efficient iterative techniques for soft decision decoding of reed solomon codes two new iterative soft decision decoding methods for reed solomon rs codes are proposed these methods are based on bit level belief propagation bp decoding in order to make bp decoding effective for rs codes we use an extended binary parity check matrix with a lower density and reduced number of 4 cycles compared to the original binary parity check matrix of the code in the first proposed method we take advantage of the cyclic structure of rs codes based on this property we can apply the belief propagation algorithm on any cyclically shifted version of the received symbols with the same binary parity check matrix for each shifted version of received symbols the distribution of reliability values will change and deterministic errors can be avoided this method results in considerable performance improvement of rs codes compared to hard decision decoding the performance is also superior to some popular soft decision decoding methods the second method is based on information correction in bp decoding it means that we determine least reliable bits and by changing their channel information the convergence of the decoder is improved compared to the first method this method needs less bp iterations less complexity but its performance is not as good

extending the bruhat order and the length function from the weyl group to the weyl monoid abstract for a symmetrizable kac moody algebra the category of admissible representations of the category o is an analogue of the category of finite dimensional representations of a semisimple lie algebra the monoid which is associated to this category and the category of restricted duals by a tannaka krein reconstruction contains the kac moody group as open dense unit group it has similar properties as a reductive algebraic monoid in particular there are bruhat and birkhoff decompositions the weyl group replaced by the weyl monoid c mokler an analogue of a reductive algebraic monoid whose unit group is a kac moody group arxiv math ag 0204246 mem amer math soc in press we determine the closure relations of the bruhat and birkhoff cells which give extensions of the bruhat order from the weyl group to the weyl monoid we show that the bruhat and birkhoff cells are irreducible and principal open in their closures we give product decompositions of the bruhat and birkhoff cells we define extended length functions which are compatible with the extended bruhat orders we show a generalization of some of the tits axioms for twin bn pairs

the geometry of turbo decoding dynamics the spectacular performance offered by turbo codes sparked intense interest in them a considerable amount of research has simplified formalized and extended the ideas inherent in the original turbo code construction nevertheless the nature of the relatively simple ad hoc turbo decoding algorithm has remained something of a mystery we present a geometric interpretation of the turbo decoding algorithm the geometric perspective clearly indicates the relationship between turbo decoding and maximum likelihood decoding analysis of the geometry leads to new results concerning existence of fixed points conditions for uniqueness conditions for stability and proximity to maximum likelihood decoding

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

efficient linear programming decoding of hdpc codes we propose several improvements for linear programming lp decoding algorithms for high density parity check hdpc codes first we use the automorphism groups of a code to create parity check matrix diversity and to generate valid cuts from redundant parity checks second we propose an efficient mixed integer decoder utilizing the branch and bound method we further enhance the proposed decoders by removing inactive constraints and by adapting the parity check matrix prior to decoding according to the channel observations based on simulation results the proposed decoders achieve near ml performance with reasonable complexity

semidefinite representation of convex sets let s x in mathbb r n g 1 x geq 0 ldots g m x geq 0 be a semialgebraic set defined by multivariate polynomials g i x assume s is convex compact and has nonempty interior let s i x in mathbb r n g i x geq 0 and s resp s i be the boundary of s resp s i this paper as does the subject of semidefinite programming sdp concerns linear matrix inequalities lmis the set s is said to have an lmi representation if it equals the set of solutions to some lmi and it is known that some convex s may not be lmi representable helton and vinnikov in commun pure appl math 60 5 654 674 2007 a question arising from nesterov and nemirovski siam studies in applied mathematics society for industrial and applied mathematics siam philadelphia 1994 see helton and vinnikov in commun pure appl math 60 5 654 674 2007 and nemirovski in plenary lecture international congress of mathematicians icm madrid spain 2006 is given a subset s of mathbb r n does there exist an lmi representable set in some higher dimensional space mathbb r n n whose projection down onto mathbb r n equals s such s is called semidefinite representable or sdp representable this paper addresses the sdp representability problem the following are the main contributions of this paper i assume g i x are all concave on s if the positive definite lagrange hessian condition holds i e the hessian of the lagrange function for optimization problem of minimizing any nonzero linear function l t x on s is positive definite at the minimizer then s is sdp representable ii if each g i x is either sos concave 2 g i x w x t w x for some possibly nonsquare matrix polynomial w x or strictly quasi concave on s then s is sdp representable iii if each s i is either sos convex or poscurv convex s i is compact convex whose boundary has positive curvature and is nonsingular i e g i x 0 on s i s then s is sdp representable this also holds for s i for which s i s extends smoothly to the boundary of a poscurv convex set containing s iv we give the complexity of schmudgen and putinar s matrix positivstellensatz which are critical to the proofs of i iii

the lattice of j classes of j σ irreducible monoids ii abstract a reductive monoid is an algebraic monoid with a reductive unit group generally we have the question of finding the orbits of the unit group of a reductive monoid acting on both sides of the monoid putcha and renner give a recipe to determine the orbits for j irreducible monoids according to the dynkin diagrams we obtain a similar recipe for the question to j irreducible monoids not j irreducible of type d 2 n however there is no similar answer for types a n n 4 and e 2 6 the fixed points of any j irreducible monoid under is a finite reductive monoid we obtain that any such finite reductive monoid is j irreducible then we find the orbits of these monoids under the two sided action of their unit groups

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

idempotent lattices renner monoids and cross section lattices of the special orthogonal algebraic monoids abstract let mson n is even be the special orthogonal algebraic monoid t a maximal torus of the unit group and t the zariski closure of t in the whole matrix monoid m n in this paper we explicitly determine the idempotent lattice e t the renner monoid r and the cross section lattice of mson in terms of the weyl group and the concept of admissible sets see definition 3 1 it turns out that there is a one to one relationship between e t and the admissible subsets and that r is a submonoid of r n the renner monoid m n also is a sublattice of n the cross section lattice of m n

enhancing iterative decoding of cyclic ldpc codes using their automorphism groups for cyclic ldpc codes we propose to use their automorphism groups to improve the iterative decoding performance the basic idea is to construct nonequivalent parity check matrices via column permutations three types of iterative decoders are devised to take advantage of the code s automorphism group in this paper we focus on cyclic ldpc codes defined by a circulant parity check matrix and consider two known subgroups of the automorphism group of a cyclic code for the large class of idempotent based cyclic ldpc codes in the literature we show that the two subgroups only provide equivalent parity check matrices and thus cannot be harnessed for iterative decoding towards exploiting the automorphism group of a code we propose a new class of cyclic ldpc codes based on pseudo cyclic mds codes with two information symbols for which nonequivalent parity check matrices are obtained simulation results show that for our constructed codes of short lengths the automorphism group can significantly enhance the iterative decoding performance

convexity in semialgebraic geometry and polynomial optimization we review several and provide new results on the theory of moments sums of squares and basic semialgebraic sets when convexity is present in particular we show that under convexity the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence a highly desirable feature as convex problems are in principle easier to solve in addition if a basic semialgebraic set mathbf k is convex but its defining polynomials are not we provide two algebraic certificates of convexity which can be checked numerically the second is simpler and holds if a sufficient and almost necessary condition is satisfied it also provides a new condition for mathbf k to have semidefinite representation for this we use and extend some of the recent results from the author and helton and nie math program to appear finally we show that when restricting to a certain class of convex polynomials the celebrated jensen s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures

reduced complexity iterative decoding of low density parity check codes based on belief propagation two simplified versions of the belief propagation algorithm for fast iterative decoding of low density parity check codes on the additive white gaussian noise channel are proposed both versions are implemented with real additions only which greatly simplifies the decoding complexity of belief propagation in which products of probabilities have to be computed also these two algorithms do not require any knowledge about the channel characteristics both algorithms yield a good performance complexity trade off and can be efficiently implemented in software as well as in hardware with possibly quantized received values

infinite dimensional reductive monoids associated to highest weight representations of kac moody groups abstract starting with a highest weight representation of a kac moody group over the complex numbers we construct a monoid whose unit group is the image of the kac moody group under the representation multiplied by the nonzero complex numbers we show that this monoid has similar properties to those of a j irreducible reductive linear algebraic monoid in particular the monoid is unit regular and has a bruhat decomposition and the idempotent lattice of the generalized renner monoid of the bruhat decomposition is isomorphic to the face lattice of the convex hull of the weyl group orbit of the highest weight

np hardness of deciding convexity of quartic polynomials and related problems we show that unless p np there exists no polynomial time or even pseudo polynomial time algorithm that can decide whether a multivariate polynomial of degree four or higher even degree is globally convex this solves a problem that has been open since 1992 when n z shor asked for the complexity of deciding convexity for quartic polynomials we also prove that deciding strict convexity strong convexity quasiconvexity and pseudoconvexity of polynomials of even degree four or higher is strongly np hard by contrast we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time

faces and maximizer subsets of highest weight modules abstract in this paper we study general highest weight modules v over a complex finite dimensional semisimple lie algebra g we present three formulas for the set of weights of a large family of modules v which include but are not restricted to all simple modules and all parabolic verma modules these formulas are direct and do not involve cancellations and were not previously known in the literature our results extend the notion of the weyl polytope to general highest weight g modules v we also show that for all simple modules the convex hull of the weights is a w j invariant polyhedron for some parabolic subgroup w j we compute its vertices faces and symmetries more generally we also do this for all parabolic verma modules and for all modules v with highest weight not on a simple root hyperplane to show our results we extend the notion of convexity to arbitrary additive subgroups a r of coefficients our techniques enable us to completely classify weak a faces of the support sets wt v in the process extending classical results of satake borel tits vinberg and casselman as well as modern variants by chari dolbin ridenour and cellini marietti to general highest weight modules

random edge local complementation with applications to iterative decoding of high density parity check codes we describe the application of edge local complementation elc to a tanner graph associated with a binary linear code c various properties of elc are described mainly the special case of isomorphic elc operations and the relationship to the automorphism group of the code aut c and the generalization of elc to weight bounding elc wb elc operations under which the number of edges remains upper bounded elc generates all systematic parity check matrices the orbit of the code so wb elc facilitates a restriction to low weight matrices of this orbit we propose using elc and wb elc as a source of diversity to improve iterative soft input soft output decoding of high density parity check hdpc codes with the sum product algorithm spa a motivation of elc enhanced spa decoding is locality that diversity is achieved by local graph action and is well suited to the local actions that constitute the spa and allows for parallel software implementation simulation data on the error rate performance of the proposed spa elc and spa wbelc iterative decoding algorithms is shown for several hdpc codes a gain is reported over spa decoding and over a recently proposed algorithm to decode hdpc codes using permutations from aut c elc enhanced decoding extends the scope of iterative decoding to codes with trivial aut c

the extraction and complexity limits of graphical models for linear codes two broad classes of graphical modeling problems for codes can be identified in the literature constructive and extractive problems the former class of problems concern the construction of a graphical model in order to define a new code the latter class of problems concern the extraction of a graphical model for a fixed given code the design of a new low density parity check code for some given criteria e g target block length and code rate is an example of a constructive problem the determination of a graphical model for a classical linear block code that implies a decoding algorithm with desired performance and complexity characteristics is an example of an extractive problem this work focuses on extractive graphical model problems and aims to lay out some of the foundations of the theory of such problems for linear codes the primary focus of this work is a study of the space of all graphical models for a fixed given code the tradeoff between cyclic topology and complexity in this space is characterized via the introduction of a new bound the forest inducing cut set bound fi csb the proposed bound provides a more precise characterization of this tradeoff than that which can be obtained using existing tools e g the csb and can be viewed as a generalization of the square root bound for tail biting trellises to graphical models with arbitrary cyclic topologies searching the space of graphical models for a given code is then enabled by introducing a set of basic graphical model transformation operations that are shown to span this space finally heuristics for extracting novel graphical models for linear block codes using these transformations are investigated

actions of the face monoid associated to a kac moody group on its building abstract we described in c mokler an analogue of a reductive algebraic monoid whose unit group is a kac moody group mem amer math soc 823 2005 90 pp a monoid g acting on the integrable highest weight modules of a symmetrizable kac moody algebra it has similar structural properties as a reductive algebraic monoid with unit group a kac moody group g now we find natural extensions of the action of the kac moody group g on its building to actions of the monoid g on these extensions are partly motivated by representation theory and the combinatorics of the faces of the tits cone

generic hecke algebra for renner monoids abstract to each renner monoid r we associate a generic hecke algebra h r over z q which is a deformation of the monoid z algebra of r if m is a finite reductive monoid with borel subgroup b and associated renner monoid r then we obtain the associated iwahori hecke algebra h m b by specialising q in h r and tensoring by c over z as in the classical case of finite reductive groups

iterative soft input soft output decoding of reed solomon codes by adapting the parity check matrix an iterative algorithm is presented for soft input soft output siso decoding of reed solomon rs codes the proposed iterative algorithm uses the sum product algorithm spa in conjunction with a binary parity check matrix of the rs code the novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the spa is applied at each iteration the proposed algorithm can be geometrically interpreted as a two stage gradient descent with an adaptive potential function this adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and therefore significantly improves the performance simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding hdd and compare favorably with other popular soft decision decoding methods

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

types of reductive monoids abstract let m be a reductive monoid with a reductive unit group g clearly there is a natural g g action on m the orbits are the j classes in the sense of semigroup theory and form a finite lattice the general problem of finding the lattice remains open in this paper we study a new class of reductive monoids constructed by multilined closure we obtain a general theorem to determine the lattices of these monoids we find that the j irreducible monoids of suzuki type and ree type belong to this new class using the general theorem we then list all the lattices and type maps of the j irreducible monoids of suzuki type and ree type

a complete characterization of the gap between convexity and sos convexity our first contribution in this paper is to prove that three natural sum of squares sos based sufficient conditions for convexity of polynomials via the definition of convexity its first order characterization and its second order characterization are equivalent these three equivalent algebraic conditions henceforth referred to as sos convexity can be checked by semidefinite programming whereas deciding convexity is np hard if we denote the set of convex and sosconvex polynomials in n variables of degree d with cn d and cn d respectively then our main contribution is to prove that cn d cn d if and only if n 1 or d 2 or n d 2 4 we also present a complete characterization for forms homogeneous polynomials except for the case n d 3 4 which is joint work with g blekherman and is to be published elsewhere our result states that the set cn d of convex forms in n variables of degree d equals the set cn d of sos convex forms if and only if n 2 or d 2 or n d 3 4 to prove these results we present in particular explicit examples of polynomials in c2 6 c2 6 and c3 4 c3 4 and forms in c3 6 c3 6 and c4 4 c4 4 and a general procedure for constructing forms in cn d 2 cn d 2 from nonnegative but not sos forms in n variables and degree d although for disparate reasons the remarkable outcome is that convex polynomials resp forms are sos convex exactly in cases where nonnegative polynomials resp forms are sums of squares as characterized by hilbert

pseudocodewords of tanner graphs this paper presents a detailed analysis of pseudocodewords of tanner graphs pseudocodewords arising on the iterative decoder s computation tree are distinguished from pseudocodewords arising on finite degree lifts lower bounds on the minimum pseudocodeword weight are presented for the bec bsc and awgn channel some structural properties of pseudocodewords are examined and pseudocodewords and graph properties that are potentially problematic with min sum iterative decoding are identified an upper bound on the minimum degree lift needed to realize a particular irreducible lift realizable pseudocodeword is given in terms of its maximal component and it is shown that all irreducible lift realizable pseudocodewords have components upper bounded by a finite value t that is dependent on the graph structure examples and different tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed the results obtained provide some insights in relating the structure of the tanner graph to the pseudocodeword distribution and suggest ways of designing tanner graphs with good minimum pseudocodeword weight

improved random redundant iterative hdpc decoding an iterative algorithm for soft input soft output siso decoding of classical algebraic cyclic block codes is presented below inspired by other approaches for high performance belief propagation bp decoding this algorithm requires up to 10 times less computational complexity than other methods that achieve similar performance by utilizing multiple bp decoders and using random permutation taken from the permutation group of the code this algorithm reaches near maximum likelihood performance a computational complexity comparison of the proposed algorithm versus other methods is presented as well this includes complexity versus performance analysis allowing one to trade between the former and the latter according to ones needs

transactions letters random redundant iterative soft in soft out decoding this letter presents an iterative soft in soft out siso decoding algorithm based on redundant tanner graphs that is applicable to arbitrary linear block codes the proposed algorithm utilizes the permutation group of a code in order to efficiently and randomly generate redundant parity checks