quantum algebras as quantizations of dual poisson lie groups a systematic computational approach for the explicit construction of any quantum hopf algebra uz g z starting from the lie bialgebra g that gives the first order deformation of the coproduct map z is presented the procedure is based on the well known quantum duality principle namely the fact that any quantum algebra can be viewed as the quantization of the unique poisson lie structure g g on the dual group g which is obtained by exponentiating the lie algebra g defined by the dual map from this perspective the coproduct for uz g is just the pull back of the group law for g and the poisson analogues of the quantum commutation rules for uz g are given by the unique poisson lie structure g on g whose linearization is the poisson analogue of the initial lie algebra g this approach is shown to be a very useful technical tool in order to solve the lie bialgebra quantization problem explicitly since once a lie bialgebra g is given the full dual poisson lie group g can be obtained either by applying standard poisson lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs as a consequence the quantization of g will give rise to the full uz g quantum algebra provided that ordering problems are appropriately fixed through the choice of certain local coordinates on g whose coproduct fulfils a precise quantum symmetry property the applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant lie algebras such as the 2 1 anti de sitter algebra so 2 2 and the poincare algebra in 3 1 dimensions

averages and higher moments for the ell torsion in class groups we prove upper bounds for the average size of the ell torsion mathrm cl k ell of the class group of k as k runs through certain natural families of number fields and ell is a positive integer we refine a key argument used in almost all results of this type which links upper bounds for mathrm cl k ell to the existence of many primes splitting completely in k that are small compared to the discriminant of k our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument in conjunction with new counting results for these invariants this leads to significantly improved upper bounds for the average and sometimes even higher moments of mathrm cl k ell for many families of number fields k considered in the literature for example for the families of all degree d fields for d in 2 3 4 5 and non d 4 if d 4 as an application of the case d 2 we obtain the best upper bounds for the number of d p fields of bounded discriminant for primes p 3

strengthening the gilbert varshamov bound abstract the paper discusses some ways to strengthen nonasymptotically the gilbert varshamov bound for linear codes the unifying idea is to study a certain graph constructed on vectors of low weight in the cosets of the code which we call the varshamov graph various simple estimates of the number of its connected components account for better lower bounds on the minimum distance of codes some of them known in the literature

relativistic and newtonian κ space times the deformations of the galilei algebra and their associated noncommutative newtonian space times are investigated this is done by analyzing the possible nonrelativistic limits of an eleven generator pseudo extended poincare algebra p and their implications for the existence of a first order differential calculus the additional one form needed to achieve a consistent calculus on minkowski space is shown to be related to the additional central generator entering in the p hopf algebra in the process deformations of the extended galilei and galilei algebras are introduced which have respectively a cocycle and a bicrossproduct structure

class groups of number fields numerical heuristics extending previous work of h w lenstra jr and the first author we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four given the signature and the galois group of the galois closure the theoretical justifications for these conjectures will appear elsewhere but the agreement with the existing tables is quite good 1 introduction and notations in 3 h w lenstra jr and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree in a forthcoming paper 4 we shall show that this technique can be extended to a much wider class of number fields and also to relative extensions the aim of the present paper is to rapidly make available the numerical conjec tures obtained for people not really interested in our heuristic reasoning or not wanting to wait for 4 to appear hence apart from a total lack of justifications for the conjectures that we present this paper is essentially self contained the plan is as follows in the rest of this section we present the notations used in the sequel some of them being nonstandard and in general differing from the notations of 3 we urge the reader to read the notations carefully before applying the conjectures in the next section we present templates for the subsequent conjectures and then the conjectures themselves illustrated by numerical examples first for their own sake and second as a double check for the reader to understand the templates these conjectures are given for all types of fields of degree less than or equal to four in the final section we comment on the consistency of the conjectures with existing tables which is quite good combinatorial notations if x is a set xi denotes its cardinality

asymptotic improvement of the gilbert varshamov bound on the size of binary codes given positive integers n and d let a sub 2 n d denote the maximum size of a binary code of length n and minimum distance d the well known gilbert varshamov bound asserts that a sub 2 n d spl ges 2 sup n v n d l where v n d spl sigma sub i 0 sup d sub i sup n is the volume of a hamming sphere of radius d we show that in fact there exists a positive constant c such that a sub 2 n d spl ges c2 sup n v n d 1 log sub 2 v n d 1 whenever d n spl les 0 499 the result follows by recasting the gilbert varshamov bound into a graph theoretic framework and using the fact that the corresponding graph is locally sparse generalizations and extensions of this result are briefly discussed

relative genus theory and the class group of l extensions the structure of the relative genus field is used to study the class group of relative i extensions application to class field towers of cyclic i extensions of the rationals are given given a number field e one tries to understand the unramified abelian extensions of e and so by class field theory derive information about the class group of e ce one way of doing this is to ask for the maximal abelian unramified extension of e of the form ef where f is an abelian extension of f c e this field is called the relative genus field and we denote it by e if f q we speak of the absolute genus field this paper will use relative genus theory to study the class group of certain i extensions of q henceforth i will always be an odd prime in particular we give applications to class field towers of such fields we also study the i ranks of the class group of number fields e containing a cyclic subextension of degree 1 this will lead to a slight improvement on a bound of iwasawa 13 this technique is then used to give bounds on both the i rank and the order of the class group of elementary abelian i extensions of q we let he respectively he denote the hilbert class field of e respectively the l hilbert class field also we refer simply to the i class rank to mean the i rank of the class group proposition 1 let e f be normal and f the fixed field of the commutator subgroup of gal he f then e ef proof any abelian extension of f whose composite with e is contained in he is contained in f since f e is contained in the tower e c f e c he it is unramified over e the result follows proposition 2 suppose e f is normal and both fields are normal over some subfield l of f then e is also normal over l proof we know he l is galois and gal he f is a normal subgroup of this group the commutator subgroup of gal he f is a characteristic subgroup of received by the editors february 26 1982 and in revised form may 25 1982 1980 mathematics subject classification primary 12a65 secondary 12a50 this paper is based on an improved version of part of the authors ph d thesis at brown university written under the direction of professor m rosen support has been graciously provided by the vaughn foundation 1 983 american mathematical society 0002 9947 82 0000 0584 03 00

solving the two dimensional constant quantum yang baxter equation a detailed analysis of the constant quantum yang baxter equation rk1k2j1j2 rl1k3k1j3rl2l3k2k3 rk2k3j2j3 rk1l3j1k3rl1l2k1k2 in two dimensions is presented leading to an exhaustive list of its solutions the set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the grobner basis methods each solution was then transformed into a canonical form based on the various trace matrices of r for final elimination of duplicates and subcases if we use homogeneous parametrization the solutions can be combined into 23 distinct cases modulo the well known c p and t reflections and rotations and scalings r q q r q q 1

the asymptotic behavior of grassmannian codes the iterated johnson bound is the best known upper bound on the size of an error correcting code in the grassmannian gq n k the iterated schonheim bound is the best known lower bound on the size of a covering code in gq n k we prove that both bounds are asymptotically attained for fixed k and fixed radius as n approaches infinity our methods rely on results from the theory of quasi random hypergraphs which are proved using probabilistic techniques we also determine the asymptotics of the size of the best grassmannian codes and covering codes when n k and the radius are fixed as n approaches infinity

group codes outperform binary coset codes on nonbinary symmetric memoryless channels typical minimum distances and error exponents are analyzed on the 8 psk gaussian channel for two capacity achieving code ensembles with different algebraic structure it is proved that the typical group code over the cyclic group of order eight achieves both the gilbert varshamov bound and the expurgated error exponent on the other hand the typical binary coset codes under any labeling is shown to be bounded away both from the gilbert varshamov bound at any rate and the expurgated exponent at low rates the reason for this phenomenon is shown to rely on the symmetry structure of the 8 psk constellation which is known to match the cyclic group of order eight but not the direct product of three copies of the binary group the presented results indicate that designing group codes matching the symmetry of the channel guarantees better typical code performance than designing codes whose algebraic structure does not match the channel this contrasts the well known fact that the average binary coset code achieves both the capacity and the random coding error exponent of any discrete memoryless channel

nonbinary stabilizer codes over finite fields one formidable difficulty in quantum communication and computation is to protect information carrying quantum states against undesired interactions with the environment to address this difficulty many good quantum error correcting codes have been derived as binary stabilizer codes fault tolerant quantum computation prompted the study of nonbinary quantum codes but the theory of such codes is not as advanced as that of binary quantum codes this paper describes the basic theory of stabilizer codes over finite fields the relation between stabilizer codes and general quantum codes is clarified by introducing a galois theory for these objects a characterization of nonbinary stabilizer codes over fq in terms of classical codes over fq 2 is provided that generalizes the well known notion of additive codes over f4 of the binary case this paper also derives lower and upper bounds on the minimum distance of stabilizer codes gives several code constructions and derives numerous families of stabilizer codes including quantum hamming codes quadratic residue codes quantum melas codes quantum bose chaudhuri hocquenghem bch codes and quantum character codes the puncturing theory by rains is generalized to additive codes that are not necessarily pure bounds on the maximal length of maximum distance separable stabilizer codes are given a discussion of open problems concludes this paper

integral points on elliptic curves and 3 torsion in class groups we give new bounds for the number of integral points on elliptic curves the method may be said to interpolate between approaches via diophantine techniques bp hbr and methods based on quasi orthogonality in the mordell weil lattice sil6 gs he we apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3 torsion part of the class group of a quadratic field the same ideas can be used to count rational points on curves of higher genus

the generalized gilbert varshamov bound is implied by turan s theorem code construction the generalization of the gilbert varshamov bound due to gu and fuja 1993 is a direct consequence of turan s theorem on the existence of a clique in a graph with many edges turan s theorem allows a slight improvement of gu and fuja s result this improved generalized gilbert varshamov bound is in fact equivalent to turan s theorem

counting arithmetic lattices and surfaces we give estimates on the number al h x of conjugacy classes of arithmetic lattices of covolume at most x in a simple lie group h in particular we obtain a first concrete estimate on the number of arithmetic 3 manifolds of volume at most x our main result is for the classical case h psl 2 where we show that limx log al h x x log x 1 2 the proofs use several different techniques geometric bounding the number of generators of as a function of its covolume number theoretic bounding the number of maximal such and sharp estimates on the character values of the symmetric groups to bound the subgroup growth of

nonlinear codes from algebraic curves improving the tsfasman vladut zink bound in the present paper we construct a class of nonlinear codes by making use of higher order derivatives of certain functions of algebraic curves it turns out that the asymptotic bound derived from the goppa geometry codes can be improved for the entire interval 0 1 in particular the tsfasman vladut zink tvz bound is ameliorated for the entire interval 0 1

quantum poincare group related to the kappa poincare algebra the classical r matrix implied by the quantum kappa poincare algebra of lukierski nowicki and ruegg is used to generate a poisson structure on the poincare group a quantum deformation of the poincare group on the hopf algebra level is obtained by a trivial quantization

some results on the existence of binary linear codes corresp a lower bound is given on the size of a linear code of given length and distance which improves the varshamov bound in certain cases though not asymptotically specific triples n k d are given with values larger than those guaranteed by the varshamov bound

new lower bounds for secure codes and related hash families a hypergraph theoretical approach various kinds of secure codes and their related hash families are broadly studied combinatorial structures for protecting copyrighted materials the codewords in such a structure can be regarded as a subset of q n the set of all q ary vectors of given length n satisfying some constraints we use a hypergraph model to characterize the combinatorial structure by applying a result of duke et al on the lower bound of the independence number of hypergraphs we provide a new approach to evaluate the lower bounds for several kinds of secure codes and related hash families in particular the general method is illustrated via the examples of existence results on some perfect hash families frameproof codes and separable codes

classical r matrices for the osp 2 2 lie superalgebra the co lie structures compatible with the osp 2 2 lie superalgebra structure are investigated and found to be all of the coboundary type the corresponding classical r matrices are classified into several disjoint families the osp 1 2 u 1 lie superbialgebras are also classified

on some haar measures on reductive groups we give a characteristic free proof of a theorem of b h gross comparing two haar measures on a reductive group over a local field as a consequence of this theorem we obtain an integration formula for adele groups over global function fields

group presentation p adic analytic groups and lattices in mathrm sl _2 mathbf c groups by using their pro p completions this method is based on the observation that if x r is a presentation of the finitely generated f g group f then x r considered as a presentation for a pro p group is actually a presentation for t the pro p completion of f first we prove a generalization of a theorem of golod and shafarevich cf h theorem let g be a p adic analytic pro p group different from zp the p adic integers if x r is a minimal presentation of g then 1x 2 iri 2

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

counting isospectral manifolds abstract given a simple lie group h of real rank at least 2 we show that the maximum cardinality of a set of isospectral non isometric h locally symmetric spaces of volume at most x grows at least as fast as x c log x log log x 2 where c c h is a positive constant in contrast with the real rank 1 case this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of belolipetsky and lubotzky 2 our proof uses sunada s method results of 2 and some deep results from number theory we also discuss an open number theoretical problem which would imply an even faster growth estimate

an improvement on the gilbert varshamov bound for permutation codes permutation codes have been shown to be useful in power line communications block ciphers and multilevel flash memory models construction of such codes is extremely difficult in fact the only general lower bound known is the gilbert varshamov type bound in this paper we establish a connection between permutation codes and independent sets in certain graphs using the connection we improve the gilbert varshamov bound asymptotically by a factor log n when the code length n goes to infinity

drinfel d superdoubles and poisson lie t plurality in low dimensions defining the real lie superalgebra as real z2 graded vector space we classify real manin supertriples and drinfel d superdoubles of superdimensions 2 2 4 2 and 2 4 the drinfel d doubles of the superdimension 2 2 are then used for construction of the simplest models related by poisson lie t plurality

quantum 1 1 extended galilei algebras from lie bialgebras to quantum r matrices and integrable systems the lie bialgebras of the 1 1 extended galilei algebra are obtained and classified into four multiparametric families their quantum deformations are obtained together with the corresponding deformed casimir operators for the coboundary cases quantum universal r matrices are also given applications of the quantum extended galilei algebras to classical integrable systems are explicitly developed

quantum heisenberg weyl algebras all lie bialgebra structures on the heisenberg weyl algebra are classified and explicitly quantized the complete list of quantum heisenberg weyl algebras so obtained includes new multiparameter deformations most of them being of the non coboundary type

1 1 schrodinger lie bialgebras and their poisson lie groups all lie bialgebra structures for the 1 1 dimensional centrally extended schrodinger algebra are explicitly derived and proved to be of coboundary type therefore since all of them come from a classical r matrix the complete family of schrodinger poisson lie groups can be deduced by means of the sklyanin bracket all possible embeddings of the harmonic oscillator extended galilei and gl 2 lie bialgebras within the schrodinger classification are studied as an application new quantum hopf algebra deformations of the schrodinger algebra including their corresponding quantum universal r matrices are constructed

counting lattices in simple lie groups the positive characteristic case in this article we prove a conjecture of a lubotzky let g g0 k where k is a local field of characteristic p 5 g0 is a simply connected absolutely almost simple k group of k rank at least 2 we give the rate of growth of x g g a lattice in g vol g x where 1 2 if and only if there is an abstract automorphism of g such that 2 1 we also study the rate of subgroup growth sx of any lattice in g as a result we show that these two functions have the same rate of growth which proves lubotzky s conjecture along the way we also study the rate of growth of the number of equivalence classes of maximal lattices in g with covolume at most x

unramified subextensions of ray class field towers abstract fix a prime l in this paper we explore various senses in which the ramification in an infinite l extension of number fields can be small in particular 1 we construct infinitely many pairs of primes p q distinct from l such that q admits an infinite l extension unramified outside p q and 2 we explore the possibility of finding infinite unramified subextensions inside infinite ramified l extensions of number fields

quantum e 2 groups and lie bialgebra structures lie bialgebra structures on e 2 are classified for two lie bialgebra structures which are not coboundaries i e those which are not determined by a classical r matrix we solve the cocycle condition find the lie poisson brackets and obtain quantum group relations there is one to one correspondence between lie bialgebra structures on e 2 and possible quantum deformations of u e 2 and e 2

classification of low dimensional lie super bialgebras a thorough analysis of lie super bialgebra structures on lie super algebras osp 1 2 and super e 2 is presented a combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied in all the cases poisson lie brackets on supergroups are found the possibility of quantizing them in order to obtain quantum groups is discussed it turns out to be straightforward for all but one structure for the super e 2 group

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

lie bi algebra structures for centrally extended two dimensional galilei algebra and their lie poisson counterparts all bi algebra structures for centrally extended galilei algebra are classified the corresponding lie poisson structures on centrally extended galilei group are found

classification of the quantum deformations of the superalgebra gl 1 1 we present a classification of the possible quantum deformations of the supergroup and its lie superalgebra in each case the super commutation relations and the hopf structures are explicitly computed for each r matrix one finds two inequivalent co products whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras in the braided case one recovers the classical algebra for suitable limits of the deformation parameters but this is no longer true in the unbraided case

on the classification of quantum poincare groups using the general theory of 10 quantum poincare groups without dilatations are described and investigated the description contains a set of numerical parameters which satisfy certain polynomial equations for most cases we solve them and give the classification of quantum poincare groups each of them corresponds to exactly one quantum minkowski space the poincare series of these objects are the same as in the classical case we also classify possibler matrices for the fundamental representation of the group

asymptotic improvement of the gilbert varshamov bound for linear codes the gilbert varshamov gv bound states that the maximum size a2 n d of a binary code of length n and minimum distance d satisfies a2 n d ges2n v n d 1 where v n d sigmai 0 d i n stands for the volume of a hamming ball of radius d recently jiang and vardy showed that for binary nonlinear codes this bound can be improved to a2 n d gescn2n v n d 1 for c a constant and d nges0 499 in this paper we show that certain asymptotic families of linear binary n n 2 random double circulant codes satisfy the same improved gv bound

twist deformation of the rank one lie superalgebra the drinfeld twist is applied to deform the rank one orthosymplectic lie superalgebra the twist element is the same as for the sl 2 lie algebra due to the embedding of the sl 2 into the superalgebra the r matrix has the direct sum structure in the irreducible representations of the dual quantum group is defined using the frt formalism it includes the jordanian quantum group as subalgebra and grassmann generators as well

cavity approach to sphere packing in hamming space in this paper we study the hard sphere packing problem in the hamming space by the cavity method we show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of gilbert and varshamov consistently with known numerical results the replica symmetric equations also suggest a crystalline solution where for even diameters the spheres are more likely to be found in one of the subspaces even or odd of the hamming space these crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultra metric space finally we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres known maximum packings are reproduced efficiently i n non trivial ranges of dimensions and number of spheres

reconstruction of sequences over non identical channels motivated by the error behavior in the dna storage channel in this paper we extend the previously studied sequence reconstruction problem by levenshtein the reconstruction problem studies the model in which the information is read through multiple noisy channels and the decoder which receives all channel estimations is required to decode the information for the combinatorial setup the assumption is that all the channels cause at most some t errors levenshtein considered the case in which all the channels have the same behavior and we generalize this model and assume that the channels are not identical thus different channels may cause different maximum numbers of errors for example we assume that there are n channels which cause at most t 1 or t 2 errors where t 1 and the number of channels with at most t 1 errors is at least lceil pn rceil for some fixed 0 if the information codeword belongs to a code with minimum distance d the problem is then to find the minimum number of channels that guarantees successful decoding in the worst case a different problem we study in this paper is where the number of channels is fixed and the question is finding the minimum distance d that provides exact reconstruction we study these problems and show how to apply them for the cases of substitutions and transpositions

the number of extensions of a number field with fixed degree and bounded discriminant we give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant x these bounds improve on work of schmidt we also prove various related results such as lower bounds for the number of extensions and upper bounds for galois extensions

classification of four and six dimensional drinfel d superdoubles using adjoint representation we first classify two and three dimensional lie superbialgebras obtained from decomposable lie superalgebras in this way we complete the classification obtained by eghbali et al j math phys 51 073503 2010 e print arxiv 0901 4471 math ph then we classify all four and six dimensional drinfel d superdoubles

counting and effective rigidity in algebra and geometry the purpose of this article is to produce effective versions of some rigidity results in algebra and geometry on the geometric side we focus on the spectrum of primitive geodesic lengths resp complex lengths for arithmetic hyperbolic 2 manifolds resp 3 manifolds by work of reid this spectrum determines the commensurability class of the 2 manifold resp 3 manifold we establish effective versions of these rigidity results by ensuring that for two incommensurable arithmetic manifolds of bounded volume the length sets resp the complex length sets must disagree for a length that can be explicitly bounded as a function of volume we also prove an effective version of a similar rigidity result established by the second author with reid on a surface analog of the length spectrum for hyperbolic 3 manifolds these effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras to prove these effective rigidity results we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest

a nonlinear model of heat conduction a nonlinear model of heat propagation is presented from which a new heat conduction equation is derived an exact similarity solution in closed form of this equation is obtained which reveals the travelling wave characteristics for the transient temperature distribution it is shown that the temperature disturbances propagate with finite velocity which is a monotonically decreasing function of time

quantum deformation of the poincare supergroup and kappa deformed superspace the classical r matrix for the n 1 super poincare algebra given by lukierski et al 1993 is used to describe the graded poisson structure on the n 1 poincare supergroup the standard correspondence principle between the even odd poisson brackets and anti commutators leads to the consistent quantum deformation of the super poincare group with the deformation parameter q described by the fundamental mass parameter k k 1 in q the k deformation of n 1 superspace as dual to the k deformed supersymmetry algebra is discussed

manifolds counting and class field towers in burger et al 2002 12 and goldfeld et al 2004 17 it was conjectured that if h is a simple lie group of real rank at least 2 then the number of conjugacy classes of arithmetic lattices in h of covolume at most x is x h o 1 logx loglogx where h is an explicit constant computable from the absolute root system of h in this paper we prove that this conjecture is false in fact we show that the growth is at rate xclogx a crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of golod and shafarevich on class field towers

a group theoretic characterization of linear groups let r be a group when is f linear this is an old problem the first to study this question systematically was malcev in 1940 m who essentially reduced the problem to finitely generated groups note that a finitely generated group is linear over some field of characteristic zero if and only if it can be embedded in cl c for some n very little progress was made since that paper of malcev although as linear groups are a quite special type of group many necessary conditions were obtained e g r should be residually finite and even virtually residually p for almost all primes p f should be virtually torsion free and if not solvable by finite it has a free non abelian subgroup etc cf z of course none of these properties characterizes the finitely generated linear groups over in this paper we give such a characterization using the congruence structure of r first some definitions for a group h d h denotes the minimal number of generators for h

a generalized gilbert varshamov bound derived via analysis of a code search algorithm a generalization of the gilbert varshamov bound that is applicable to block codes whose codewords must be drawn from irregular sets is derived the bound improves by a factor of four a similar result derived by v d kolesnik and v y krachkovsky 1991 this generalization is derived by analysing a code search algorithm referred to as the altruistic algorithm this algorithm iteratively deletes potential codewords so that at each iteration the candidate is removed the bound is derived by demonstrating that as the algorithm proceeds the average volume of a sphere of a given radius approaches the maximum such volume and so a bound previously expressed in terms of the maximum volume can in fact be expressed in terms of the average volume examples of applications where the bound is relevant include error correcting d k codes and binary codes for code division multiple access

noncommutative geometry of super jordanian osph 2 1 covariant quantum space extending a recently proposed procedure of construction of various elements of differential geometry on noncommutative algebras we obtain these structures on noncommutative superalgebras as an example a quantum superspace covariant under the action of super jordanian osph 2 1 is studied it is shown that there exist a two parameter family of torsionless connections and the curvature computed from this family of connections is bilinear it is also shown that the connections are not compatible with the metric

the threshold probability of a code we define and estimate the threshold probability spl theta of a linear code using a theorem of margulis 1974 originally conceived for the study of the probability of disconnecting a graph we then apply this concept to the study of the erasure and z channels for which we propose linear coding schemes that admit simple decoding we show that spl theta is particularly relevant to the erasure channel since linear codes achieve a vanishing error probability as long as p spl lesspl theta where p is the probability of erasure in effect spl theta can be thought of as a capacity notion designed for codes rather than for channels binomial codes haven the highest possible spl theta and achieve capacity as for the z channel a subcapacity is derived with respect to the linear coding scheme for a transition probability in the range log 3 2 1 we show how to achieve this subcapacity as a by product we obtain improved constructions and existential results for intersecting codes linear sperner families which are used in our coding schemes

generating functions and lower bounds on rates for limited error correcting codes runlength limited rll and digital sum limited dsl codes are considered for these codes the finite and asymptotic lower bounds on achievable rates for the given minimum hamming distance are derived using generating functions and trellis diagram techniques the authors prove the existence of rll and dsl codes of rate r and minimum distance d delta n such that r or 2 log lambda sub 1 0 min log lambda sub 1 z delta log z 0

super jordanian algebra as a contraction limit we develop a generic representation independent contraction procedure for obtaining for instance matrices and l operators of arbitrary dimensions for the quantized super jordanian algebra from the pertinent quantities of the standard q deformed algebra

averaging bounds for lattices and linear codes general random coding theorems for lattices are derived from the minkowski hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out a new version of the minkowski hlawka theorem itself is obtained as the limit for p spl rarr spl infin of a simple lemma for linear codes over gf p used with p level amplitude modulation the relation between the combinatorial packing of solid bodies and the information theoretic soft packing with arbitrarily small but positive overlap is illuminated the soft packing results are new when specialized to the additive white gaussian noise channel they reduce to a version of the de buda poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1 2 log sub 2 p n

lie bialgebra quantizations of the oscillator algebra and their universal r matrices all coboundary lie bialgebras and their corresponding poisson lie structures are constructed for the oscillator algebra generated by quantum oscillator algebras are derived from these bialgebras by using the lyakhovsky and mudrov formalism and for some cases quantizations at both algebra and group levels are obtained including their universal r matrices

discrete subgroups acting transitively on vertices of a bruhat tits building we describe all the discrete subgroups of ad g0 f o aut f which act transitively on the set of vertices of b b f g0 the bruhat tits building of a pair f g0 of a characteristic zero non archimedean local eld and a simply connected absolutely almost simple f group if b is of dimension at least 4 in fact we classify all the maximal such subgroups we show that there are exactly eleven families of such subgroups and explicitly construct them moreover we show that four of these families act simply transitively on the vertices in particular we show that there is no such actions if either the dimension of the building is larger than 7 f is not isomorphic to qp for some prime p or the building is associated to sln d0 where d0 is a non commutative division algebra along the way we also give a new proof of siegel klingen theorem on the rationality of certain dedekind zeta functions and l functions 1

on the accuracy of the binomial approximation to the distance distribution of codes the binomial distribution is a well known approximation to the distance spectra of many classes of codes we derive a lower estimate for the deviation from the binomial approximation

heisenberg xxz model and quantum galilei group the 1d heisenberg spin model with an anisotropy of the xxz type is analysed in terms of the symmetry given by the quantum galilei group gamma q 1 for a chain with an infinite number of sites the authors show that the magnon excitations and the s 1 2 n magnon bound states are determined by the algebra in this case the gamma q 1 symmetry provides a description naturally compatible with the bethe ansatz the recurrence relations determined by gamma q 1 permit one to express the energy of the n magnon bound states in a closed form in terms of tchebischeff polynomials

improved gilbert varshamov bound for constrained systems nonconstructive existence results are obtained for block error correcting codes whose codewords lie in a given constrained system each such system is defined as a set of words obtained by reading the labels of a finite directed labeled graph for a prescribed constrained system and relative minimum distance delta the new lower bounds on the rate of such codes improve on those derived recently by v d kolesnik and v y krachkovsky 1991 the better bounds are achieved by considering a special subclass of sequences in the constrained system namely those having certain empirical statistics determined by delta

on the distance distribution of codes the distinct distribution of a binary code c is the sequence b sub i sub i 0 sup n defined as follows let b sub i w be the number of codewords at distance i from the codeword w and let b sub i be the average of b sub i w over all w in c in this correspondence we study the distance distribution for codes of length n and minimal distance spl delta n with spl delta 0 fixed and n spl rarr spl infin our main aim is to relate the size of the code with the distribution of distances near the minimal distance

hopf superalgebra contractions and r matrix for fermions different contractions are applied to the quantum superalgebra ospq 1 mod 2 in the first of them the graded analogue s hq 1 of the one dimensional heisenberg quantum algebra is obtained and its r matrix explicitly calculated in the second contraction a z2 graded version of the q oscillator is proposed and finally the supersymmetric two dimensional euclidean quantum algebra s eq 2 is found

distance spectrum formulas on the largest minimum distance of block codes a general formula for the asymptotic largest minimum distance in block length of deterministic block codes under generalized distance functions not necessarily additive symmetric and bounded is presented as revealed in the formula the largest minimum distance can be fully determined by the ultimate statistical characteristics of the normalized distance function evaluated under a properly chosen random code generating distribution interestingly the new formula has an analogous form to the general information spectrum expressions of the channel capacity and the optimistic channel capacity respectively derived by verdu and han 1994 and chen and alajaji 1998 1999 as a result a minor class of distance functions for which the largest minimum distance can be derived is characterized a general varshamov gilbert lower bound is next addressed some discussions on the tightness of the general varshamov gilbert bound are also provided finally lower bounds on the largest minimum distances for several specific block coding schemes are rederived in terms of the new formulas followed by comparisons with the known results devoted to the same codes

duals of quasitriangular hopf superalgebras and the classical limit the duality of quasitriangular hopf superalgebras and superbialgebras with so called main commutation relations mcr which is established by means of bilinear forms with values in a finite dimensional representation of the universal r matrix r is used to calculate relations of quantum supercoordinates for several examples the tensor product of representations of r with respect to the comultiplication then leads to the construction of an accordingly deformed tensor product of mcr type superbialgebras the correspondingly deformed tensor product of quantum vector superspaces is then established in the classical limit of lie superbialgebras one similarly obtains supercommutation relations which may be understood in case of quantum groups with deformation parameter q 1 as mcr in the first order of ln q again the deformed tensor product of such semiclassical mcr type lie superbialgebras is constructed

quantum 2 1 kinematical algebras a global approach in this paper we give an approach to quantum deformations of the 2 1 kinematical lie algebras within a scheme that simultaneously describes all groups of motions of classical geometries in n 3 dimensions we cover at once all the kinematical geometries including the quantum versions of inonu wigner contractions which are defined in a natural way and relate q deformations as expected we thus obtain some q deformations previously known for the three dimensional euclidean and 2 1 poincare algebras and also some new q deformations for these and other kinematical algebras such as the 2 1 de sitter galilei and newton hooke algebras

on kissing numbers and spherical codes in high dimensions abstract we prove a lower bound of d 3 2 2 3 d on the kissing number in dimension d this improves the classical lower bound of chabauty shannon and wyner by a linear factor in the dimension we obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle in high dimensions

counting maximal arithmetic subgroups we study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semisimple lie group using an extension of the method developed by borel and prasad

improving the gilbert varshamov bound for q ary codes given positive integers q n and d denote by a sub q n d the maximum size of a q ary code of length n and minimum distance d the famous gilbert varshamov bound asserts that a sub q n d 1 spl ges q sup n v sub q n d where v sub q n d spl sigma sub i 0 sup d sub i sup n q 1 sup i is the volume of a q ary sphere of radius d extending a recent work of jiang and vardy on binary codes we show that for any positive constant spl alpha less than q 1 q there is a positive constant c such that for d spl les spl alpha n a sub q n d 1 spl ges cq sup n v sub q n d n this confirms a conjecture by jiang and vardy

classification of two and three dimensional lie superbialgebras using adjoint representation of lie superalgebras we obtain the matrix form of super jacobi and mixed super jacobi identities of lie superbialgebras by direct calculations of these identities and use of automorphism supergroups of two and three dimensional lie superalgebras we obtain and classify all two and three dimensional lie superbialgebras