a 2 adic approach to the analysis of cyclic codes this paper describes how 2 adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over z sub 2 a a spl ges 2 the ring of integers modulo 2 sup a it provides a 2 adic proof of a theorem of mceliece that characterizes the possible hamming weights that can appear in a binary cyclic code a generalization of this theorem is derived that applies to cyclic codes over z sub 2 a that are obtained from binary cyclic codes by a sequence of hensel lifts this generalization characterizes the number of times a residue modulo 2 sup a appears as a component of an arbitrary codeword in the cyclic code the limit of the sequence of hensel lifts is a universal code defined over the 2 adic integers this code was first introduced by calderbank and sloane 1995 and is the main subject of this paper binary cyclic codes and cyclic codes over z sub 2 a are obtained from these universal codes by reduction modulo some power of 2 a special case of particular interest is cyclic codes over z sub 4 that are obtained from binary cyclic codes by means of a single hensel lift the binary images of such codes under the gray isometry include the kerdock preparata and delsart goethals codes these are nonlinear binary codes that contain more codewords than any linear code presently known fundamental understanding of the composition of codewords in cyclic codes over z sub 4 is central to the search for more families of optimal codes this paper also constructs even unimodular lattices from the hensel lift of extended binary cyclic codes that are self dual with all hamming weights divisible by 4 the leech lattice arises in this way as do extremal lattices in dimensions 32 through 48

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

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

localizations on complex networks we study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix the probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the euclidean distance is invalid several quantities are used to describe the localization properties of the representative states such as the participation ratio the structural entropy and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks whole cell networks in the real world and the watts strogatz small world and barabasi albert scale free networks are considered the networks have nontrivial localization properties due to the nontrivial topological structures it is found that the ascending order ranked series of the occurrence probabilities at the nodes behave generally multifractally this characteristic can be used as a structural measure of complex networks

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

current flow in random resistor networks the role of percolation in weak and strong disorder we study the current flow paths between two edges in a random resistor network on a ltimes l square lattice each resistor has resistance e ax where x is a uniformly distributed random variable and a controls the broadness of the distribution we find a the scaled variable uequiv l a nu where nu is the percolation connectedness exponent fully determines the distribution of the current path length ell for all values of u for ugg 1 the behavior corresponds to the weak disorder limit and ell scales as ellsim l while for ull 1 the behavior corresponds to the strong disorder limit with ellsim l dscriptsize opt where dscriptsize opt 1 22pm0 01 is the optimal path exponent b in the weak disorder regime there is a length scale xisim a nu below which strong disorder and critical percolation characterize the current path

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

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

limiting curves for i i d records we consider the concentration of measure for n i i d two dimensional random variables under the conditioning that they form a record under mild conditions we show that all random variables tend to concentrate as n rightarrow infty around limiting curves which are the solutions of an appropriate variational problem we also show that the same phenomenon occurs without the records conditioning for the longest increasing subsequence in the sample

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

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

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

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

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

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

analysis of aeroplane boarding via spacetime geometry and random matrix theory we show that aeroplane boarding can be asymptotically modelled by two dimensional lorentzian geometry boarding time is given by the maximal proper time among curves in the model discrepancies between the model and simulation results are closely related to random matrix theory the models can be used to explain why some commonly practised airline boarding policies are ineffective and even detrimental

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

records the maximal layer and uniform distributions in monotone sets abstract consider a nondecreasing nonnegative integrable function f on 0 1 draw an independent sample x 1 y 1 x n y n of size n from the uniform distribution under f and let n n be the number of records in the sample where x i y i is a record when for all j i either x j x i or y j y i we study the dependence upon f of the constant c in the asymptotic formula e n n c n and show that whenever 0 1 f 0 n n e n n 1 in probability the results are related to the expected time analysis of algorithms for finding the collection of all records i e the maximal layer

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

optimal paths in disordered media scaling of the crossover from self similar to self affine behavior we study optimal paths in disordered energy landscapes using energy distributions of the type p mathrm log 10 e mathrm const that lead to the strong disorder limit if we truncate the distribution so that p mathrm log 10 e mathrm const only for e mathrm min l el e mathrm max and p mathrm log 10 e 0 otherwise we obtain a crossover from self similar strong disorder to self affine moderate disorder behavior at a path length mathcal l ifmmode times else texttimes fi we find that mathcal l ifmmode times else texttimes fi ensuremath propto mathrm log 10 e mathrm max e mathrm min ensuremath kappa where the exponent ensuremath kappa has the value ensuremath kappa 1 60 ifmmode pm else textpm fi 0 03 both in d 2 and d 3 we show how the crossover can be understood from the distribution of local energies on the optimal paths

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

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

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

transport and percolation theory in weighted networks we study the distribution p of the equivalent conductance for erdos renyi er and scale free sf weighted resistor networks with n nodes each link has conductance geax where x is a random number taken from a uniform distribution between 0 and 1 and the parameter a represents the strength of the disorder we provide an iterative fast algorithm to obtain p and compare it with the traditional algorithm of solving kirchhoff equations we find both analytically and numerically that p for er networks exhibits two regimes i a low conductance regime for eapc where pc 1 k is the critical percolation threshold of the network and k is the average degree of the network in this regime p is independent of n and follows the power law p where 1 k a ii a high conductance regime for eapc in which we find that p has strong n dependence and scales as pf apc n 1 3 for sf networks with degree distribution pkk kminkkmax we find numerically also two regimes similar to those found for er networks

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

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

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

scaling of optimal path lengths distribution in complex networks we study the distribution of optimal path lengths in random graphs with random weights associated with each link disorder with each link i we associate a weight i expari where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we suggest in an analogy with the average length of the optimal path that the distribution of optimal path lengths has a universal form that is controlled by the expression 1 pc a where is the optimal path length in strong disorder a and pc is the percolation threshold this relation is supported by numerical simulations for erd s renyi and scale free graphs we explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network

universal behavior of optimal paths in weighted networks with general disorder we study the statistics of the optimal path in both random and scale free networks where weights are taken from a general distribution p w we find that different types of disorder lead to the same universal behavior specifically we find that a single parameter s defined as al 1 v for d dimensional lattices and s defined as an 1 3 for random networks determines the distributions of the optimal path length including both strong and weak disorder regimes here v is the percolation connectivity exponent and a depends on the percolation threshold and p w we show that for a uniform p w poisson or gaussian the crossover from weak to strong does not occur and only weak disorder exists

optimal back to front airplane boarding the problem of finding an optimal back to front airplane boarding policy is explored using a mathematical model that is related to the 1 1 polynuclear growth model with concave boundary conditions and to causal sets in gravity we study all airplane configurations and boarding group sizes optimal boarding policies for various airplane configurations are presented detailed calculations are provided along with simulations that support the main conclusions of the theory we show that the effectiveness of back to front policies undergoes a phase transition when passing from lightly congested airplanes to heavily congested airplanes the phase transition also affects the nature of the optimal or near optimal policies under what we consider to be realistic conditions optimal back to front policies lead to a modest 8 char21 12 phantom rule 0 16em 0ex improvement in boarding time over random no policy boarding using two boarding groups having more than two groups is not effective

time needed to board an airplane a power law and the structure behind it a simple model for the boarding of an airplane is studied passengers have reserved seats but enter the airplane in arbitrary order queues are formed along the aisle as some passengers have to wait to reach the seats for which they have reservation we label a passenger by the number of his or her reserved seat in most cases the boarding process is much slower than for the optimal situation where passenger and seat orders are identical we study this dynamical system by calculating the average boarding time when all permutations of n passengers are given equal weight to first order the boarding time for a given permutation ordering of the passengers is given by the number s of sequences of monotonically increasing values in the permutation we show that the distribution of s is symmetric on 1 n which leads to an average boarding time n 1 2 we have found an exact expression for s and have shown that the full distribution of s approaches a normal distribution as n increases however there are significant corrections to the first order results due to certain correlations between passenger ordering and the substrate seat ordering this occurs for some cases in which the sequence of the seats is partially mirrored in the passenger ordering these cases with correlations have a boarding time that is lower than predicted by the first order results the large number of cases with reduced boarding times have been classified we also give some indicative results on the geometry of the correlations with sorting into geometry groups with increasing n both the number of correlation types and the number of cases belonging to each type increase rapidly using enumeration we find that as a result of these correlations the average boarding time behaves like n with 0 69 as compared with 1 0 for the first order approximation

epidemic spreading with nonlinear infectivity in weighted scale free networks abstract in this paper we investigate the epidemic spreading for the sir model in weighted scale free networks with the nonlinear infectivity and weighted transmission rate concretely we introduce the infectivity exponent and the weight exponent into the epidemic system then examine the impact of and on the epidemic spreading we show that one can adjust the values of and to rebuild a nonzero finite epidemic threshold furthermore we also find the infectivity exponent has a stronger effect not only on the epidemic threshold but also on the epidemic prevalence in addition it is also interesting to see that the absence of the epidemic threshold appears not very dejected since the prevalence grows much more slowly as the transmission rate increases

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

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

effect of disorder strength on optimal paths in complex networks we study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path opt in a disordered erd s renyi er random network and scale free sf network each link i is associated with a weight ti expsarid where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we find that for any finitea there is a crossover network size n sad at which the transition occurs for n n sad the scaling behavior of opt is in the strong disorder regime with opt n 1 3 for er networks and for sf networks with lo 4 and opt n sl 3d sl 1d for sf networks with 3 l 4 for n n sad the scaling behavior is in the weak disorder regime with opt ln n for er networks and sf networks with l 3 in order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder we propose a scaling ansatz for this measure and demonstrate its validity we proceed to derive the scaling relation between n sad and a we find thatn sa d a 3 for er networks and for sf networks with lo 4 and n sa d a sl 1d sl 3d for sf networks with 3 l 4

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

z4 linear codes obtained as projections of kerdock and delsarte goethals codes abstract the kerdock and delsarte goethals codes can be very simply constructed as binary images under a certain natural map called the gray map of linear codes over z 4 the integers modulo 4 we consider the gray images of linear codes over z 4 obtained from the kerdock and delsarte goethals codes by projection on a hyperplane for m odd certain gray images have the same weight distribution as duals of extended binary bch codes of length 2 m but are not equivalent to these codes inequivalence follows from a general theorem identifying binary linear codes that are not gray images of linear codes over z 4