**Contemporary Mathematics**

Volume: 637;
2015;
306 pp;
Softcover

MSC: Primary 11; 14; 94;

**Print ISBN: 978-1-4704-1461-0
Product Code: CONM/637**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-2339-1
Product Code: CONM/637.E**

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# Algorithmic Arithmetic, Geometry, and Coding Theory

Share this page *Edited by *
*Stéphane Ballet; Marc Perret; Alexey Zaytsev*

This volume contains the proceedings of the
14th International Conference on Arithmetic, Geometry, Cryptography,
and Coding Theory (AGCT), held June 3–7, 2013, at CIRM, Marseille,
France.

These international conferences, held every two years, have been a
major event in the area of algorithmic and applied arithmetic geometry
for more than 20 years.

This volume contains 13 original research articles covering geometric
error correcting codes, and algorithmic and explicit arithmetic
geometry of curves and higher dimensional varieties. Tools used in
these articles include classical algebraic geometry of curves,
varieties and Jacobians, Suslin homology, Monsky–Washnitzer
cohomology, and \(L\)-functions of modular forms.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry and related topics of coding theory.

# Table of Contents

## Algorithmic Arithmetic, Geometry, and Coding Theory

- Cover Cover11 free
- Title page iii2 free
- Contents iii4 free
- Preface v6 free
- Geometric error correcting codes 18 free
- On products and powers of linear codes under componentwise multiplication 310
- 1. Introduction 512
- \qquadBasic definitions 512
- \qquadLink with tensor constructions 714
- \qquadRank functions 815
- \qquadGeometric aspects 1118
- 2. Basic structural results and miscellaneous properties 1623
- \qquadSupport 1623
- \qquadDecomposable codes 1724
- \qquadRepeated columns 2027
- \qquadExtension of scalars 2229
- \qquadMonotonicity 2532
- \qquadStable structure 2633
- \qquadAdjunction properties 2734
- \qquadSymmetries and automorphisms 2936
- 3. Estimates involving the dual distance 3239
- 4. Pure bounds 3643
- \qquadThe generalized fundamental functions 3643
- \qquadAn upper bound: Singleton 3845
- \qquadLower bounds for 𝑞 large: AG codes 4148
- \qquadLower bounds for 𝑞 small: concatenation 4249
- 5. Some applications 4653
- \qquadMultilinear algorithms 4653
- \qquadConstruction of lattices from codes 4855
- \qquadOblivious transfer 5057
- \qquadDecoding algorithms 5158
- \qquadAnalysis of McEliece-type cryptosystems 5259
- Appendix \thesection: A criterion for symmetric tensor decomposition 5360
- \qquadFrobenius symmetric maps 5360
- \qquadTrisymmetric and normalized multiplication algorithms 5865
- Appendix \thesection: On symmetric multilinearized polynomials 6269
- \qquadPolynomial description of symmetric powers of an extension field 6370
- \qquadEquidistributed beads on a necklace 6572
- Appendix \thesection: Review of open questions 7380
- References 7582

- Higher weights of affine Grassmann codes and their duals 7986
- Algorithmic: special varieties 93100
- The geometry of efficient arithmetic on elliptic curves 95102
- 2–2–2 isogenies between Jacobians of hyperelliptic curves 111118
- Part 1. 2...2 isogenies and theta functions 112119
- Part 2. Correspondences between family (f-2) and family (f-3) 118125
- Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians 127134
- 1. Introduction 127134
- 2. Relations between quadratic orders 130137
- 3. General two-dimensional decompositions for elliptic curves 131138
- 4. Shrinking the basis (or expanding the sublattice) to fit \G 132139
- 5. Decompositions for GLV endomorphisms 133140
- 6. Decompositions for the GLS endomorphism 135142
- 7. Decompositions for reductions of \QQ-curves 136143
- 8. Four-dimensional decompositions for GLV+GLS 136143
- 9. Decompositions for the Guillevic–Ionica construction 138145
- 10. Two-dimensional decompositions in genus 2 138145
- References 140147

- Algorithmic: point counting 143150
- A point counting algorithm for cyclic covers of the projective line 145152
- 1. Introduction 145152
- 2. Cyclic covers of the projective line 147154
- 3. Monsky–Washnitzer cohomology for cyclic covers and the action of Frobenius 148155
- 4. Adaptation of the Gaudry–Gürel algorithm to general cyclic covers 152159
- 5. Complexity 154161
- 6. Bounds on precision 158165
- 7. The choice of the set of differentials 161168
- 8. Numerical experiments 168175
- References 171178

- Point Counting on Non-Hyperelliptic Genus 3 Curves with Automorphism Group ℤ/2ℤ using Monsky-Washnitzer Cohomology 173180
- Wiman’s and Edge’s sextic attaining Serre’s bound II 191198
- Algorithmic: general 205212
- Genetics of polynomials over local fields 207214
- Introduction 207214
- 1. MacLane valuations 209216
- 2. Okutsu equivalence of prime polynomials 216223
- 3. Types over (𝐾,𝑣) 218225
- 4. OM representations of square-free polynomials 225232
- 5. Computation of the genetics of a polynomial: the Montes algorithm 229236
- 6. Algorithmic applications of polynomial genetics 233240
- References 240247

- Explicit algebraic geometry 243250
- Explicit equations of optimal curves of genus 3 over certain finite fields with three parameters 245252
- Smooth Embeddings for the Suzuki and Ree Curves 251258
- 1. Introduction 251258
- 2. Preliminaries 252259
- 3. The Smooth Embeddings for the Hermitian and Suzuki Curves 260267
- 4. The Defining Equations and Automorphism group of the Ree Curve 267274
- 5. Smooth Embedding for the Ree Curve 276283
- 6. Relation to the Previous Work on the Embeddings of the Deligne–Lusztig Curves 281288
- 7. Representation of the Ree Group 283290
- 8. Further Properties of the Ree Curve 284291
- Appendix A. The Action of the Group 𝐺_{\fr}(𝑃_{∞}) on \Da’ 288295
- References 289296

- Arithmetic geometry 293300
- Uniform distribution of zeroes of 𝐿-functions of modular forms 295302
- A survey on class field theory for varieties 301308
- Back Cover Back Cover1316