harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

chord arc surfaces with small constant ii good parameterizations abstract in a companion paper we introduced a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for planar curves in this paper we address the problem of finding good parameterizations for these surfaces in particular we show that these surfaces must be homeomorphic to r d and we build parameterizations with l p bounds on the gradient and the gradient of the inverse with p as the chord arc constant goes to 0 when d 2 we use the uniformization theorem to find a quasisymmetric parameterization we do not know how to build bilipschitz parameterizations when d 2 or quasisymmetric parameterizations when d 3 the author conjectures that they do not exist in general however we can build bilipschitz reflections in all dimensions and these reflections will satisfy certain extra estimates in terms of carleson measures

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

a free boundary problem for the parabolic poisson kernel abstract we study parabolic chord arc domains introduced by hofmann lewis and nystrom 14 and prove a free boundary regularity result below the continuous threshold more precisely we show that a reifenberg flat parabolic chord arc domain whose poisson kernel has logarithm in vmo must in fact be a vanishing chord arc domain i e satisfies a vanishing carleson measure condition this generalizes to the parabolic setting a result of kenig and toro 26 and answers in the affirmative a question left open in the aforementioned paper of hofmann et al a key step in this proof is a classification of flat blowups for the parabolic problem

new quantum mds codes from negacyclic codes let q be an odd prime power based on classical negacyclic codes we construct two classes of quantum maximum distance separable mds codes with parameters q2 1 q2 2d 3 d q where q 1 mod 4 and 2 d q 1 is even and q2 1 2 q2 1 2 2d 2 d q where 3 d q is odd some of these quantum mds codes are new in the sense that their parameters are different from all the previously known ones

optimal doubling reifenberg flatness and operators of p laplace type in this paper we consider operators of p laplace type of the form a x u 0 concerning a we assume for p 1 fixed an appropriate ellipticity type condition holder continuity in x and that a x p 1 a x whenever x r n and r n 0 let r n be a bounded domain let d be a compact subset of we say that u up d is the a capacitary

nonbinary quantum stabilizer codes we define and show how to construct nonbinary quantum stabilizer codes our approach is based on nonbinary error bases it generalizes the relationship between self orthogonal codes over f sub 4 and binary quantum codes to one between self orthogonal codes over f q sup 2 and q ary quantum codes for any prime power q

enlargement of calderbank shor steane quantum codes it is shown that a classical error correcting code c n k d which contains its dual c sup spl perp spl sube c and which can be enlarged to c n k k 1 d can be converted into a quantum code of parameters n k k n min d 3d 2 this is a generalization of a previous construction it enables many new codes of good efficiency to be discovered examples based on classical bose chaudhuri hocquenghem bch codes are discussed

construction of partial unit memory mds convolutional codes maximum distance separable mds convolutional codes form an optimal family of convolutional codes the study of which is of great importance there are very few general algebraic constructions of mds convolutional codes in this paper we construct a large family of partial unit memory mds convolutional codes over mathbb f q with flexible parameters compared with the previous work the field size q required to define these codes is much smaller the construction also leads to many new strongly mds convolutional codes an important subclass of mds convolutional codes some examples are presented at the end of this paper

weighted lq estimates for stationary stokes system with partially bmo coefficients abstract we prove the unique solvability of solutions in sobolev spaces to the stationary stokes system on a bounded reifenberg flat domain when the coefficients are partially bmo functions i e locally they are merely measurable in one direction and have small mean oscillations in the other directions using this result we establish the unique solvability in muckenhoupt type weighted sobolev spaces for the system with partially bmo coefficients on a reifenberg flat domain we also present weighted a priori l q estimates for the system when the domain is the whole euclidean space or a half space

convolutional codes with maximum distance profile abstract maximum distance profile codes are characterized by the property that two trajectories which start at the same state and proceed to a different state will have the maximum possible minimum distance from each other relative to any other convolutional code of the same rate and degree in this paper we use methods from systems theory to characterize maximum distance profile codes algebraically the main result shows that maximum distance profile codes form a generic set inside the variety which parametrizes the set of convolutional codes of a fixed rate and a fixed degree

convolutional and tail biting quantum error correcting codes rate n 2 n unrestricted and css type quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical self orthogonal rate 1 n f4 linear and binary linear convolutional codes respectively these codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes rate n 2 n block stabilizer codes with the same rate and error correction capability and essentially the same decoding complexity are derived from these convolutional codes via tail biting

on negacyclic mds convolutional codes new families of classical and quantum optimal negacyclic convolutional codes are constructed in this paper this optimality is in the sense that they attain the classical quantum generalized singleton bound the constructions presented in this paper are performed algebraically and not by computational search

on an inverse type problem for the heat equation in parabolic regular graph domains in this paper we prove some results concerning inverse free boundary type problems below the continuous threshold for the heat equation in the setting of parabolic regular graph domains

harmonic measure on locally flat domains we will review work with tatiana toro yielding a characterizati on of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation

strongly mds convolutional codes maximum distance separable mds convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree in this paper a class of mds convolutional codes is introduced whose column distances reach the generalized singleton bound at the earliest possible instant such codes are called strongly mds convolutional codes they also have a maximum or near maximum distance profile the extended row distances of these codes will also be discussed briefly

chord arc surfaces with small constant i abstract we introduce a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for curves in the plane we give various equivalent operator theoretic function theoretic and geometrical characterizations of these surfaces we not know any nice potential theoretic characterizations in a companion paper we address the issue of finding good parameterizations for these surfaces

on optimal constacyclic codes abstract in this paper we construct new families of maximum distance separable mds convolutional codes derived from classical constacyclic codes additionally we also construct new families of almost mds block and convolutional codes most of the new convolutional codes constructed here are unit memory and have degree 2 moreover based on a different method we construct families of asymmetric quantum codes known in the literature all these results are mainly derived from the investigation of suitable properties on cyclotomic cosets of such codes

free boundary regularity for harmonic measures and poisson kernels one of the basic aims of this paper is to study the relationship between the geometry of hypersurface like subsets of euclidean space and the properties of the measures they support in this context we show that certain doubling properties of a measure determine the geometry of its support a radon measure is said to be doubling with constant c if c times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center we prove that if the doubling constant of a measure on r n 1 is close to the doubling constant of the ndimensional lebesgue measure then its support is well approximated by ndimensional a ne spaces provided that the support is relatively at to start with primarily we consider sets which are boundaries of domains in r n 1 the n dimensional hausdorfi measure may not be deflned on the boundary of a domain in r n 1 thus we turn our attention to the harmonic measure which is well behaved under minor assumptions see section 3 we obtain a new characterization of locally at domains in terms of the doubling properties of their harmonic measure see section 3 along these lines we investigate how the weak regularity of the poisson kernel of a domain determines the geometry of its boundary sections 5 and 6 pursue this goal as in alt and cafiarelli s work see ac c1 c2 and also jerison s j in both cases the goal is to prove that under the appropriate technical conditions at at points of the boundary the oscillation of the poisson kernel controls the oscillation of the unit normal vector the difierence between our work and the work in ac is that we measure the oscillation in an integral sense bmo estimates

cauchy integrals calderon projectors and toeplitz operators on uniformly rectifiable domains abstract we develop properties of cauchy integrals associated to a general class of first order elliptic systems of differential operators d on a bounded uniformly rectifiable ur domain in a riemannian manifold m we show that associated to such cauchy integrals are analogues of hardy spaces of functions on annihilated by d and we produce projections of calderon type onto subspaces of l p consisting of boundary values of elements of such hardy spaces we consider toeplitz operators associated to such projections and study their index properties of particular interest is a cobordism argument which often enables one to identify the index of a toeplitz operator on a rough ur domain with that of one on a smoothly bounded domain

variable lorentz estimate for conormal derivative problems of stationary stokes system with partially bmo coefficients abstract we prove a global calderon zygmund type estimate in the framework of lorentz spaces for the variable power of the gradients to the solution pair u p of the conormal derivative problems of stationary stokes system it is mainly assumed that the leading coefficients are merely measurable in one of the spatial variables and have sufficiently small bounded mean oscillation seminorm in the other variables the boundary of domain belongs to the reifenberg flatness and the variable exponents p x satisfy the log holder continuity

caloric measure in parabolic flat domains in this paper we define parabolic chord arc domains and show that in a parabolic chord arc domain with vanishing constant the logarithm of the density of caloric measure with respect to a certain projective lebesgue measure is of vanishing mean oscillation we also obtain a partial converse to this result our work generalizes to the heat equation recent work of kenig and toro for the laplacian

constructions of mds convolutional codes maximum distance separable mds convolutional codes are characterized through the property that the free distance attains the generalized singleton bound the existence of mds convolutional codes was established by two of the authors by using methods from algebraic geometry this correspondence provides an elementary construction of mds convolutional codes for each rate k n and each degree spl delta the construction is based on a well known connection between quasi cyclic codes and convolutional codes

on classes of convolutional codes that are not asymptotically catastrophic the author denotes by w sub 0 the minimum average weight per edge over all nonzero cycles in the state diagram for a convolutional code and assumes that a technique is available for generating canonical parity check matrices for codes with increasing degree m the obtained class of codes is asymptotically catastrophic if w sub 0 approaches zero for large m we prove the existence of convolutional code classes that are not asymptotically catastrophic by providing explicit constructions of codes with nonzero w sub 0 for all m

regularity and free boundary regularity for the p laplace operator in reifenberg flat and ahlfors regular domains regularity and free boundary regularity for the p laplace operator in reifenberg flat and ahlfors regular domains

on cyclotomic cosets and code constructions abstract new properties of q ary cyclotomic cosets modulo n q m 1 where q 3 is a prime power are investigated in this paper based on these properties the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed as an application new families of nonbinary calderbank shor steane css quantum codes as well as new families of convolutional codes are constructed in this work these new css codes have parameters better than the ones available in the literature the convolutional codes constructed here have free distance greater than the ones available in the literature

divergence form operators in reifenberg flat domains we study the boundary regularity of solutions of elliptic operators in divergence form with c 0 coefficients or operators which are small perturbations of the laplacian in non smooth domains we show that as in the case of the laplacian there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain

a numerical algorithm for the stokes problem based on an integral equation for the pressure via conformal mappings in this paper we present an algorithm for solving numerically the stokes problem in the plane the known algorithms are all based on certain discretization schemes for the analytic equations in contrast to this recent work our algorithm uses an explicit analytic solution of a certain approximating problem which can easily be solved numerically up to machine accuracy on the one hand this analytic formula is based on a complex representation of all solutions of the stokes differential equations and on the other hand it is based on the conformal mapping of the given domain on the unit disc therefore a central prerequisite of our corresponding program is a program for computing this conformal mapping

linear system modelization of concatenated block and convolutional codes in this paper we characterize four models of concatenation of a block code and a convolutional code from a linear systems theory viewpoint we provide the input state output representation of these models and we give conditions in order to get a non catastrophic concatenated convolutional code with minimal representation lower bounds on the free distances of the concatenated codes are also developed

some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes abstract quantum maximal distance separable mds codes are an important class of quantum codes recently many scholars utilize constacyclic codes to construct some quantum mds codes in this paper several new families of optimal asymmetric quantum codes and optimal quantum convolutional codes are constructed by using constacyclic codes moreover these quantum codes constructed in this paper are different from the ones in the literature

pseudocyclic maximum distance separable codes the n k pseudocyclic maximum distance separable mds codes modulo x sup n a over gf q are considered suppose that n is a divisor of q 1 if n is odd pseudocyclic mds codes exist for all k however if n is even nontrivial pseudocyclic mds codes exist for odd k but not for even k if a is a quadratic residue in gf q and they exist for even k but not for odd k if a is not a quadratic residue in gf q also considered is the case when n is a divisor of q 1 and it is shown that pseudocyclic mds codes exist if and only if the multiplicative order of a divides q 1 n and that when this condition is satisfied such codes exist for all k if the condition is not satisfied every pseudocyclic code of length n is the result of interleaving a shorter pseudocyclic code