On Principal Units of Qp(ζp

Arithm´e tique,g´e om´e trie et th´e orie des codes Arithmetic,geometry and coding theory
R´e sum´e s des expos´e s
Abstract of talks
On Maximally Non-Linear Functions
Miriam Abd´o n
In order to study maximally non-linear functions,we introduce a transform which is rather similar to a Fourier transform,and we express the Hamming distance between a general function f and a linear function using this transform.This distance depends on certain numbers N(v,t)(f) which are solutions of a linear system of equations.The numbers N(v,t)(f)can also be obtained as the intersection number of certain hyperplanes at the point(v,t).We study some properties of the maximal distance from a function to the generalized order one Reed-Muller code and we obtain a lower bound for the covering ray of the generalized order one Reed-Muller code by studying quadratics functions.This is a joint work(in progress)with R.Rolland.
On Principal Units of Q p(ζp n)
Bruno Angl`e s
Let p be an odd prime,p≥5.Let U n=1+(ζp n−1)Z p[ζp n]be the group of principal units of Q p(ζp n).In this talk,we will determine a basis”`a la Leopoldt”of the Galois module Log p(U n). As a consequence,we recover Iwasawa’s famous result on cyclotomic units without the use of logarithmic derivatives”`a la Coleman”and we also obtain a similar result for the minus part of the principal units.This talk is based on a joint work with Thomas Herreng.
Cyclic codes with few different weights
Yves AUBRY,
We consider the question whether an irreducible cyclic code c(p,m,v)with two weights can be non semiprimitive.We focus on the case where there is three p-cyclotomic cosets modulo v and show that,in this case,they are always semiprimitive if v≡3(mod8).
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Linear programming bounds for codes in grassmannian spaces Christine Bachoc
We show how to apply linear programming method to obtain bounds for the size of codes on the real Grassmannian space,endowed with the chordal distance,using the associated zonal polynomials.We obtain an asymptotic bound that improves for some range the one obtained from the Hamming bound given by Barg and Nogin.We obtain a generalization of the results due to G.Kabatianski and V.Levenshtein on codes in rank one symmetric compact spaces.
A simple proof for the limit of the Bezerra-Garcia-Stichtenoth tower over cubicfinitefields
Alp Bassa
Determining the limit of a tower usually involves a lot of technical computations.Recently, Garcia and Stichtenoth showed how these computations could be avoided in some Artin-Schreier towers and their Galois closures.Unfortunately,this method is not directly applicable to the Bezerra-Garcia-Stichtenoth tower over cubicfinitefields,since in this tower the steps are not even Galois.In this talk,I will explain how their idea can still be used in this case to simplify the proof of the limit of the tower and to obtain a lower bound for the limit of the Galois closure of the tower.
Lower bounds for the minimum distance of algebraic geometry codes
Peter Beelen
A one-point AG-code is an algebraic geometry code based on a divisor whose support consists of one point.Since the discovery of the Feng-Rao lower bound for the minimum distance,there has been a renewed interest in such codes.This lower bound is also called the order bound.An alternative description of these codes in terms of order domains has been found.
In my talk I will indicate how one can use the ideas behind the order bound to obtain a lower bound for the minimum distance of any AG-code.After this I will compare this generalized order bound with other known lower bounds,such as the Goppa bound,the Feng-Rao bound and the Kirfel-Pellikaan bound.
I willfinish my talk by giving several examples.Especially for two-point codes,the generalized order bound is fairly easy to compute.As an illustration,I will indicate how a lower bound can be obtained for the minimum distance of some two-point codes coming from the Hermitian curve and compare the outcome with some recent results of Kim and Homma.
3 Maximal set of ideals without coprimes
Vladimir Blinovsky
We prove that for all sufficiently large N0the maximal set of ideals of the maximal order of the algebraic numberfield,such that any pair of ideals from this set is not coprime and norm of each ideal does not exceed N0is the set E(N0)={θ:N(θ)≤N0,θ=η1u},where{η1,η2,...}is the set of prime ideals of the maximal order and N(η3)>2.
Asymptotically good towers and differential equations
Irene Bouw
In this talk I report on joint work with Peter Beelen.
Let k=F q be afinitefield of characteristic p>0.If X/k is a smooth projective curve which is absolutely irreducible,we write N q(X)for its number of F q-rational curves.
In this talk,we consider infinite towers
T=(...→X m→ (X0)
of curves over afixedfinitefield k=F q.We suppose moreover,that the maps X m+1→X m are finite and separable.It is known that
A(T):=lim
m→∞N q(X m)
g(X m)
≤
√
q−1.
A tower T is called asymptotically optimal(resp.asymptotically good)if A(T)=√
q−1
(resp.A(T)>0).Garcia–Stichtenoth and their co-authors constructed many examples of asymptotically optimal and good towers.Elkies showed that all known asymptotically optimal towers are towers of(elliptic or Drinfeld)modular curves or Shimura curves.
In this talk,we give a new approach to constructing such towers.We formulate a general set-up, which is not just applicable to towers of modular or Shimura curves.A key ingredient is the use of solutions of Fuchsian differential equations in characteristic p.Our construction gives many new examples of asymptotically good towers.One might hope that this new approach allows a more systematical search for new optimal towers.
Genus2curves for cryptography
Iwan Duursma
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Computing in the Jacobian of C ab Curves of Genus3
Jorge Estrada
with
-R´e gis Blache.Universit´e de Polyn´e sie Fran¸c aise,France.
-Maria Petkova,IRM-Humboldt University zu Berlin,Germany.
-Wilfredo Hern´a ndez Lezca,Havana University,Cuba.
In the last years some families of algebraic curves has been studied with special interest infinding efficient algorithms for the addition in the Jacobian Variety of the curves as well as forfinding curves in the family whose Jacobian Variety has almost prime order.
The family of C ab curves,whichfirst appeared in the context of algebraic geometry codes,is a wide family of curves possesing good properties for performing computations explicitly.
There have been obtained some methods for computing the zeta function as well as for computing the reduction and/or the addition of divisors in the Jacobian Variety of C ab curves of some subfamilies(elliptic,hyperelliptic,Picard curves,etc.).It will be worth to check up the complete effectiveness of some of these methods and also to compare their complexity.
In the case of genus3non-hyperelliptic C ab curves,we found out an efficient algorithm to compute the addition of two divisors.Due the nice geometric interpretation of the algorithm, we are able to show that it is effective,has low complexity and it does not stop when non-generic divisors appear.
Keywords.-Jacobian Variety,Addition of Divisors,non-hyperelliptic C ab curves
On the Galois closure of towers.
Arnaldo Garcia
Joint work with H.Stichtenoth.
For an Artin-Schreier extension of degree p,the smallest different exponent at a ramified place is 2(p−1).Let E denote the compositum of two such extensions E1and E2of a functionfield F, and let P be a place of F that is ramified in both E1and E2,having both different exponents equal to2(p−1).Then either the place of E1above P is unramified in the extension E over E1or it has again different exponent equal to2(p−1).This is a very simple result,which we call the Key Lemma,that can be used to show that certain towers of functionfields overfinite fields are asymptotically good.This Key Lemma allows a determination of good lower bounds for the limits of certain towers,without the(usually very technical)calculation of the genus of thefields in the towers.We will determine the limit of the Galois closure of certain good towers, using this Key Lemma.
5 On the minimum distance of one-point geometric Goppa codes Olav Geil
(joint work with H.E.Andersen and C.Thommesen)
Consider an algebraic functionfield F/F q with a single place P∞at infinity.Let D=P1+···+P n where P1,...,P n are pairwise distinct places of degree1not equal to P∞.We consider one-point geometric Goppa codes C L(D,mP∞)and their duals C L(D,mP∞)⊥=CΩ(D,mP∞).
The Feng-Rao bound is an improvement to the usual bound from algebraic geometry on the minimum distance of C L(D,mP∞)⊥.From the Feng-Rao bound it is clear how to often improve on the performance of C L(D,mP∞)⊥by leaving out certain rows in the parity check matrix. We present a bound on the minimum distance of C L(D,mP∞)that is also an improvement to the usual bound from algebraic geometry.From our bound it is clear how to often improve on the performance of C L(D,mP∞)by including certain rows in the generator matrix.
All results are easily translated to the more general set-up of codes from order domains and all results can easily be reformulated to deal with generalized Hamming weights.Finally,we establish the connection to the set-up of affine variety codes.
References
[1]H.E.Andersen,O.Geil,The Missing Evaluation Codes from Order Domain Theory,(2004),submitted.
[2]G.-L.Feng and T.R.N.Rao,Decoding of algebraic geometric codes up to the designed minimum distance,
IEEE Trans.Inf.Theory,39,(1993)37-46.
[3]O.Geil,C.Thommesen,On the Feng-Rao Bound for Generalized Hamming Weights,(2005),submitted.
To be announced
Sudhir Ghorpade
Iteration of polynomials andfinitely ramified tree representations of Galois groups
Farshid Hajir
Joint work with Christian Maire(Toulouse)and Wayne Aitken(CSU San Marcos).
Galois groups withfinite ramification over a globalfield K are the“fundamental groups”of number theory.Most of what we know about them stems from their action on certain p-adic vector spaces.In this talk,I will describe their action on certain trees which promises to throw a different kind of light on fundamental groups.Let K be a globalfield and f∈K[x]be a polynomial whose critical points are preperiodic under iteration of f.Then every K-rational specialization of the tower of iterates of f:P1→P1isfinitely ramified.This leads to a number of open problems about the nature of the corresponding“iterated monodromy”representations of the Galois group of K.
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Non-isomorphic maximal curves over a finite field
James Hirschfeld
joint work with M.Giulietti,G.Korchm´a ros and F.Torres
A maximal curve F over the finite field F q is an algebraic curve attaining the Hasse–Weil upper bound,q +1+2g √q,
where g is the genus of F .
The genus of F satisfies the inequality,
g ≤12(q −√q ),
where equality is achieved if and only if F is isomorphic to the Hermitian curve H q ,given by the form,
X √q +10+X √q +11+X √q +12.
If a curve is a quotient of H q ,then it is maximal.A family of quotient curves of H q with genus √q −1is considered.The members of the family have many similar properties but provide many non-isomorphic maximal curves.
The Euler-Kronecker invariants of global fields and primes with small norms
Yasutaka Ihara
Codes associated to Schubert unions in flag varieties
Trygve.Johnsen
Pick a partial flag variety F with respect to a finite dimensional vector space.We will study unions of Schubert varieties in F with respect to a fixed flag in this vector space.We will give some code theoretical applications of this study in the situation where F is a Grassmannian,and investigate how much of results in the Grassmannian case that carry over to the case where F is an arbitrary flag variety.Parts of the talk will be based on joint work with Johan P.Hansen and Kristian Ranestad.
To be announced
Gregory Kabatiansky
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Computing zeta functions of surfaces
Kiran S.Kedlaya
We describe an algorithm for computing the characteristic polynomial of Frobenius on the Monsky-Washnitzer(p-adic)cohomology of a smooth projective hypersurface over afinitefield, using the Griffiths-Dwork description of the de Rham cohomology of such a hypersurface.While this algorithm turns out to be computationally infeasible in general,we(with MIT students Tim Abbott and David Roe)have succeeded in computing low-precision approximations to the Frobenius matrix of a generic smooth quintic surface over F2.We will explain how,thanks to the easy direction of Tate’s conjecture,such approximations are typically sufficient to determine whether the geometric Picard number of such a surface is equal to1.This uses the fact(due to Mazur and Ogus)that the elementary divisors of the Frobenius matrix on a basis of crystalline cohomology coincide with Hodge numbers,so the Frobenius matrix is“nearly”divisible by p. Modular forms,the Klein quartic,and discrete logarithms
Gilles Lachaud
In one of his notebooks,Ramanujan introduced three theta functions of order7.We describe the automorphic character of a vector valued mapping made from these theta series.This provides a systematic way to establish old and new identities on modular forms for the congruence subgroup of level7,above all a parametrization of the Klein quartic.As an application,we introduce a L-series in four different ways,generating the number of points of the Klein quartic overfinite fields.
The nonsingular projective plane curve X defined over afield k,with equation
ax3y+by3z+cz3x=0
where a,b,c are three elements in k×is a twisted form of the Klein quartic.When k is afinite field with q elements,we give some formulas for the order of the group of rational points of the Jacobian J X of X.When q is a prime number congruent to1modulo7,the number of points of the group J X(k)is prime in a significant number of occurences.This provides cyclic groups which seem to be accurate for cryptographic applications.
Monomial bent functions
Philippe LANGEVIN,
Joint work with Gregor Leander.
Let n be an even integer,say n=2k.Let L be afinite extention of degree n of F2and letµL the additive character.A polynomial f(X)∈L[X]defines a bent function when all its Fourier coefficients have absolute value2k:
∀a∈L, f(a)=
x∈L µL
f(x)+ax
=±2k.
(1)
For a such f,there exists a Boolean function f∗such that f(a)=(−1)f∗(a)2k.This dual function is completely determined by dyadic approximations of order k+2.The determination of bent
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functions of the simplest form that is monomial of the formαx d was initiated by Dillon in his PhD Thesis.According to the table below,one knows a very short list of exponents d leading to monomial bent function.In this talk,we write the Fourier coefficients ofαx d in term of Gauss
Type Exponent d Conditions
Gold2r+1gcd(d,2n−1)=1,α/∈L d
Dillon2k−1Kloos K(α)=−1
Kasami22r−2r+1gcd(r,n)=1,α/∈L d
MF(2r+1)2n=4r,r odd, α =F∗2r
conjecture22r+2r+1n=6r,α∈?
Table1.All known bent exponents(up to cyclotomy)for n≤24.
sums in order to analyse the bentness of monomial by means of Stickelberger’s congruences. Our approach applies very well to Dillon,Gold and Kasami cases.Most notably we are able to give a very short proof for the Kasami exponents that includes the case where n is not coprime to3and show that the dual of the bent functions of the Kasami exponent are not monomial bent functions.
Characterisations of pseudocodewords of LDPC codes
Winnie Li and Judy Walker
Low density parity-check(LDPC)codes are distinguished by their fast decoding algorithm.A drawback of this algorithm is that,instead of a codeword,sometimes it yields a pseudocodeword, which is a codeword of afinite cover of the original associated Tanner graph.In this talk we shall discuss various ways to characterize pseudo-codewords of an LDPC code.This is based on the joint work of Koetter,Li,Vontobel,and Walker.
Explicit constructions of algebraic-geometric codes from towers Hiren Maharaj
There are many explicit constructions of recursively defined asymptotically good towers of func-tionfields.Thefirst few levels of these towers provide excellent examples of curves with many points for code construction.Since the code length from towers increase exponentially with the level,only codes from thefirst few levels are expected to be of current practical interest.In this talk,we present a simple technique to produce easily described bases for a class of high performance codes from thefirst few levels of any given recursively defined tower.It is an open problem whether these codes are asymptotically good.
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Pro-p-extensions of numberfields and cohomological dimension Christian Maire
In this talk,we will discuss on the relationships between the arithmetic of pro-p-extensions of numberfields and the cohomological dimension of their groups.This work is motivated by some observations of Nigel Boston and a very recent paper of John Labute.
This talk is closely related to the talk by Farshid Hajir.
Parameters of relative Reed-Muller codes
Tohru Nakashima
Let X be a smooth projective variety defined over afinitefield F q.Let X(F q)={P1,...,P n} denote the set of F q-rational points of X.Assume that X is equipped with afibrationπ:X→C over a smooth projective curve and let L be aπ-very ample line bundle on X.We define the relative Reed-Muller code C(X,L)to be the image of the natural evaluation map
α:H0(X,L)→
n
i=1
L P
i
∼=F n
q
.
In this talk we will explain how the parameters of C(X,L)are related to those of Reed-Muller codes onfibers.In particular,we will give a lower bound for the minimum distance of C(X,L) when X is isomorphic to the projective bundle P(E)associated to a vector bundle E on C and L has the form L=aO(1)+bf,where O(1)is the tautological bundle and f is aπ-fiber. Jacobians in isogeny classes of abelian surfaces overfinitefields Enric Nart
Joint work with E.Howe and C.Ritzenthaler
An isogeny class A of abelian surfaces over afinitefield F q is determined by a Weil polynomial of the form x4+ax3+bx2+aqx+q2∈Z[x].For which values of(a,b)there is a projective smooth genus2curve over F q whose Jacobian lies in A?
In this talk we describe different procedures that provide a complete answer to this question: 1.Determine for what values of(a,b)there is a principally polarized surface in A.This is based essentially in work developed by Howe,who expressed the obstruction to the existence of principal polarizations in terms of the vanishing of an element in a group constructed in a categorical framework of objects analogous to Deligne modules.After a classical result of Weil, this solves the problem for those A that are simple over F q2.
2.For simple ordinary surfaces that split over F q2the problem was solved by Howe and Maisner by counting arguments.They found independent formulas(or bounds)for the total number of principally polarized surfaces(A,λ)with A belonging to A,and for the number of them that are not Jacobians.Both formulas involve arithmetic invariants of the biquadraticfield generated by the Weil polynomial and they can be compared using the Brauer relations;when the numbers coincide,A does not contain Jacobians.
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3.For simple supersingular surfaces that split over F q2we use results of Shimura,Oort and
Ibukiyama-Katsura-Oort on supersingular abelian surfaces and their polarizations.We descend
principally polarized surfaces from¯F q to F q and we control the isogeny class of the descended surface;to this end we use results of Hashimoto and Ibukiyama on mass formulas for quaternion
hermitian forms with a given group of automorphisms.
4.For the split case one uses a result of Kani characterizing when two elliptic curves can be tied
along respective torsion subgroups to get a common covering by a curve of genus two.In some
cases we need to study twists of Dieudonn´e modules of supersingular elliptic curves to check
that certain curves belonging to different isogeny classes cannot be tied along their p-torsion
subgroups.
Nonbinary Quantum Goppa Codes Exeeding the Quantum Gilbert-Varshamov Bound
Annika Niehage
In this talk I will give an explicit construction for nonbinary quantum Goppa codes exceeding
the quantum Gilbert-Varshamov bound.First quantum codes and especially quantum Goppa
codes are introduced with their specific properties.Next an example of hyperelliptic curves is
given to show the idea of the later code construction.Finally I use a tower of functionfields by
H.Stichtenoth to construct a class of quantum Goppa codes that are better than the quantum
Gilbert-Varshamov bound.
Further improvements on asymptotic bounds for codes
Ferruh Ozbudak
Joint work with Harald Niederreiter.
For a prime power q,letαq be the standard function in the asymptotic theory of codes,that
is,αq(δ)is the largest asymptotic information rate that can be achieved for a given asymptotic
relative minimum distanceδof q-ary codes.In recent years the Tsfasman-Vl˘a dut¸-Zink lower bound onαq(δ)wasfirst improved by Xing and later by Niederreiter and¨Ozbudak.In this paper we show further improvements on these bounds by using distinguished divisors of global functionfields.
On Prym varieties overfinitefields
Marc Perret
Prym varieties form a class of abelian varieties larger than the class of Jacobian.Of course,Weil theorem gives estimates for their number of rational points over afinitefield.We will see that better estimates can be given.
11 Schubert unions in Grassmannian varieties
Kristian Ranestad
in collaboration with Trygve Johnsen and Johan P.Hansen.
Schubert cycles form a natural basis for the homology of Grassmann varieties,and are therefore well studied.Less systematically are studies of unions of Schubert cycles with respect to afixed flag.I will present some basic properties such as dimension and cardinality overfinitefields, and the duality of such unions.In particular there is a simple geometric duality of such unions in Grassmann varieties of lines that corresponds to a combinatorial duality of partitions.
On Zeta–functions of Kummer surfaces overfinitefields.
Sergey Rybakov
On codes with large weights simultaneously for the Rosenbloom-Tsfasman and Hamming metrics
Maxim Skriganov
We show that MDS codes,or more generally nearly MDS codes,for the Rosenbloom-Tsfasman metric can also meet the Gilbert-Varshamov bound for thier Hamming weights.
New directions in convolutional codes
Patrick Sole
Regarding convolutional codes as polynomial analogues of geometry of numbers lattices,we derive a MacWilliams formula for their trivariate weight enumerator.The proof is based on harmonic analysis on locally compact abelian groups as developed in Tates thesis to derive the functional equation of the zeta function(joint work with Dimitrii Zinoviev).As noted by MacEliece in1977such a formula cannot exist for a code-like duality(vanishing scalar product) but we know now that it exists for a lattice-like duality(integral scalar product).By analogy with Construction A of lattices I build binary convolutional codes from codes from block codes over an extensionfield.Specifically,from an[n,k,d]code over GF(2s)a binary nonrecursive convolutional code of rate k/n,memory m≤s−1and free distance d is constructed.There exist long binary convolutional codes of unit memory obtained from this construction that improve on the Costello bound for R≤0.35.
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Transitive codes meet the TVZ bound
Henning Stichtenoth
A major problem in coding theory is the question if the class of cyclic codes is asymptotically good.In this talk I introduce-as a generalization of cyclic codes-the notion of transitive codes, and I show that the class of transitive codes is asymptotically good.Even more,transitive codes attain the Tsfasman-Vladut-Zink bound over F q,for all squares q=l2.I also show that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound,thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound.The main tool is a new asymptotically optimal tower E0⊆E1⊆E2⊆...of functionfields over F q(with q= 2),where all extensions E n/E0are Galois and where exactly one place of thefield E0 splits completely in allfields E n.
This talk is closely related to the talk by Arnaldo Garcia:The Galois closure of some good towers.
On the Gray image of linear cyclic codes over Galois rings Horacio Tapia-Recillas
After the work of Hammons-Kumar-Calderbank-Sloane-Sol´e(IEEE,1994)in which,among other results,the authors show that the(non-linear)binary Kerdock and Preparata codes are the Gray image of a linear Z Z4-code,the study of linear and cyclic codes defined overfinite rings has received attention from many researchers.
The ring Z Z p k where p is a prime and k a positive integer has been of particular interest.A natural extension of the Gray map on these rings has been given and results on the linearity and cyclicity of the Gray image of Z Z p k-codes appear now in the literature.Codes over the ring Z Z p2have received special attention.
However,the ring Z Z p k is a special case of a Galois ring.In this talk,results on the linearity and cyclicity of the Gray image of a class of codes defined over the Galois ring GR(p2,m)will be presented,generalizing those given for Z Z p2-codes.The ring of Witt vectors is an important tool in the proof.
Nonlinear codes from points of bounded height
JeffThunder
Joint work with Chris Hurlburt.
Using ideas and methods from Diophantine geometry,we generalize a construction of nonlinear error-correcting codes due to Elkies.The result is a larger family of codes with transmission rates and error detection rates similar to those codes constructed by Elkies.With this gener-alization we produce a precise average code size.Moreover,we exhibit a connection between these nonlinear codes and solutions to simple homogeneous linear equations defined over certain functionfields.
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Asymptotic behaviour of the Euler-Kronecker invariant
Michael Tsfasman
Algebraic-Geometric codes on surfaces
Felipe Voloch
In this talk we will present some results obtained by my student M.Zarzar and myself on Algebraic Geometric codes on surfaces.In particular we will discuss the minimal distance of these codes and decoding algorithms.To obtain bounds for the minimal distance we needed to understand how zero sets of functions on surfaces decompose in irreducible components and this led to some geometric results.We discuss a decoding algorithm which uses the LDPC structure of these codes and extends some work of Luby and Mitzenmacher on decoding LDPC codes over large alphabets.
Del Pezzo surfaces,cubic curves and jacobians.
Yuri ZARHIN
The generalized Brauer-Siegel theorem
Alexey Zykin.。