all the stabilizer codes of distance 3 we give necessary and sufficient conditions for the existence of stabilizer codes n k 3 of distance 3 for qubits n k log2 3n 1 n where n 1 if n 84m 1 3 1 2 or n 4m 2 1 3 1 2 3 for some integer m 1 and n 0 otherwise or equivalently a code n n r 3 exists if and only if n 4r 1 3 4r 1 3 n 1 2 3 for even r and n 8 4r 3 1 3 8 4 3 1 3 n 1 for odd r given an arbitrary length n we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

duality in entanglement assisted quantum error correction the dual of an entanglement assisted quantum error correcting eaqec code is defined from the orthogonal group of a simplified stabilizer group from the poisson summation formula this duality leads to the macwilliams identities and linear programming bounds for eaqec codes we establish a table of upper and lower bounds on the minimum distance of any maximal entanglement eaqec code with length up to 15 channel qubits

on the constructions of constant weight codes two methods of constructing binary constant weight codes from 1 codes over gf q and 2 constant weight codes over gf q are presented several classes of binary optimum constant weight codes are derived from these methods in general we show that binary optimum constant weight codes which achieve the johnson bound can be constructed from optimum codes over gf q which achieve the plotkin bound finally several classes of optimum constant weight codes over gf q are constructed

the minimum distance problem for two way entanglement purification entanglement purification takes a number of noisy epr pairs 00 11 and processes them to produce a smaller number of more reliable pairs if this is done with only a forward classical side channel the procedure is equivalent to using a quantum error correcting code qecc we instead investigate entanglement purification protocols with two way classical side channels 2 epps for finite block sizes in particular we consider the analog of the minimum distance problem for qeccs and show that 2 epps can exceed the quantum hamming bound and the quantum singleton bound we also show that 2 epps can achieve the rate k n 1 t n log sub 2 3 h t n o 1 n asymptotically reaching the quantum hamming bound where the epp produces at least k good pairs out of n total pairs with up to t arbitrary errors and h x xlog sub 2 x 1 x log sub 2 1 x is the usual binary entropy in contrast the best known lower bound on the rate of qeccs is the quantum gilbert varshamov bound k n spl ges 1 2t n log sub 2 3 h 2t n indeed in some regimes the known upper bound on the asymptotic rate of good qeccs is strictly below our lower bound on the achievable rate of 2 epps

linear size optimal q ary constant weight codes and constant composition codes an optimal constant composition or constant weight code of weight w has linear size if and only if its distance d is at least 2w 1 when d 2w the determination of the exact size of such a constant composition or constant weight code is trivial but the case of d 2w 1 has been solved previously only for binary and ternary constant composition and constant weight codes and for some sporadic instances this paper provides a construction for quasicyclic optimal constant composition and constant weight codes of weight w and distance 2w 1 based on a new generalization of difference triangle sets as a result the sizes of optimal constant composition codes and optimal constant weight codes of weight w and distance 2w 1 are determined for all such codes of sufficiently large lengths this solves an open problem of etzion the sizes of optimal constant composition codes of weight w and distance 2w 1 are also determined for all w 6 except in two cases

nonbinary quantum codes we present several results on quantum codes over general alphabets that is in which the fundamental units may have more than two states in particular we consider codes derived from finite symplectic geometry assumed to have additional global symmetries from this standpoint the analogs of calderbank shor steane codes and of gf 4 linear codes turn out to be special cases of the same construction this allows us to construct families of quantum codes from certain codes over number fields in particular we get analogs of quadratic residue codes including a single error correcting code encoding one letter in five for any alphabet size we also consider the problem of fault tolerant computation through such codes generalizing ideas of gottesman see phys rev a vol 57 no 1 p127 37 1998

on the macwilliams identity for classical and quantum convolutional codes the weight generating functions associated with convolutional codes ccs are based on state space realizations or the weight adjacency matrices wams the macwilliams identity for ccs on the wams was first established by gluesing luerssen and schneider in the case of minimal encoders and generalized by forney we consider this problem in the viewpoint of constraint codes and obtain a simple and direct proof of this macwilliams identity in the case of minimal encoders for our purpose we choose a different representation for the exact weight generating function ewgf of a block code by defining it as a linear combination of orthonormal vectors in dirac bra ket notation this representation provides great flexibility so that general split weight generating functions and their macwilliams identities can be easily obtained from the macwilliams identity for ewgfs as a result we also obtain the macwilliams identity for the input parity wams of a systematic convolutional code and its dual finally paralleling the development of the classical case we establish the macwilliams identity for quantum ccs

kerdock codes determine unitary 2 designs the non linear binary kerdock codes are known to be gray images of certain extended cyclic codes of length n 2 m over mathbb z 4 we show that exponentiating these mathbb z 4 valued codewords by i triangleq sqrt 1 produces stabilizer states that are quantum states obtained using only clifford unitaries these states are also the common eigenvectors of commuting hermitian matrices forming maximal commutative subgroups mcs of the pauli group we use this quantum description to simplify the derivation of the classical weight distribution of kerdock codes next we organize the stabilizer states to form n 1 mutually unbiased bases and prove that automorphisms of the kerdock code permute their corresponding mcs thereby forming a subgroup of the clifford group when represented as symplectic matrices this subgroup is isomorphic to the projective special linear group psl 2 n we show that this automorphism group acts transitively on the pauli matrices which implies that the ensemble is pauli mixing and hence forms a unitary 2 design the kerdock design described here was originally discovered by cleve et al 2016 but the connection to classical codes is new which simplifies its description and translation to circuits significantly sampling from the design is straightforward the translation to circuits uses only clifford gates and the process does not require ancillary qubits finally we also develop algorithms for optimizing the synthesis of unitary 2 designs on encoded qubits i e to construct logical unitary 2 designs software implementations are available at https github com nrenga symplectic arxiv18a which we use to provide empirical gate complexities for up to 16 qubits

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 semidefinite program for distillable entanglement we show that the maximum fidelity obtained by a positive partial transpose p p t distillation protocol is given by the solution to a certain semidefinite program this gives a number of new lower and upper bounds on p p t distillable entanglement and thus new upper bounds on 2 locally distillable entanglement in the presence of symmetry the semidefinite program simplifies considerably becoming a linear program in the case of isotropic and werner states using these techniques we determine the p p t distillable entanglement of asymmetric werner states and maximally correlated states we conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1 local distillation

stochastic cascades and 3 dimensional navier stokes equations in this article we study the incompressible navier stokes equations in 3 the non linear integral equation satisfied by the fourier transform of the laplacian of the velocity field can be interpreted in terms of a branching process and a composition rule along the associated tree we derive from this representation new classes where global existence and uniqueness can be proven

ergodicity of the 2d navier stokes equations with random forcing we consider the navier stokes equation on a two dimensional torus with a random force acting at discrete times and analytic in space for arbitrarily small viscosity coefficient we prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time

high performance single error correcting quantum codes for amplitude damping we construct families of high performance quantum amplitude damping codes all of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension one family is built from nonlinear error correcting codes for classical asymmetric channels with which we systematically construct quantum amplitude damping codes with parameters better than any prior construction known for any block length n 8 except n 2r 1 we generalize this construction to employ classical codes over gf 3 with which we numerically obtain better performing codes up to length 14 because the resulting codes are of the codeword stabilized cws type conceptually simple though potentially computationally expensive encoding and decoding circuits are available

on the units of bipartite entanglement is sixteen ounces of entanglement always equal to one pound in a good physical theory dimensionless quantities such as the ratio mp me of the mass of the proton to the mass of the electron do not depend on the system of units being used this paper demonstrates that one widely used method for defining measures of entanglement violates this principle specifically in this approach dimensionless ratios e e of entanglement measures may depend on what state is chosen as the basic unit of entanglement this observation leads us to suggest three novel approaches to the quantification of entanglement these approaches lead to unit free definitions for the entanglement of formation and the distillable entanglement and suggest natural measures of entanglement for multipartite systems we also show that the behaviour of one of these novel measures the entanglement of computation is related to some open problems in computational complexity

constructions for q ary constant weight codes this paper introduces a new combinatorial construction for q ary constant weight codes which yields several families of optimal codes and asymptotically optimal codes the construction reveals intimate connection between q ary constant weight codes and sets of pairwise disjoint combinatorial designs of various types

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

optimal ternary constant weight codes of weight four and distance six recently chee and ling constructions for q ary constant weight codes ieee trans inf theory vol 53 no 1 135 146 jan 2007 introduced a new combinatorial construction for q ary constant weight codes which reveals a close connection between q ary constant weight codes and sets of pairwise disjoint combinatorial designs in this paper we study the problem of constructing optimal ternary constant weight codes with hamming weight four and minimum distance six using this approach the construction here exploits completely reducible super simple designs and group divisible codes the problem is solved leaving only two cases undetermined previously the sizes of constant weight codes of weight four and distance six were known only for those of length no greater than 10

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

existence and uniqueness of stationary solutions for 3d navier stokes system with small random forcing via stochastic cascades we consider the 3d navier stokes system in the fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition we explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with fourier transform majorized by a certain function h moreover we prove the following one force one solution principle the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t our explicit construction of the solution is based upon the stochastic cascade representation

the sizes of optimal q ary codes of weight three and distance four a complete solution this correspondence introduces two new constructive techniques to complete the determination of the sizes of optimal q ary codes of constant weight three and distance four

new asymptotic bounds for self dual codes and lattices we give an independent proof of the krasikov litsyn bound d n spl lsim 1 5 sup 1 4 2 on doubly even self dual binary codes the technique used a refinement of the mallows odlyzko sloane approach extends easily to other families of self dual codes modular lattices and quantum codes in particular we show that the krasikov litsyn bound applies to singly even binary codes and obtain an analogous bound for unimodular lattices we also show that in each case our bound differs from the true optimum by an amount growing faster than o spl radic n

codeword stabilized quantum codes we present a unifying approach to quantum error correcting code design that encompasses additive stabilizer codes as well as all known examples of nonadditive codes with good parameters we use this framework to generate new codes with superior parameters to any previously known in particular we find 10 18 3 and 10 20 3 codes we also show how to construct encoding circuits for all codes within our framework

quantum error detection i statement of the problem this paper is devoted to the problem of error detection with quantum codes we show that it is possible to give a consistent definition of the undetected error event to prove this we examine possible problem settings for quantum error detection our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes we discuss possible transmission protocols with stabilizer and unrestricted quantum codes the set of results proved in the paper shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators we examine polynomial invariants of quantum codes and show that coefficients of rains s see ibid vol44 p 1388 94 1998 unitary weight enumerators are known for classical codes under the name of binomial moments of the distance distribution as in the classical situation these enumerators provide an alternative expression for the probability of undetected error

two infinite families of nonadditive quantum error correcting codes we construct explicitly two infinite families of genuine nonadditive 1 error correcting quantum codes and prove that their coding subspaces are 50 larger than those of the optimal stabilizer codes of the same parameters via the linear programming bound all these nonadditive codes can be characterized by a stabilizer like structure and thus their encoding circuits can be designed in a straightforward manner

stochastic tamed 3d navier stokes equations existence uniqueness and ergodicity in this paper we prove the existence of a unique strong solution to a stochastic tamed 3d navier stokes equation in the whole space as well as in the periodic boundary case then we also study the feller property of solutions and prove the existence of invariant measures for the corresponding feller semigroup in the case of periodic conditions moreover in the case of periodic boundary and degenerated additive noise using the notion of asymptotic strong feller property proposed by hairer and mattingly ann math 164 993 1032 2006 we prove the uniqueness of invariant measures for the corresponding transition semigroup

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

optimal quaternary constant weight codes with weight four and distance five constant weight codes play an important role in coding theory the problem of determining the sizes for optimal quaternary constant weight codes with length n weight 4 and minimum hamming distance 5 n 5 4 4 codes has been investigated in several papers although some constructions and several infinite families for such codes with length n 0 1 mod 4 have been given the problem is still far from complete in this paper we determine the size of an optimal n 5 4 4 code for each integer n 4 leaving 55 lengths unsolved especially for length n 0 1 mod 4 the existence problem of the equivalent combinatorial object namely the generalized steiner system is solved leaving only seven values undetermined

quantum error detection ii bounds for pt i see ibid vol 46 no 3 p 778 88 2000 in part i of this paper we formulated the problem of error detection with quantum codes on the depolarizing channel and gave an expression for the probability of undetected error via the weight enumerators of the code in this part we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent the lower existence bound is proved for stabilizer codes by a counting argument for classical self orthogonal quaternary codes upper bounds are proved by linear programming first we formulate two linear programming problems that are convenient for the analysis of specific short codes next we give a relaxed formulation of the problem in terms of optimization on the cone of polynomials in the krawtchouk basis we present two general solutions of the problem together they give an upper bound on the exponent of undetected error the upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1

polynomial invariants of quantum codes the weight enumerators shor and laflamme 1997 of a quantum code are quite powerful tools for exploring its structure as the weight enumerators are quadratic invariants of the code this suggests the consideration of higher degree polynomial invariants we show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one to one correspondence with s sub k sup n we then present a number of equations and inequalities in these invariants in particular we give a higher order generalization of the shadow enumerator of a code and prove that its coefficients are nonnegative we also prove that the quartic invariants of a 4 4 2 sub 2 code are uniquely determined an important step in a proof that any 4 4 2 sub 2 code is additive rains 1999

complex codebooks from combinatorial designs codebooks also called signal sets meeting the welch bounds are desirable in code division multiple access cdma systems in 2003 binary codebooks meeting welch s bounds were constructed using difference sets by ding golin and kloslashve recently a generic construction of codebooks with cyclic difference sets meeting welch s bound on the maximum cross correlation amplitude was developed by xia in this correspondence the idea of xia is extended and related constructions of optimal codebooks with both cyclic and noncyclic difference sets are presented these codebooks are optimal in the sense that they also meet this welch bound in addition complex codebooks that almost meet welch s bound on the maximum cross correlation amplitude are also constructed with almost difference sets

reduced dimension multiuser detection we present a reduced dimension multiuser detector rd mud structure for synchronous systems that significantly decreases the number of required correlation branches at the receiver front end while still achieving performance similar to that of the conventional matched filter mf bank rd mud exploits the fact that in some wireless systems the number of active users may be small relative to the total number of users in the system hence the ideas of analog compressed sensing may be used to reduce the number of correlators the correlating signals used by each correlator are chosen as an appropriate linear combination of the users spreading waveforms we derive the probability of symbol error when using two methods for recovery of active users and their transmitted symbols the reduced dimension decorrelating rdd detector which combines subspace projection and thresholding to determine active users and sign detection for data recovery and the reduced dimension decision feedback rddf detector which combines decision feedback matching pursuit for active user detection and sign detection for data recovery we derive probability of error bounds for both detectors and show that the number of correlators needed to achieve a small probability of symbol error is on the order of the logarithm of the number of users in the system the theoretical performance results are validated via numerical simulations

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

the need for structure in quantum ldpc codes the existence of quantum ldpc codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information despite years of research good quantum ldpc codes are not known to exist but at the very least it is known they cannot be defined on very regular topologies like low dimensional grids in this work we establish a complementary result showing that good quantum css codes which are sparsely generated require structure in the local terms that constrain the code space so as not to be too random in a well defined sense to show this we prove a weak converse to a theorem of krasikov and litsyn on weight distributions of classical codes due to which may be of independent interest subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight tantamount to having a non negligible approximate minimal distance while they may not have a large minimal non zero weight they still have very few words of low hamming weight

parallel quantum computation and quantum codes we study the class qnc of efficient parallel quantum circuits the quantum analog of nc we exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth including circuits for encoding and decoding standard quantum error correcting codes or more generally any circuit consisting of controlled not gates controlled pi shifts and hadamard gates finally while we note the exact quantum fourier transform can be parallelized to linear depth we conjecture that neither it nor a simpler staircase circuit can be parallelized to less than this

monotonicity of the quantum linear programming bound the most powerful technique known at present for bounding the size of quantum codes of prescribed minimum distance is the quantum linear programming bound unlike the classical linear programming bound it is not immediately obvious that if the quantum linear programming constraints are satisfiable for dimension k then the constraints can be satisfied for all lower dimensions we show that the quantum linear programming bound is monotonic in this sense and give an explicitly monotonic reformulation

a family of asymptotically good quantum codes based on code concatenation we explicitly construct an infinite family of asymptotically good concatenated quantum stabilizer codes where the outer codes use quantum reed solomon codes and the inner codes use a set of special quantum codes in the field of quantum error correcting codes this is the first time that a family of asymptotically good quantum codes is derived from bad codes its occurrence supplies a gap in quantum coding theory

classical and quantum evaluation codes at the trace roots we introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function we give conditions for self orthogonality of these codes and their subfield subcodes with respect to the hermitian inner product they allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature for the binary case we obtain records at http codetables de moreover we obtain several classical linear codes over the field mathbb f 4 which are records at http codetables de

a nonadditive quantum code every good quantum error correcting code discovered thus far such as those known as stabilizer or additive codes has had the structure of an eigenspace of an abelian group generated by tensor products of pauli matrices in this letter we present the first example of a code that is better than any code of this type it encodes six states in five qubits and can correct the erasure of any single qubit

a coupling approach to randomly forced nonlinear pde s ii we consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure the result obtained applies in particular to the 2d navier stokes system and multidimensional complex ginzburg landau equation with random kick force

quantum codes of minimum distance two it is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2 thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize with this in mind we present a number of results on codes of minimum distance 2 we first compute the linear programming bound on the dimension of such a code then show that this bound can only be attained when the code either is of even length or is of length 3 or 5 we next consider questions of uniqueness showing that the optimal code of length 2 or 1 is unique implying that the well known one qubit in five single error correcting code is unique and presenting nonadditive optimal codes of all greater even lengths finally we compute the full automorphism group of the more important distance 2 codes allowing us to determine the full automorphism group of any gf 4 linear code

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

optimal ternary constant weight codes with weight 4 and distance 5 constant weight codes cwcs play an important role in coding theory the problem of determining the sizes for optimal ternary cwcs with length n weight 4 and minimum hamming distance 5 n 5 4 3 code has been settled for all positive integers n 10 or n 10 and n 1 mod 3 with n 13 52 58 undetermined in this paper we investigate the problem of constructing optimal n 5 4 3 codes for all lengths n with the tool of group divisible codes we determine the size of an optimal n 5 4 3 code for each integer n 4 leaving the lengths n 12 13 21 27 33 39 45 52 unsolved

majorizing kernels and stochastic cascades with applications to incompressible navier stokes equations a general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d navier stokes equations the basic idea generalizes a probabilistic approach introduced by lejan and sznitman 1997 to obtain weak solutions whose fourier transform may be represented by an expected value of a stochastic cascade a functional analytic framework is also developed which partially connects stochastic iterations and certain picard iterates some local existence and uniqueness results are also obtained by contractive mapping conditions on the picard iteration

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

quantum shadow enumerators in a previous paper shor and laflamme see phys rev lett vol 78 p 1600 02 1997 define two weight enumerators for quantum error correcting codes connected by a macwilliams 1977 transform and use them to give a linear programming bound for quantum codes we extend their work by introducing another enumerator based on the classical theory of shadow codes that tightens their bounds significantly in particular nearly all of the codes known to be optimal among additive quantum codes codes derived from orthogonal geometry can be shown to be optimal among all quantum codes we also use the shadow machinery to extend a bound on additive codes to general codes obtaining as a consequence that any code of length can correct at most n 1 6 errors

communicating over adversarial quantum channels using quantum list codes in this correspondence we study quantum communication in the presence of adversarial noise in this setting communicating with perfect fidelity requires a quantum code of bounded minimum distance for which the best known rates are given by the quantum gilbert varshamov qgv bound asking only for arbitrarily high fidelity and letting the sender and receiver use a secret key of length logarithmic in the number of qubits sent we find a dramatic improvement over the qgv rates in fact our protocols allow high fidelity transmission at noise levels for which perfect fidelity is impossible to achieve such rates we introduce fully quantum list codes which may be of independent interest

quantum weight enumerators in a recent paper shor and laflamme see phys rev lett vol 78 p 1600 2 1997 defined two weight enumerators for quantum error correcting codes connected by a macwilliams transform and used them to give a linear programming bound for quantum codes we introduce two new enumerators which while much less powerful at producing bounds are useful tools nonetheless the new enumerators are connected by a much simpler duality transform clarifying the duality between shor and laflamme s enumerators we also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance and to extend the enumerator theory to codes with block size greater than 2

a lower bound for ternary constant weight codes we derive a lower bound for ternary constant weight codes with minimum hamming distance three the bound is similar to a bound for binary constant weight codes with minimum distance four described by graham and sloane 1980 it improves upon the gilbert 1952 bound and coincides asymptotically with the johnson 1962 bound for fixed weight as the codeword length tends to infinity

on quasi orthogonal signatures for cdma systems sum capacity optimal signatures in synchronous code division multiple access cdma systems are functions of the codebook length as well as the number of active users a new signature set must be assigned every time the number of active users changes this correspondence considers signature sets that are less sensitive to changes in the number of active users equiangular signature sequences are proven to solve a certain max min signal to interference plus noise problem which results from their interference invariance unions of orthonormal bases have subsets that come close to satisfying the welch bound bounds on the maximum number of bases with minimum maximum correlation are derived and a new construction algorithm is provided connections are made between these signature design problems grassmannian line packing frame theory and algebraic geometry

linear programming bounds for entanglement assisted quantum error correcting codes by split weight enumerators linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes in this paper we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator after deriving the macwilliams identities for these enumerators we are able to prove algebraic linear programming bounds such as the singleton bound the hamming bound and the first linear programming bound our singleton bound and hamming bound are more general than the previous bounds for entanglement assisted quantum stabilizer codes in addition we show that the first linear programming bound improves the hamming bound when the relative distance is sufficiently large on the other hand we obtain additional constraints on the size of pauli subgroups for quantum codes which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length in particular we show that there is no 27 15 5 or 28 14 6 stabilizer code we also discuss the existence of some entanglement assisted quantum stabilizer codes with maximal entanglement as a result the upper and lower bounds on the minimum distance of maximal entanglement quantum stabilizer codes with length up to 20 are significantly improved

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen