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

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

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

statistical inference for stochastic dominance and for the measurement of poverty and inequality we derive the asymptotic sampling distribution of various estimators frequently used to order distributions in terms of poverty welfare and inequality this includes estimators of most of the poverty indices currently in use as well as estimators of the curves used to infer stochastic dominance of any order these curves can be used to determine whether poverty inequality or social welfare is greater in one distribution than in another for general classes of indices and for ranges of possible poverty lines we also derive the sampling distribution of the maximal poverty lines up to which we may confidently assert that poverty is greater in one distribution than in another the sampling distribution of convenient dual estimators for the measurement of poverty is also established the statistical results are established for deterministic or stochastic poverty lines as well as for paired or independent samples of incomes our results are briefly illustrated using data for four countries drawn from the luxembourg income study data bases

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

statistical inference for poverty measures with relative poverty lines relative poverty lines such as one half median income have been increasingly used in poverty studies this paper contributes to the literature by developing statistical inference for testing decomposable poverty measures with relative poverty lines the poverty lines we consider are percentages of mean income and percentages of quantiles we show that the estimates of poverty indices with relative poverty lines are asymptotically normally distributed and that the covariance structure can be consistently estimated as a consequence asymptotically distribution free statistical inference can be established in a straightforward manner

on the overlapping resonances concept of acoustic transmission through an elastic plate ii numerical examples and physical implications abstract in a companion paper 1 the fiorito madigosky and uberall approximate theory of transmittivity of an infinite isotropic elastic plate in terms of lamb mode resonances 2 is examined and extended by validating coherent addition of these overlapping resonances the present paper provides numerical and heuristic tests which yield convincing proof of the applicability of the overlapping resonances concept at various thicknesses of a water immersed steel plate including the half space and zero thickness limits the behaviour is shown to be compatible with known exact behaviour overlapping resonances explain the reduction towards zero of the transmittivity near the coincidence angles of the zero order lamb modes as these converge to the rayleigh angle contrary to earlier interpretations both zero order modes survive at the half space limit the a 0 mode is shown to contribute below fd critical becoming then highly damped and effectively non propagating the interpretation provided of the transition of the a 0 mode in this region bridges the gap between conventional lamb mode theory and thin plate theory

generalized hamming weights of trace codes linear codes over f sub p often admit a natural representation as trace codes of codes that are defined over an extension field f sub p m in the paper the authors obtain estimates for the weights of subcodes of such trace codes their main result is a far reaching generalization of the carlitz uchiyama bound for the duals of binary bch codes in particular they prove sharp bounds for the generalized hamming weights of a large class of codes including duals of bch codes classical goppa codes melas codes and arbitrary cyclic codes of length n p sup m 1 the main tool is the theory of algebraic functions over finite fields in particular the hasse weil bound for the number of places of degree one

some generalized functions for the size distribution of income many distributions have been used as descriptive models for the size distribution of income this paper considers two generalized beta distributions which include many of these models as special or limiting cases these generalized distributions have not been used as models for the distribution of income and provide a unified method of comparing many models previously considered

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

quantile estimation with a complex survey design estimation of the finite population distribution function and related statistics such as the median and interquartile range is considered large sample properties of estimators constructed from stratified cluster samples and properties of large sample confidence intervals are established the results are obtained within the context of a sequence of finite populations generated from a superpopulation

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

internal and external fields of insonified plates along lamb mode branches of zero order a detailed investigation is presented of the reflection and transmission coefficients and of the internal displacement field along the branches of the a 0 and s 0 lamb modes of a water immersed infinite steel plate insonified by a plane wave the extensive numerical results presented are based on exact elastic theory the results for the transmission and reflection coefficients exhibit some previously unknown phenomena in the low fd regions of both zero order branches for example it is demonstrated that a plate only 0 0001 shear wavelengths thick can behave opaquely over a small angular region near the s 0 branch a lamb mode is shown not to imply a fixed pattern of vibration but one which evolves along the branch this evolution is interpreted in terms of the concept that the behaviour along such a branch is influenced by other modes besides the local one displacement magnitudes are encountered within the plate which are up to the order of 100 times the displacement magnitude of the incident wave

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

wald criteria for jointly testing equality and inequality restrictions on montre comment le test de wald s applique a des ensembles de restrictions d egalite et d inegalite et on presente la distribution pour de grands echantillons du test sous l hypothese nulle

elementary 2 group character codes we describe a class of codes over gf q where q is a power of an odd prime these codes are analogs of the binary reed muller codes and share several features in common with them we determine the minimum weight and properties of these codes for a subclass of codes we find the weight distribution and prove that the minimum nonzero weight codewords give 1 designs

on binary constructions of quantum codes recently steane 1998 proposed an enlargement of the calderbank shor steane construction here we further improve on the parameters obtained from his construction present examples of new codes and analyze asymptotical non constructive bounds

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

a finite gilbert varshamov bound for pure stabilizer quantum codes a finite gilbert varshamov gv bound for pure stabilizer binary and nonbinary quantum error correcting codes is presented in analogy to the gv bound for classical codes by using several enumerative results in finite unitary geometry from this quantum gv bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes

weight formulas for ternary melas codes in this paper we derive a formula for the frequencies of the weights in ternary melas codes and we illustrate this formula by computing a table of examples

sparse graph codes for quantum error correction sparse graph codes appropriate for use in quantum error correction are presented quantum error correcting codes based on sparse graphs are of interest for three reasons first the best codes currently known for classical channels are based on sparse graphs second sparse graph codes keep the number of quantum interactions associated with the quantum error correction process small a constant number per quantum bit independent of the block length third sparse graph codes often offer great flexibility with respect to block length and rate we believe some of the codes we present are unsurpassed by previously published quantum error correcting codes

beyond stabilizer codes i nice error bases nice error bases have been introduced by knill 1996 as a generalization of the pauli basis these bases are shown to be projective representations of finite groups we classify all nice error bases of small degree and all nice error bases with abelian index groups we show that in general an index group of a nice error basis is necessarily solvable

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

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

cyclic quantum error correcting codes and quantum shift registers we transfer the concept of linear feedback shift registers to quantum circuits it is shown how to use these quantum linear shift registers for encoding and decoding cyclic quantum error correcting codes

quantum codes 6 2 3 sub p and 7 3 3 sub p p spl ges 3 exist we prove that nonbinary quantum stabilizer codes with parameters n k d sub p 6 2 3 sub p and 7 3 3 sub p exist for all odd primes p by using graph machinery given by schlingemann and werner see phys rev a vol 65 no 012308 2001 with a little number theory and combinatorics

on the overlapping resonances concept of acoustic transmission through an elastic plate i an examination of properties abstract the fiorito madigosky and uberall approximate representation 1 of the acoustic transmission coefficient of an infinite isotropic elastic plate by resonances associated with lamb modes is examined and clarification of its properties is obtained some theoretical extension is provided which validates coherent addition of these overlapping resonances minor limitations shown are that the theory is inapplicable near grazing incidence and that it gives only a partial solution at incidence beyond the second critical angle when the frequency thickness product is the variable existing overlapping resonances theory is shown not to yield a unique relationship between the incident wave and a set of physical motions within the plate and an approach to overcome this non uniqueness is suggested an equivalence is demonstrated between the 3 db resonance half width of each lamb mode and the imaginary part of the modal wavenumber per unit wavenumber of the fluid and this facilitates the numerical checking of the theory which is carried out in a companion paper 2 and yields convincing proof of the applicability of the approach

testing inequality constraints in linear econometric models abstract this paper develops three asymptotically equivalent tests for examining the validity of imposing linear inequality restrictions on the parameters of linear econometric models first we consider the model y x e where e is n 0 and the hypothesis test h r r versus k rk later we generalize this testing framework to the linear simultaneous equations model we show that the joint asymptotic distribution of these test statistics and the test statistics from the hypothesis test h r r versus k r r is a weighted sum of two sets of independent 2 distributions we also derive a useful duality relation between the multivariate inequality constraints test developed here and the multivariate one sided hypothesis test in small samples these three test statistics satisfy inequalities similar to those derived by berndt and savin 1977 for the case of equality constraints the paper also contains an illustrative application of this testing technique

inference on inequality from household survey data abstract i develop a theory of asymptotic inference for the lorenz curve and the gini coefficient for testing economic inequality when the data come from stratified and clustered household surveys with large number of clusters per stratum using the asymptotic framework of bhattacharya asymptotic inference from multi stage surveys journal of econometrics 126 1 145 171 i derive a weak convergence result for the continuously indexed lorenz process even when the underlying density is not uniformly bounded away from zero i provide analytical formulae for the asymptotic covariance functions that are corrected for both stratification and clustering and develop consistent tests for lorenz dominance inference on the gini coefficient follows as a corollary the methods are applied to per capita household expenditure data from the complexly designed indian national sample survey to test for changes in inequality before and after the reforms of the early 1990s ignoring the survey design is seen to produce qualitatively different results especially in the urban sector where the population sorts more completely into rich and poor neighborhoods

some good quantum error correcting codes from algebraic geometric codes it is shown that quantum error correction can be achieved by the use of classical binary codes or additive codes over f sub 4 in this correspondence with the help of some algebraic techniques the theory of algebraic geometric codes is used to construct an asymptotically good family of quantum error correcting codes and other classes of good quantum error correcting codes our results are compared with the known best quantum codes