on the inequivalence of generalized preparata codes if m is odd and sigma in aut gf 2 m is such that x rightarrow x sigma 2 1 is 1 1 there is a 2 m 1 1 2 m l 2m 2 nonlinear binary code p sigma having minimum distance 5 all the codes p sigma have the same distance and weight enumerators as the usual preparata codes which rise as p sigma when x sigma x 2 it is shown that p sigma and p tau are equivalent if and only if tau sigma pm 1 and aut p sigma is determined

performance of reed muller codes and a maximum likelihood decoding algorithm for ofdm the performance of reed muller encoding and a maximum likelihood decoding algorithm for orthogonal frequency division multiplexing is presented the example codes have a tightly bounded peak to mean envelope power ratio while simultaneously enabling powerful error correction we present a maximum likelihood decoder that makes use of a distance preserving map and multiple fast hadamard transforms its operation is described in detail and its performance is assessed under realistic channel conditions

construction of hadamard mathbb z _ 2 mathbb z _ 4 q _ 8 codes for each allowable value of the rank and dimension of the kernel this paper deals with hadamard mathbb z 2 mathbb z 4 q 8 codes which are binary codes after a gray map from a subgroup of direct products of mathbb z 2 mathbb z 4 and q 8 where q 8 is the quaternionic group in a previous work these codes were classified in five shapes in this paper we analyze the allowable range of values for the rank and dimension of the kernel which depends on the particular shape of the code we show that all these codes can be represented in a standard form from a set of generators which can help in understanding the characteristics of each shape the main results we present are the characterization of hadamard mathbb z 2 mathbb z 4 q 8 codes as a quotient of a semidirect product of mathbb z 2 mathbb z 4 linear codes and the construction of hadamard mathbb z 2 mathbb z 4 q 8 codes with each allowable pair of values for the rank and dimension of the kernel

on symplectic cobordisms in this note we make several observations concerning symplectic cobordisms among other things we show that every contact 3 manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3 manifolds are symplectic cobordism equivalent

adjustment by minimum discriminant information minimum discriminant information adjustment has primarily been used in the analysis of multinomial data however no such restriction is necessary let p be a distribution on ra and let be a convex set of distributions on ra let xi 1 c i c n be independent and identically distributed observations with common distribution p the minimum discriminant information adjustment mdia of p relative to is the element q of that is closest to p in the sense of kullback leibler discriminant information if pn is the empirical distribution of the xi 1 s i c n and qn is the mdia of pn relative to then qn is the maximum likelihood estimate in y let y consist of distributions a on r such that f t da t where t is a measurable transformation from r to rb and t e rb it is shown that under mild regularity conditions qn converges weakly to q the mdia of the true p with probability 1 and that en d f ddqn is an asymptotically normal and asymptotically unbiased estimate of e d f d dq

coding for the lee and manhattan metrics with weighing matrices this paper has two goals the first one is to discuss two related packing problems in the lee and manhattan metrics one is to find good codes for error correction i e packings of lee spheres and the other is to transform the space in a way that volumes are preserved and each lee sphere or scaled cross polytope will be transformed into a shape inscribed in a small cube the second goal is to consider weighing matrices for some of these coding problems weighing matrices have been used as building blocks for codes in the hamming metric in various constructions in this paper we will consider mainly two types of weighing matrices namely conference matrices and hadamard matrices to construct codes in the lee and manhattan metric we will show that these matrices have some desirable properties when considered as generator matrices for codes in these metrics

generalized hamming weights of nonlinear codes and the relation to the z sub 4 linear representation we give a new definition of generalized hamming weights of nonlinear codes and a new interpretation connected with it these generalized weights are determined by the entropy length profile of the code we show that this definition characterizes the performance of nonlinear codes on the wire tap channel of type ii the new definition is invariant under translates of the code it satisfies the property of strict monotonicity and the generalized singleton bound we check the relations between the generalized weight hierarchies of z sub 4 linear codes and their binary image under the gray map we also show that the binary image of a z sub 4 linear code is a symmetric not necessarily rectangular code moreover if this binary image is a linear code then it admits a twisted squaring construction

zrm codes quaternary zrm r m codes were defined so that their binary images via gray map are reed muller codes for some specific values of in the literature two different definitions of such codes can be found they will be denoted zrm r m and zrm r m codes in this correspondence we show that both definitions are equivalent exactly for those values of r such that their binary images are reed muller codes moreover we prove that for all r these binary images are linear codes in the case of zrm r m but they are not if we use the definition of zrm r m in this last case we compute the rank and the dimension of the kernel of these codes

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

cyclic and negacyclic codes over finite chain rings the structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring r are established when n is not divisible by the characteristic of the residue field r some cases where n is divisible by the characteristic of the residue field r are also considered namely the structure of negacyclic codes of length 2 sup t over spl zopf sub 2 sup m and that of their duals are derived

mubs inequivalence and affine planes there are fairly large families of unitarily inequivalent complete sets of n 1 mutually unbiased bases mubs known in cn for various prime powers n the number of such sets is not bounded above by any polynomial as a function of n while it is standard that there is a superficial similarity between complete sets of mubs and finite affine planes there is an intimate relationship between these large families and affine planes this note briefly summarizes old results that do not appear to be well known concerning known families of complete sets of mubs and their associated planes

decoding of linear codes over galois rings we present a method for decoding an arbitrary linear code over a galois ring spl rscr by a process of lifting decoding algorithms for a family of linear codes over a finite field spl kscr forming an spl alpha element chain where spl kscr is the quotient field of spl rscr and spl rscr has characteristic p sup spl alpha as a new result this method also works for linear codes over spl rscr which are nonfree spl rscr modules

duadic codes over z sub 2k duadic codes constitute a well known family of binary cyclic codes they are generalized in this correspondence to the setting of abelian codes over the ring z sub 2k self duality isoduality and type ii properties are studied

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

right veering diffeomorphisms of compact surfaces with boundary we initiate the study of the monoid of right veering diffeomorphisms on a compact oriented surface with nonempty boundary the monoid strictly contains the monoid of products of positive dehn twists we explain the relationship to tight contact structures and open book decompositions

on minimum lee weights of hensel lifts of some binary bch codes motivated by the paper of calderbank mcguire kumar and helleseth see ibid vol 42 no 1 p 217 26 jan 1996 we prove the following result for any given positive integer l spl ges 3 the minimum lee weights of hensel lifts to z sub 4 of binary primitive bch codes of length 2 sup m 1 and designed distance 2 sup l 1 is just 2 sup l 1 when a m can be divided by l or b m is sufficiently large for hensel lifts of binary primitive bch codes of arbitrary designed distance spl delta spl ges 4 we also prove that their minimum lee weight d sub l spl les 2 log sub 2 spl delta 1 1 when m is sufficiently large moreover a result about minimum lee weights of certain z sub 4 codes defined by galois rings which is similar to the result in calderbank et al is proved

on lee association schemes over z4 and their terwilliger algebra abstract let f 0 1 2 3 and define the set k k 0 k 1 k 2 of relations on f such that x y k i if and only if x y i mod 4 let n be a positive integer we consider the lee association scheme l n over z 4 which is the extension of length n of the initial scheme f k let t denote the terwilliger algebra of l n with respect to the zero codeword of length n we show that t is generated by a homomorphic image of the universal enveloping algebra of the lie algebra sl 3 c and the center z t furthermore we determine the irreducible modules for t using the schur weyl duality

handlebody construction of stein surfaces the topology of stein surfaces and contact 3 manifolds is studied by means of handle decompositions a simple characterization of homeomorphism types of stein surfaces is obtained they correspond to open handlebodies with all handles of index 2 an uncountable collection of exotic l4 s is shown to admit stein structures new invariants of contact 3 manifolds are produced including a complete and computable set of invariants for determining the homotopy class of a 2 plane field on a 3 manifold these invariants are applicable to seiberg witten theory several families of oriented 3 manifolds are examined namely the seifert fibered spaces and all surgeries on various links in s3 and in each case it is seen that most members of the family are the

constacyclic codes of length p s over fpm ufpm abstract for any prime p all constacyclic codes of length p s over the ring r f p m u f p m are considered the units of the ring r are of the forms and u where and are nonzero elements of f p m which provides p m p m 1 such constacyclic codes first the structure and hamming distances of all constacyclic codes of length p s over the finite field f p m are obtained they are used as a tool to establish the structure and hamming distances of all u constacyclic codes of length p s over r we then classify all cyclic codes of length p s over r and obtain the number of codewords in each of those cyclic codes finally a one to one correspondence between cyclic and constacyclic codes of length p s over r is constructed via ring isomorphism which carries over the results regarding cyclic codes corresponding to constacyclic codes of length p s over r

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

on the weight hierarchy of goethals codes over z sub 4 the rth generalized hamming weight d sub r m j of the goethals code spl gscr sub m j of length 2 sup m over z sub 4 is considered in this correspondence in the case that m spl ges 3 is an odd integer d sub r m j is exactly determined for r 0 5 1 1 5 2 2 5 and 3 0 for a composite m we give an upper bound d sub r m j using the lifting technique

negacyclic codes over z sub 4 of even length this paper generalizes the results from wolfmann see ibid vol 45 p 2527 2532 nov 1999 and vol 47 p 1773 1779 july 2001 classifying all negacyclic codes over z sub 4 of even length using a transform approach it is then shown which linear binary cyclic codes are images of negacyclic codes under the gray map in the process the concatenated structure of both negacyclic codes and binary repeated root cyclic codes is given

several classes of codes and sequences derived from a bbz_ 4 valued quadratic form let m and k be positive integers with m gcd m k being odd for a r and b l the exponential sum x litr ax 2bx2k 1 is studied systematically in this paper where i 1 r gr 4 m is a galois ring l is the teichmuller set of r and tr is the trace function from the galois ring r to z4 through the discussions on the solutions of certain equations and the newly developed theory of z4 valued quadratic forms the distribution of the exponential sum is completely determined as its applications we can determine the lee weight and hamming weight distributions of a class of codes ck over z4 and the correlation distribution of a quaternary sequence family uk respectively furthermore the hamming weight distributions of the binary codes obtained from ck under the most significant bit msb and gray maps are also determined for the msb map sequences of uk the nontrivial maximal correlation value is given and the correlation distribution is determined for the gray map sequences of uk it should be noted that the distribution of the exponential sum for the case gcd m k 1 is obtained for the first time and then the corresponding codes and sequences are novel

complementary sets generalized reed muller codes and power control for ofdm the use of error correcting codes for tight control of the peak to mean envelope power ratio pmepr in orthogonal frequency division multiplexing ofdm transmission is considered in this correspondence by generalizing a result by paterson it is shown that each q phase q is even sequence of length 2m lies in a complementary set of size 2k 1 where k is a nonnegative integer that can be easily determined from the generalized boolean function associated with the sequence for small k this result provides a reasonably tight bound for the pmepr of q phase sequences of length 2 m a new 2h ary generalization of the classical reed muller code is then used together with the result on complementary sets to derive flexible ofdm coding schemes with low pmepr these codes include the codes developed by davis and jedwab as a special case in certain situations the codes in the present correspondence are similar to paterson s code constructions and often outperform them

optimal phases for a family of quadriphase cdma sequences signature sequences with good even even and odd or polyphase correlations are crucial for asynchronous code division multiple access cdma when the data sequence is random the even and odd or polyphase correlations are equally important however for most known signature sequences only their even correlations were analyzed it appears that determining the odd or the polyphase correlations is generally a very hard problem since the odd or the polyphase correlations depend on the phases of the signature sequences sole 1989 boztas hammons and kumar 1992 found a family of quadriphase sequences that are asymptotically optimal these sequences gain a factor spl radic 2 in terms of their maximum periodic even correlations when compared with the best possible binary phase shift keying bpsk sequences we find the optimal phases of these sequences the optimality is in the sense that at these phases the mean square values of the even odd and the polyphase correlations are minimal and achieve the welch 1974 bound equality simultaneously furthermore we show that at these phases the average user interference of these sequences is always smaller than that of the ideal random signature sequences comprehensive analytical and numerical results show that good phase sequences can offer a nonnegligible amount of gain over bad phase sequences at modest and high signal to noise ratios

the predictable degree property and row reducedness for systems over a finite ring motivated by applications in communications we consider linear discrete time systems over the finite ring z p r we solve the open problem of deriving a theory of row reduced representations for these systems we introduce a less restrictive form of representation than the adapted form introduced by fagnani and zampieri we call this form the composed form we define the concept of p predictable degree property and p row reduced we demonstrate that these concepts coupled with the composed form provide a natural setting that extends several classical results from the field case to the ring case in particular the classical rank test on the leading row coefficient matrix is generalized we show that any annihilator of a behavior mathfrak b of a pre specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p row reduced composed form the underlying theory is the theory of reduced p basis for submodules of z q p r xi that is developed in this paper we show how to construct a p row reduced kernel representation in composed form by constructing a reduced p basis for the module mathfrak b perp

linear constructions for dna codes in this paper we translate in terms of coding theory constraints that are used in designing dna codes for use in dna computing or as bar codes in chemical libraries we propose new constructions for dna codes satisfying either a reverse complement constraint a gc content constraint or both that are derived from additive and linear codes over four letter alphabets we focus in particular on codes over gf 4 and we construct new dna codes that are in many cases better sometimes far better than previously known codes we provide updated tables up to length 20 that include these codes as well as new codes constructed using a combination of lexicographic techniques and stochastic search

the preparata and goethals codes trellis complexity and twisted squaring constructions the trellis complexity of the preparata and goethals codes is examined it is shown that at least for a given set of permutations these codes are rectangular upper bounds on the state complexity profiles of the preparata and goethals codes are given the upper bounds on the state complexity of the preparata and goethals codes are determined by the dimension length profiles dlp of the extended primitive double and triple error correcting bch codes respectively a twisted squaring construction for the preparata and goethals codes is given based on the double and triple error correcting extended primitive bch codes respectively

binary sequences with gold like correlation but larger linear span a new construction of optimal binary sequences identical to the well known family of gold sequences in terms of maximum nontrivial correlation magnitude and family size but having larger linear span is presented the distribution of correlation values is determined for every odd integer spl tau spl ges 3 the construction provides a family that contains 2 sup spl tau 1 cyclically distinct sequences each of period 2 sup spl tau 1 the maximum nontrivial correlation magnitude equals 2 spl tau 1 sup 2 1 with one exception each of the sequences in the family has linear span at least spl tau sup 2 spl tau 2 compared to 2 spl tau for gold sequences the sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity

p adic valuation of weights in abelian codes over spl zopf p sup d counting polynomial techniques introduced by wilson are used to provide analogs of a theorem of mceliece mceliece s original theorem relates the greatest power of p dividing the hamming weights of words in cyclic codes over gf p to the length of the smallest unity product sequence of nonzeroes of the code calderbank li and poonen presented analogs for cyclic codes over zopf 2d using various weight functions hamming lee and euclidean weight as well as count of occurrences of a particular symbol some of these results were strengthened by wilson who also considered the alphabet zopf pd for p an arbitrary prime these previous results new strengthened versions and generalizations are proved here in a unified and comprehensive fashion for the larger class of abelian codes over zopf pd with p any prime for abelian codes over zopf 4 combinatorial methods for use with counting polynomials are developed these show that the analogs of mceliece s theorem obtained by wilson for hamming weight lee weight and symbol counts and the analog obtained here for euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose fourier transforms have a specified support

quaternionic line sets and quaternionic kerdock codes abstract when n is even orthogonal spreads in an orthogonal vector space of type o 2n 2 2 are used to construct line sets of size 2n 1 1 2n 2 in h2n 2 all of whose angles are 90 or cos 1 2 n 2 2 these line sets are then used to obtain quaternionic kerdock codes these constructions are based on ideas used by calderbank cameron kantor and seidel in real and complex spaces

on the closedness of the sum of ranges of operators ak with almost compact products ai aj abstract let h 1 h n h be complex hilbert spaces and a k h k h be a bounded linear operator with closed range ran a k k 1 n it is known that if a i a j is compact for all i j then k 1 n ran a k is closed we show that if all the products a i a j i j are almost compact in a certain sense then the subspaces ran a 1 ran a n are essentially linearly independent and their sum is closed

decoding of cyclic codes over f sub 2 uf sub 2 we give a simple decoding algorithm to decode linear cyclic codes of odd length over the ring r f sub 2 uf sub 2 0 1 u u u 1 where u sup 2 0 a spectral representation of the cyclic codes over r is given and a bch like bound is given for the lee distance of the codes the ring r shares many properties of z sub 4 and f sub 4 and admits a linear gray map

canonical analysis relative to a closed subspace first a maximum coefficient notion between two real closed hilbertian subspaces relative to a third one is given many examples illustrate the study when the third subspace is trivial or not then a relative canonical analysis between two closed subspaces relative to a third one is introduced it leads to relative or partial association measures and allows a notion of relative information between subspaces

a universal ribbon surface in b 4 we construct an orientable ribbon surface f subset b 4 which is universal in the following sense any orientable 4 manifold m cong b 4 cup text 1 handles cup text 2 handles can be represented as a cover of b 4 branched over f

compact stein surfaces with boundary as branched covers of b 4 we prove that stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of b 4 whose branch set is a positive braided surface as a consequence we have that a smooth oriented 3 manifold is stein fillable iff it has a positive open book decomposition

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

macwilliams type identities on the lee and euclidean weights for linear codes over zl abstract motivated by the works of shiromoto 3 and shi et al 4 we study the existence of the macwilliams type identities with respect to the lee and euclidean weight enumerators for linear codes over the ring z l necessary and sufficient conditions for the existence of the macwilliams type identities with respect to the lee and euclidean weight enumerators for linear codes over z l are given some examples about such macwilliams type identities are also presented

on the recovery of joint distributions from limited information the paper considers the role of entropy and other information theoretic concepts in the description and formation of joint distributions for m random variables the discussion includes a review of methods for constructing discrete and continuous joint distributions from the component marginal distributions we then propose a minimum cross entropy approach that recovers continuous joint distributions from the joint and marginal moments and the marginal densities the large sample properties of the associated estimator are outlined and a simple demonstration problem that highlights the advantages and drawbacks of the proposed method is presented

the nordstrom robinson code representation over gf 4 and efficient decoding we show that the nordstrom robinson 1968 code may be represented as the union of binary images of two isomorphic linear 4 2 3 codes over gf 4 certain symmetries of the unique quaternary 4 2 3 quadracode are discussed it is shown how the properties of the nordstrom robinson code itself such as weight distribution distance invariance and the well known representation as the union of eight cosets of the 16 5 8 reed muller code may be easily rederived from these symmetries in addition we introduce a decoding algorithm for the nordstrom robinson code which involves projecting its codewords onto the codewords of the quadracode the algorithm is simple enough to enable hard decision decoding of the nordstrom robinson code by hand furthermore we present an algorithm for maximum likelihood soft decision decoding of the nordstrom robinson code based on its representation over gf 4 the complexity of the proposed algorithm is at most 205 real operations this is to the best of our knowledge less than the complexity of any existing decoder and twice as efficient as the fast maximum likelihood decoder of adoul 1987 we also investigate several related topics such as bounded distance decoding sphere packings and the 20 2048 6 code

binary images of cyclic codes over z sub 4 we determine all linear cyclic codes over z sub 4 of odd length whose gray images are linear codes or equivalently whose nechaev gray 1989 image are linear cyclic codes or are linear cyclic codes

4 phase sequences with near optimum correlation properties two families of four phase sequences are constructed using irreducible polynomials over z sub 4 family a has period l 2 sup r 1 size l 2 and maximum nontrivial correlation magnitude c sub max

on the hamming distance of linear codes over a finite chain ring let r be a finite chain ring e g a galois ring k its residue field and c a linear code over r we prove that d c the hamming distance of c is d c spl alpha where c spl alpha is a submodule quotient spl alpha is a certain element of r and denotes the canonical projection to k these two codes also have the same set of minimal codeword supports we explicitly construct a generator matrix polynomial of c spl alpha from the generator matrix polynomials of c we show that in general d c spl les d c with equality for free codes i e for free r submodules of r sup n and in particular for hensel lifts of cyclic codes over k most of the codes over rings described in the literature fall into this class we characterize minimum distance separable mds codes over r and prove several analogs of properties of mds codes over finite fields we compute the hamming weight enumerator of a free mds code over r

orbit and coset analysis of the golay and related codes let b be a code of length n over a field f with automorphism group g b sub w denotes the subset of codewords of weight w the goal is to classify the vectors of f sup n into orbits under g and to determine their distances from the various subcodes b sub w this is done for the first order reed muller nordstrom robinson and hamming codes of length 16 the golay and shortened golay codes of lengths 22 23 24 and the ternary golay code of length 12

positive factorizations of pseudoperiodic homeomorphisms we generalize a classical result concerning smooth germs of surfaces by proving that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs admit a positive factorization as a consequence of this and a topological characterization of these monodromies by anne pichon we conclude that a pseudoperiodic homeomorphism on an oriented surface with boundary with positive fractional dehn twist coefficients and screw numbers admits a positive factorization we use the main theorem to give a sufficiency criterion for certain pseudoperiodic homeomorphisms with negative screw numbers to admit a positive factorization

fft based bp decoding of general ldpc codes over abelian groups we introduce a wide class of low density parity check ldpc codes large enough to include ldpc codes over finite fields rings or groups as well as some nonlinear codes a belief propagation decoding procedure with the same complexity as for the decoding of ldpc codes over finite fields is also presented moreover an encoding procedure is developed

on the intersection of bbz _ 2 bbz _ 4 additive perfect codes the intersection problem for z2z4 additive extended and nonextended perfect codes i e which are the possibilities for the number of codewords in the intersection of two z2z4 additive codes c1 and c2 of the same length is investigated lower and upper bounds for the intersection number are computed and for any value between these bounds codes which have this given intersection value are constructed for all these z2z4 additive codes c1 and c2 the abelian group structure of the intersection codes c1 cap c2 is characterized the parameters of this abelian group structure corresponding to the intersection codes are computed and lower and upper bounds for these parameters are established finally for all possible parameters between these bounds constructions of codes with these parameters for their intersections are given

a note on a basic exact sequence for the lee and euclidean weights of linear codes over zl abstract this paper is devoted to presenting two counterexamples to shiromoto s 1999 results 2 on macwilliams type identities with respect to lee weight enumerators and euclidean weight enumerators over z l

on the correlation distributions of the optimal quaternary sequence family cal u and the optimal binary sequence family cal v recently new optimal families s and u of quaternary sequences have been presented and the optimal binary sequence family v obtained from family s under gray map has been investigated as well the two sequence families u and v are optimal with respect to the well known sidelnikov bound and welch bound but their exact correlation distributions are not known until now in this paper their exact correlation distributions are completely determined in some cases by making use of exponential sums and the theory of z4 valued quadratic forms

kaplan meier estimate on the plane estimation of the bivariate survival function from censored data is considered the product integral representation of univariate survival functions is generalized to the bivariate case and used to determine identifiability of the survival function of the partially observed data a bivariate analogue of the kaplan meier estimate is introduced and its almost sure consistency is studied extensions to the general multivariate case are sketched

the leech lattice the octacode and decoding algorithms new multilevel constructions of the golay code and the leech lattice are presented these are derived from the turyn construction and the holy construction with the octacode as the glue code further we show that the holy construction of the leech lattice with the octacode as the glue code is essentially different from the permuted turyn construction although both constructions rely on the octacode the turyn construction is based on an odd type of the octacode whereas any type of the octacode can be used in the holy construction moreover the multilevel representation of the holy construction leads to a novel lattice partition chain based on these structures we derive new bounded distance decoders for the golay code and the leech lattice whose effective error coefficient is smaller than that of any previously known bounded distance decoder we provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode

quaternary constant amplitude codes for multicode cdma a constant amplitude code is a code that reduces the peak to average power ratio papr in multicode code division multiple access mc cdma systems to the favorable value 1 in this paper quaternary constant amplitude codes codes over z 4 of length 2 m with error correction capabilities are studied these codes exist for every positive integer m while binary constant amplitude codes cannot exist if m is odd every word of such a code corresponds to a function from the binary m tuples to z4 having the bent property i e its fourier transform has magnitudes 2 m 2 several constructions of such functions are presented which are exploited in connection with algebraic codes over z4 in particular quaternary reed muller kerdock and delsarte goethals codes to construct families of quaternary constant amplitude codes mappings from binary to quaternary constant amplitude codes are presented as well

families of hadamard bbz_ 2 bbz_ 4 q_ 8 codes a z2z4q8 code is the binary image after a gray map of a subgroup of z2k1 z4k2 q8k3 where q8 is the quaternion group on eight elements such z2z4q8 codes are translation invariant propelinear codes as are the well known z4 linear or z2z4 linear codes in this paper we show that there exist pure z2z4q8 codes that is codes that do not admit any abelian translation invariant propelinear structure we study the dimension of the kernel and rank of the z2z4q8 codes and we give upper and lower bounds for these parameters we give tools to construct a new class of hadamard codes formed by several families of z2z4q8 codes we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are hadamard

minimal vectors in linear codes minimal vectors in linear codes arise in numerous applications particularly in constructing decoding algorithms and studying linear secret sharing schemes however properties and structure of minimal vectors have been largely unknown we prove basic properties of minimal vectors in general linear codes then we characterize minimal vectors of a given weight and compute their number in several classes of codes including the hamming codes and second order reed muller codes further we extend the concept of minimal vectors to codes over rings and compute them for several examples turning to applications we introduce a general gradient like decoding algorithm of which minimal vectors decoding is an example the complexity of minimal vectors decoding for long codes is determined by the size of the set of minimal vectors therefore we compute this size for long randomly chosen codes another example of algorithms in this class is given by zero neighbors decoding we discuss relations between the two decoding methods in particular we show that for even codes the set of zero neighbors is strictly optimal in this class of algorithms this also implies that general asymptotic improvements of the zero neighbors algorithm in the frame of gradient like approach are impossible we also discuss a link to secret sharing schemes

algebraic construction of cyclic codes over z sub 8 with a good euclidean minimum distance let s 8 denote the set of the eight admissible signals of an 8psk communication system the alphabet s 8 is endowed with the structure of z sub 8 the set of integers taken modulo 8 and codes are defined to be z sub 8 submodules of z sub 8 sup n three cyclic codes over z sub 8 are then constructed their length is equal to 6 8 and 7 and they respectively contain 64 64 and 512 codewords the square of their euclidean minimum distance is equal to 8 16 4 spl radic 2 and 10 2 spl radic 2 respectively the size of the codes of length 6 and 7 can be doubled while the euclidean minimum distance remains the same

on the weight hierarchy of kerdock codes over z sub 4 the rth generalized hamming weight d sub r of the kerdock code of length 2 sup m over z sub 4 is considered a lower bound on d sub r is derived for any r and d sub r is exactly determined for r 0 5 1 1 5 2 2 5 in the case of length 2 sup 2m d sub r is determined for any r where 0 spl les r spl les m and 2r is an integer in addition it is shown that it is sometimes possible to determine the generalized hamming weights of the kerdock codes of larger length using the results of d sub r for a given length the authors also provide a closed form expression for the lee weight of a kerdock codeword in terms of the coefficients in its trace expansion

a mass formula and rank of spl zopf sub 4 cyclic codes of length 2 sup e in this correspondence we study cyclic codes of length n 2 sup e over the ring r sub 4 spl zopf sub 4 x x sup n 1 in particular we derive a closed form expression for the number of these codes for a given length n we also study the rank of these codes and derive an expression for that furthermore we give an example in which we study all cyclic codes of length 8 we also study all self dual codes of length 8 and 16 and classify them according to their type

strongly fillable contact manifolds and j holomorphic foliations we prove that every strong symplectic filling of a planar contact manifold admits a symplectic lefschetz fibration over the disk and every strong filling of t 3 similarly admits a lefschetz fibration over the annulus it follows that strongly fillable planar contact structures are also stein fillable and all strong fillings of t 3 are equivalent up to symplectic deformation and blowup these constructions result from a compactness theorem for punctured j holomorphic curves that foliate a convex symplectic manifold we use it also to show that the compact v supported symplectomorphism group on t t 2 is contractible and to define an obstruction to strong fillability that yields a non gauge theoretic proof of gay s recent nonfillability result g for contact manifolds with positive giroux torsion

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

a linear construction for certain kerdock and preparata codes the nordstrom robinson kerdock and slightly modified preparata codes are shown to be linear over z4 the integers mod 4 the kerdock and preparata codes are duals over z4 and the nordstrom robinson code is self dual all these codes are just extended cyclic codes over z4 this provides a simple definition for these codes and explains why their hamming weight distributions are dual to each other firstand second order reed muller codes are also linear codes over z4 but hamming codes in general are not nor is the golay code

fast ml decoding algorithm for the nordstrom robinson code corresp the extended nordstrom robinson code is an optimum nonlinear double error correcting code of length 16 with considerable practical importance this code is also useful as a rate 1 2 vector quantizer for random waveforms such as speech linear predictive coding lpc residual a fast decoding algorithm is described for maximum likelihood decoding or nearest neighbor search in the squared error sense with 304 additions and 128 comparisons

an exponential number of generalized kerdock codes if n 1 is an odd composite integer then there are at least 2 1 2 n pairwise inequivalent binary error correcting codes of length 2 n size 2 2 n and minimum distance 2 n 1 2 1 2 n 1

symplectic semifield planes and ℤ linear codes there are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand and z 4 linear kerdock and preparata codes on the other these interrelationships lead to the construction of large numbers of objects of each type in the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields or equivalently large numbers of non isotopic semifields their numbers are not bounded above by any polynomial in the order of the plane in the coding theory context we construct and study large numbers of z 4 linear kerdock and preparata codes all of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal witt index over finite fields of characteristic 2 we also obtain large numbers of boring affine planes in the sense that the full collineation group fixes the line at infinity pointwise as well as large numbers of kerdock codes boring in the sense that each has as small an automorphism group as possible the connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes and also to determine their full automorphism groups

efficient estimation of linear functionals of a probability measure p with known marginal distributions suppose that p is the distribution of a pair of random variables x y on a product space x x v with known marginal distributions px and py we study efficient estimation of functions 0 h fh dp for fixed h x x y r under iid sampling of x y pairs from p and a regularity condition on p our proposed estimator is based on partitions of both x and v and the modified minimum chi square estimates of deming and stephan 1940 the asymptotic behavior of our estimator is governed by the projection on a certain sum subspace of l2 p or equivalently by a pair of equations which we call the ace equations

list decoding of generalized reed solomon codes over commutative rings we show that the list decoding procedure of guruswami and sudan may be used to decode generalized reed solomon rs codes defined over commutative rings with identity an algorithm for performing the first of the two phases of this decoding procedure is also given in particular we show that for generalized rs codes over chain rings the probability of picking an incorrect codeword from a given list can be reduced by increasing the size of the code alphabet while keeping the length and minimum hamming distance of the code constant

ranks and kernels of codes from generalized hadamard matrices the ranks and kernels of generalized hadamard matrices are studied it is proved that any generalized hadamard matrix h q lambda over mathbb f q q 3 or q 3 and gcd 3 lambda not 1 generates a self orthogonal code this result puts a natural upper bound on the rank of the generalized hadamard matrices lower and upper bounds are given for the dimension of the kernel of the corresponding generalized hadamard codes for specific ranks and dimensions of the kernel within these bounds generalized hadamard codes are constructed

a lefschetz fibration on minimal symplectic fillings of a quotient surface singularity in this article we construct a genus 0 or genus 1 positive allowable lefschetz fibration on any minimal symplectic filling of the link of non cyclic quotient surface singularities as a byproduct we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a sequence of rational blowdowns from its minimal resolution

a note on one weight and two weight projective mathbb z _ 4 codes in this paper we solve the open problems raised in 8 and present some examples to illustrate the obtained results moreover we work out the diophantine problem by shi and wang and then give the sufficient conditions for the nonexistence of two lee weight projective codes over mathbb z 4 with type 4 k 1 2 k 2