were identified that needed correction. The... more After initial publication of the book, various errors were identified that needed correction. The following corrections have been updated within the current version, along with all known typographical errors.
If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, a... more If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, and K on ΓA cutting those sides in the same ratio: AH : HB = BΘ : ΘΓ = ΓK : KA, then Pappus of Alexandria proved that the triangles ABΓ and HΘK have the same centroid (center of mass). We present two proofs of this result: an English translation of Pappus's original synthetic proof and a modern algebraic proof making use of Cartesian coordinates and vector concepts. Comparing the two methods, we can see that while the algebraic proof gets to the heart of the matter more efficiently, the synthetic proof does a better job of revealing hidden aspects of the geometric configuration. Moreover, as Pappus presents it, the synthetic proof provides a real element of surprise and a sense of discovering unexpected connections. We conclude with some general observations about synthetic versus algebraic techniques in geometry and in the teaching and learning of mathematics.
Preface to the Second Edition The first edition of this book was published 5 years ago. When we h... more Preface to the Second Edition The first edition of this book was published 5 years ago. When we have been asked to prepare another edition, we decided not only to correct typographical errors, update the references, and improve some of the proofs but also to add new material, some appearing in printed form for the first time. The major changes in this edition are the following: (1) A new section about non-commutative Gröbner basis is added to chapter one, written mainly by Viktor Levandovskyy. (2) Two new sections about characteristic sets and triangular sets together with the corresponding decomposition-algorithm are added to chapter four. (3) There is a new appendix about polynomial factorization containing univariate factorization over F p and Q and algebraic extensions, as well as multivariate factorization over these fields and over the algebraic closure of Q. (4) The system Singular has improved quite a lot. A new CD is included, containing the version 3-0-3 with all examples of the book and several new Singular-libraries. (5) The appendix concerning Singular is rewritten corresponding to the version 3-0-3. In particular, more examples on how to write libraries and about the communication with other systems are given. We should like to thank many readers for helpful comments and finding typographical errors in the first edition. We thank the Singular Team for the support in producing the new CD. Special thanks to
In this paper we classify the singular curves whose theta divisors in their generalized Jacobians... more In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.
This essay describes the author\u27s recent encounter with two well-known passages in Plutarch th... more This essay describes the author\u27s recent encounter with two well-known passages in Plutarch that touch on a crucial episode in the history of the Greek mathematics of the fourth century BCE involving various approaches to the problem of the duplication of the cube. One theme will be the way key sources for understanding the history of our subject sometimes come from texts that have much wider cultural contexts and resonances. Sensitivity to the history, to the mathematics, and to the language is necessary to tease out the meaning of such texts. However, in the past, historians of mathematics often interpreted these sources using the mathematics of their own times. Their sometimes anachronistic accounts have often been presented in the mainstream histories of mathematics to which mathematicians who do not read Greek must turn to learn about that history. With the original sources, the tidy and inevitable picture of the development of mathematics disappears and we are left with a m...
We study a sort of analog of the key equation for decoding Reed-Solomon and BCH codes and identif... more We study a sort of analog of the key equation for decoding Reed-Solomon and BCH codes and identify a key equation for all codes from order domains which have finitely-generated value semigroups (the field of fractions of the order domain may have arbitrary transcendence degree, however). We provide a natural interpretation of the construction using the theory of Macaulay's inverse systems and duality. O'Sullivan's generalized Berlekamp-Massey-Sakata (BMS) decoding algorithm applies to the duals of suitable evaluation codes from these order domains. When the BMS algorithm does apply, we will show how it can be understood as a process for constructing a collection of solutions of our key equation.
This chapter will study systematic methods for eliminating variables from systems of polynomial e... more This chapter will study systematic methods for eliminating variables from systems of polynomial equations. The basic strategy of elimination theory will be given in two main theorems: the Elimination Theorem and the Extension Theorem. We will prove these results using Groebner bases and the classic theory of resultants. The geometric interpretation of elimination will also be explored when we discuss the Closure Theorem. Of the many applications of elimination theory, we will treat two in detail: the implicitization problem and the envelope of a family of curves.
ABSTRACT This appendix will discuss several computer algebra systems that can be used in conjunct... more ABSTRACT This appendix will discuss several computer algebra systems that can be used in conjunction with the text. We will describe AXIOM, Maple, Mathematica and REDUCE in some detail, and then mention some other systems. These are all amazingly powerful programs, and our brief discussion will not do justice to their true capability.
A canonically-embedded curve of genus $g$ is a pure 1-dimensional, non-degenerate subscheme $C$ o... more A canonically-embedded curve of genus $g$ is a pure 1-dimensional, non-degenerate subscheme $C$ of ${\bf P}^{g-1}$ over an algebraically closed field $k$, for which ${\cal O}_C(1) \cong \omega_C$, (the dualizing sheaf)$ and $h^0(C, {\cal O}_C) = 1$, $h^0(C, \omega_C) = g$. The singularities of $C$ (if any) are Gorenstein, and $C$ is connected of degree $2g-2$ and arithmetic genus $g$. In a recent paper, Schreyer has proved that Petri's normalization of the homogeneous ideal $I(C)$ of a smooth canonically-embedded curve can be also carried out for singular curves, provided that the curve has a simple $(g-2)$-secant (a linear ${\bf P}^{g-3}$ intersecting $C$ transversely at exactly $g-2$ (smooth) points). We use the Petri normalization to study the Hilbert scheme of curves of degree $2g-2$ and arithmetic genus $g$ in ${\bf P}^{g-1}$ in the low-genus cases $g = 5,6$. The main results are that the Hilbert points of all curves for which Petri's approach applies lie on one irred...
The recent extensive work on different approaches to the Schottky problem has produced marked pro... more The recent extensive work on different approaches to the Schottky problem has produced marked progress on several fronts. At the same time, it has become apparent that there exist very close connections between the various characterizations of Jacobian varieties described in Mumford's classic lectures {\it Curves and Their Jacobians\/}. Until now, the approach via double translation manifolds has seemed to be quite different from other approaches to the Schottky problem. The purpose of this paper is to bring this last approach ``into the fold'' as it were, and to show precisely how it relates to characterizations of Jacobians based on trisecants and flexes of the Kummer variety, and the K.P. equation.
Let X be an irreducible rational nodal curve of arithmetic genus g > 2, and let S* be a non-speci... more Let X be an irreducible rational nodal curve of arithmetic genus g > 2, and let S* be a non-special, effective invertible sheaf on X. Let W(= S ?) denote the set of smooth Weierstrass points of 3? and all its positive tensor powers on I. In this paper, we study the distribution of W{3?) on X. In particular, we will show that is not dense on X, generalizing an example of R. F. Lax.
Any linear code with a nontrivial automorphism has the structure of a module over a polynomial ri... more Any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring. The theory of Griihner bases for modules gives a compact description and implementation of a systematic encoder. We present examples of algebraic-geometric Goppa codes that can be encoded by these methods, including the one-point Hermitian codes. By row operations and, if necessary, column interchanges (i.e., relabeling the points of D), we can take M to the form M' = [Ik (B] (1) where I, is a k x k identity matrix, and B is some Ic x (n-k) matrix. Multiplying the row vector w E Ft on the right by Manuscript
We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf ... more We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf 𝓛 on an irreducible rational curve X with g ≧ 2 ordinary cusps. Using an idea from B. Olsen's study of the analogous question on smooth curves, and an explicit formula for the "theta function" of a cuspidal rational curve, we show that W(𝓛) is never dense on X (in contrast to the case of smooth curves of genus g ≧ 2).
Let X,,, denote the Hermitian curve xm+' = ym + y over the field F,,,z. Let Q be the single point... more Let X,,, denote the Hermitian curve xm+' = ym + y over the field F,,,z. Let Q be the single point at infinity, and let D be the sum of the other rn3 points of X,,, rational over F,,,z, each with multiplicity 1. X,,, has a cyclic group of automorphisms of order m2-1, which induces automorphisms of each of the the one-point algebraic geometric Goppa codes &(D,uQ) and their duals. As a result, these codes have the structure of modules over the ring F,[t], and this structure can be used to good effect in both encoding and decoding. In this paper we examine the algebraic structure of these modules by means of the theory of Groebner bases. We introduce a roof diagram for each of these codes (analogous to the set of roots for a cyclic code of length q-1 over Fq), and show how the root diagram may be determined combinatorially from a. We also give a specialized algorithm for computing Groebner bases, adapted to these particular modules. This algorithm has a much lower complexity than general Groebner basis algorithms, and has been successfully implemented in the Maple computer algebra system. This permits the computation of Groebner bases and the construction of compact systematic encoders for some quite large codes (e.g. codes such as C~(D,4010Q) on the curve X16, with parameters n = 4096, k = 3891). @ 1997 Elsevier Science B.V.
John B. Little is the translator. This is a Latin to English translation of Geometria Practica by... more John B. Little is the translator. This is a Latin to English translation of Geometria Practica by Chrisopher Clavius, S.J. (1538-1612), the preeminent Jesuit mathematician and mathematical astronomer of his time. The first edition of Geometria Practica appeared in 1604. This translation is of the second edition from 1606, produced by the printshop of Johann Albin in Mainz. In preparing this translation we have made use of the electronic version of the 1606 edition of the Geometria Practica maintained by the Bayerische StaatsBibliothek. In particular, all of the figures have been copied from the scanned images here. The typesetting was done with the LaTeX system. In an attempt to duplicate the organization of the original book as much as possible, the marginal references and labels as in Clavius\u27s original have been included. References in the form Book X, Prop. Y are references to Clavius\u27s own edition of Euclid\u27s Elements. This was very influential and a standard text in J...
We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and compr... more We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and comprehensive textbook of practical geometry, whose first edition appeared in 1604. Our focus is on four particular sections from Books IV and VI where Clavius has either used his sources in an interesting way or where he has been uncharacteristically reticent about them. These include the treatments of Heron’s Formula, Archimedes’ Measurement of the Circle, four methods for constructing two mean proportionals between two lines, and finally an algorithm for computing nth roots of numbers.
Toric codes are a class of m-dimensional cyclic codes introduced recently by J. Hansen in [7], [8... more Toric codes are a class of m-dimensional cyclic codes introduced recently by J. Hansen in [7], [8], and studied in [9], [5], [10]. They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope P ⊆ R m. As such, they are in a sense a natural extension of Reed-Solomon codes. Several articles cited above use intersection theory on toric surfaces to derive bounds on the minimum distance of some toric codes with m = 2. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all m ≥ 2. Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from rectangular polytopes and simplices. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope P 1 to a second polytope P 2 , then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension.
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