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

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

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

generalized binary udaya siddiqi sequences in this correspondence we present a family of binary 2n sequences of period 2 2n 1 where n is an integer which can be seen as a generalization of nonlinear binary sequences obtained from z4 sequences and recently constructed gkw gold kasami and welch like sequences the sequences have low correlations and are useful in code division multiple access cdma communication systems and cryptography

quadriphase sequences for spread spectrum multiple access communication this paper studies constructions for quadriphase sequences that are suitable for use as signature sequences in quadriphase spread spectrum multiple access communications quadriphase sequences are closely related to biphase i e binary sequences and many of the properties of the former can be expressed in terms of properties of the latter methods of construction are presented which obtain sets of quadriphase sequences from sets of biphase sequences methods of construction of quadriphase sequences are provided based upon the properties of the multiplicative characters of the finite field gf q where q equiv 1 mod 4 these methods use codewords from low rate reed solomon codes which are mapped onto the fourth roots of unity 2q 2 sequences of period q 1 are constructed for which the maximum magnitudes of the periodic cross correlation and the periodic out of phase autocorrelation are bounded by 3 sqrt q 5

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

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

differentiability and dimension of some fractal fourier series taking into account that the frequencies of fourier expansions are integers it is not surprising that some arithmetical results play an important role in many theorems and examples in harmonic analysis this relation is even clearer from the historical point of view and hardy and littlewood can be considered at the same time as founders of a substantial part of harmonic analysis and analytic number theory for instance in ha li they considered

on the construction of binary sequence families with low correlation and large sizes in this paper we revisit a method to produce binary sequences using the most significant bit map from z4 to the binary field this method is useful for the construction of binary sequences with low correlation and large family size there may be more cases where starting with z4 could help researchers design new low correlation sequences for code division multiple access application

new classes of optimal frequency hopping sequences by interleaving techniques in this paper we construct new classes of optimal frequency hopping sequences fhss with respect to the lempel greenberger bound and the peng fan bound by interleaving techniques which are used to construct a sequence of length kn from k sequences of length n we first give two generic constructions for optimal fhs sets from some known optimal fhs sets by interleaving techniques and present some examples of new optimal fhs sets we then design new optimal fhss whose parameters include those of the known optimal constructions with length kn and frequency set size n for some positive integers k and n we also construct optimal fhs sets of length kp from power residue sequences for any odd prime p and 2 les k p in particular our constructions give several new parameters not covered in the literature which are summarized in i and ii

non vanishing elements of finite groups let g be a finite group and let irr g denote the set of irreducible complex characters of g an element x of g is non vanishing if for every in irr g we have x 0 we prove that if x is a non vanishing element of g and the order of x is coprime to 6 then x lies in the fitting subgroup of g

dispersion of discontinuous periodic waves the dynamic evolution of linearly dispersive waves on periodic domains with discontinuous initial profiles is shown to depend remarkedly upon the asymptotics of the dispersion relation at large wavenumbers asymptotically linear or sublinear dispersion relations produce slowly changing waves while those with polynomial growth exhibit dispersive quantization a k a the talbot effect being approximately quantized at rational times but a non differentiable fractal at irrational times numerical experiments suggest that such effects persist into the nonlinear regime for both integrable and non integrable systems implications for the successful modelling of wave phenomena on bounded domains and numerical challenges are discussed

the brjuno function continuously estimates the size of quadratic siegel disks if a is an irrational number yoccoz defined the brjuno function by a ni 0 a0 a1 i i i an 1 log 1 an where a0 is the fractional part of a and an 1 is the fractional part of 1 an the numbers a such that a i are called the brjuno numbers the quadratic polynomial p a z iu e2i az z2 has an indifferent fixed point at the origin if p a is linearizable we let r a be the conformal radius of the siegel disk and we set r a 0 otherwise yoccoz y proved that a i if and only if r a 0 and that the restriction of a iu a log r a to the set of brjuno numbers is bounded from below by a universal constant in bc2 we proved that it is also bounded from above by a universal constant in fact marmi moussa and yoccoz mmy conjecture that this function extends to r as a hi older function of exponent 1 2 in this article we prove that there is a continuous extension to r

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

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

non vanishing elements in finite groups abstract we show that if a is an elementary abelian normal p subgroup of a finite group g and p is a sylow p subgroup of g then no irreducible character of g vanish on any element of z p a

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

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

the quantitative behaviour of polynomial orbits on nilmanifolds a theorem of leibman asserts that a polynomial orbit g n g n z on a nilmanifold g g is always equidistributed in a union of closed sub nilmanifolds of g g in this paper we give a quantitative version of leibman s result describing the uniform distribution properties of a finite polynomial orbit g n g n n in a nilmanifold more specifically we show that there is a factorisation g eg where e n is smooth n g n z is periodic and rational and g n g n p is uniformly distributed up to a specified error d inside some subnilmanifold g g of g g for all sufficiently dense arithmetic progressions p n our bounds are uniform in n and are polynomial in the error tolerance d in a companion paper we shall use this theorem to establish the mobius and nilsequences conjecture from an earlier paper of ours

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

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

optimal biphase sequences with large linear complexity derived from sequences over z sub 4 new families of biphase sequences of size 2 sup r 1 1 r being a positive integer are derived from families of interleaved maximal length sequences over z sub 4 of period 2 2 sup r 1 these sequences have applications in code division spread spectrum multiuser communication systems the families satisfy the sidelnikov bound with equality on spl theta sub max which denotes the maximum magnitude of the periodic cross correlation and out of phase autocorrelation values one of the families satisfies the welch bound on spl theta sub max with equality the linear complexity and the period of all sequences are equal to r r 3 2 and 2 2 sup r 1 respectively with an exception of the single m sequence which has linear complexity r and period 2 sup r 1 sequence imbalance and correlation distributions are also computed

finite group elements where no irreducible character vanishes in this paper we consider elements x of a finite group g with the property that x 0 for all irreducible characters of g if g is solvable and x has odd order we show that x must lie in the fitting subgroup f g

optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings we construct new sequences over finite rings having optimal hamming correlation properties these sequences are useful in frequency hopping multiple access fhma spread spectrum communication systems our constructions can be classified into linear and nonlinear categories both giving optimal hamming correlations according to lempel greenberger 1974 bound the nonlinear sequences have large linear complexity and can be seen as a generalized version of gmw sequences over fields

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

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

non vanishing elements in finite groups abstract many results have been established about determining whether or not an element evaluates to zero on an irreducible character of a group in this note it is shown that if a group g has a normal p subgroup n and x n lies in the centre of a sylow p subgroup of g then no irreducible character of g vanishes on x

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

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

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

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

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

a new family of nonbinary sequences with three level correlation property and large linear span in this correspondence we present a new family of nonbinary sequences with three level nontrivial correlations and large linear complexity the sequences may be considered as nonlinear analogues of the well known sequences by trachtenberg and helleseth it is shown that the family is optimal with respect to the welch bound in terms of root mean square of all nontrivial correlations we also determine the correlation distribution of the new family

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

binary sequence sets with favorable correlations from difference sets and mds codes we propose new families of pseudorandom binary sequences based on hadamard difference sets and mds codes we obtain for p 4k 1 prime and t an integer with 1 spl les t spl les p 1 2 a set of p sup t binary sequences of period p sup 2 whose peak correlation is bounded by 1 2t p 1 the sequences are balanced have high linear complexity and are easily generated

wavelet analysis of fractal boundaries part 2 multifractal analysis this second part deals with the global analysis of the boundary of domains omega rd we develop methods for determining the dimensions of the sets where the local behaviors introduced in part 1 occur these methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

multifractal behavior of polynomial fourier series abstract we study the spectrum of singularities of a family of fourier series with polynomial frequencies in particular we prove that they are multifractal functions the case of degree two was treated by s jaffard in 1996 higher degrees require completely different ideas essentially because harmonic analysis techniques poisson summation are useless to study the oscillation at most of the points we introduce a new approach involving special diophantine approximations with prime power denominators and fine analytic and arithmetic aspects of the estimation of exponential sums to control the holder exponent in thin cantor like sets

pointwise analysis of riemann s nondifferentiable function we will show how to analyse the local regularity of functions with the help of the wavelet transform these results will be applied to the function of riemann where we show the existence of a dense set of points where this function is differentiable on another dense set we show the existence of local singularities of cusp type on a third set we show differentiability to the right left on the remaining set the functions will be shown to be not differentiable

on the orders of zeros of irreducible characters abstract let g be a finite group and p a prime number we say that an element g in g is a vanishing element of g if there exists an irreducible character of g such that g 0 the main result of this paper shows that if g does not have any vanishing element of p power order then g has a normal sylow p subgroup also we prove that this result is a generalization of some classical theorems in character theory of finite groups

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

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

two new families of optimal binary sequences obtained from quaternary sequences in this paper we present two optimal binary families of sequences of length 2n 1 and 2 2n 1 for odd integer n they are obtained as the images of proposed optimal quaternary sequences under the most significant bit and the gray maps the first family has 2n 1 sequences of length 2n 1 and the identical correlation distribution to that of gold sequences and gold like sequences and the second family of sequences of length 2 2n 1 has 2n sequences and the same correlation values as those of kerdock sequences

fourier series in text bmo bmo with number theoretical implications we introduce an elementary argument to bound the text bmo seminorm of fourier series with gaps giving in particular a sufficient condition for them to be in this space using finer techniques we carry out a detailed study of the series sum n 1 e 2 pi i n 2 x providing some insight into how much this text bmo fourier series differs from defining an l infty function

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

orbits of the actions of finite solvable groups abstract suppose v is a completely reducible faithful g module for a solvable group g we show g has a large orbit on v specifically there is v v such that c g v is contained in a normal subgroup of derived length 9 contained in the seventh ascending fitting subgroup of g this is in some ways best possible this is applied to generate many theorems showing that a solvable group must have characters of large degree

multifractal analysis of the brjuno function the brjuno function b is a 1 periodic nowhere locally bounded function introduced by yoccoz because it encapsulates a key information concerning analytic small divisor problems in dimension 1 we show that t p alpha regularity introduced by calderon and zygmund is the only one which is relevant in order to unfold the pointwise regularity properties of b we determine its t p alpha regularity at every point and show that it is directly related to the irrationality exponent tau x its p exponent at x is exactly 1 tau x this new example of multifractal function puts into light a new link between dynamical systems and fractal geometry finally we also determine the holder exponent of a primitive of b

improved binary codes and sequence families from z sub 4 linear codes a bound on exponential sums over galois rings is used to construct a nested chain of z sub 4 linear binary codes and binary sequences when compared with the chain of delsarte goethals 1975 codes the codes in the new chain offer a larger minimum distance for the same code size the binary sequence families constructed also make use of nechaev s 1991 construction of a cyclic version of the kerdock code for a given value of maximum correlation the binary sequences are shown to have a family size considerably larger than the best sequence families known

the brjuno functions and their regularity properties we show that various possible versions of the brjuno function based on different kinds of continued fraction developments are all equivalent and we study their regularity l p bmo and holder properties through a systematic analysis of the functional equation which they fulfill

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

complex brjuno functions 1 1 the real brjuno function let r q and let pn qn n 0 be the sequence of the convergents of its continued fraction expansion a brjuno number is an irrational number such that n 0 log qn 1 qn the importance of brjuno numbers comes from the study of one dimensional analytic small divisors problems in the case of germs of holomorphic diffeomorphisms of one complex variable with an indifferent fixed point extending a previous result of siegel s brjuno proved br1 br2 that all germs with linear part e are linearizable if is a brjuno number conversely the third author proved that this condition is also necessary yo1 similar results hold for the local conjugacy of analytic diffeomorphisms of the circle kh yo2 yo3 and for some area preserving maps ma da1 including the standard family da2 bg1 bg2 the set of brjuno numbers is invariant under the action of the modular group pgl 2 z and it can be characterized as the set where the brjuno function b r q r is finite this arithmetical function is z periodic and satisfies the remarkable functional equation

automorphic forms and differentiability properties we consider fourier series given by a type of fractional integral of automorphic forms and we study their local and global properties especially differentiability and fractal dimension of the graph of their real and imaginary parts in this way we can construct fractal objects and continuous non differentiable functions associated with elliptic curves and theta functions

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

generic construction of quaternary sequences of period 2n with low correlation from quaternary sequences of odd period n in this paper a simple but generic method is proposed for transforming any family of quaternary sequences with low correlation of any odd period n to another family of quaternary sequences of period 2n with low correlation as an application of the generic method to sequence family a a new optimal quaternary sequence family with length 2 2n 1 family size 2n 1 and maximal nontrivial correlation value 2 n 1 2 2 where n is an odd integer is obtained most notably unlike all the known optimal quaternary sequence families the new family has a unique property that the odd integers 1 3 and the even integers 0 2 are allocated alternatively in all the sequences

on vanishing class sizes in finite groups abstract let g be a finite group an element g of g is called a vanishing element if there exists an irreducible character of g such that g 0 in this case we say that the conjugacy class of g is a vanishing conjugacy class in this paper we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group

random properties of the highest level sequences of primitive sequences over z 2 sup e using the estimates of the exponential sums over galois rings we discuss the random properties of the highest level sequences spl alpha sub e 1 of primitive sequences generated by a primitive polynomial of degree n over z 2 sup e first we obtain an estimate of 0 1 distribution in one period of spl alpha sub e 1 on the other hand we give an estimate of the absolute value of the autocorrelation function c sub n h of spl alpha sub e 1 which is less than 2 sup e 1 2 sup e 1 1 spl radic 3 2 sup 2e 1 2 sup n 2 2 sup e 1 for h spl ne 0 both results show that the larger n is the more random spl alpha sub e 1 will be

on the aperiodic and odd correlations of the binary shanbhag kumar helleseth sequences a previously derived upper bound on hybrid sums over galois rings is combined with a method due to vinogradov 1954 to obtain an upper bound for the maximum aperiodic and odd correlations of the binary sequences constructed by shanbhag kumar and helleseth see proc 1995 ieee int symp on information theory 1995

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

a proof of the marmi moussa yoccoz conjecture for rotation numbers of high type marmi moussa and yoccoz conjectured that some error function upsilon related to the approximation of the size of siegel disks by some arithmetic function of the rotation number theta is a holder continuous function of theta with exponent 1 2 using the renormalization invariant class of inou and shishikura we prove this conjecture for the restriction of upsilon to a class of high type numbers

two remarks about non vanishing elements in finite groups let g be a finite group we prove that for x g we have x 0 for all irreducible characters of g iff the class sum of x in the group algebra over c is a unit from this we conclude that if g has a normal p subgroup v and a hall p subgroup then g has non vanishing elements different from 1 hence we get another proof that a finite solvable group always has non trivial non vanishing elements moreover we give an example for a finite solvable group g which has a non vanishing involution not contained in an abelian normal subgroup of g

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

wavelet analysis of fractal boundaries part 1 local exponents let be a domain of rd in part 1 of this paper we introduce new tools in order to analyse the local behavior of the boundary of classifica tions based on geometric accessibility conditions are introduced and compared they are related to analytic criteria based either on local lp regularity of the characteristic function 1 or on its wavelet coefficients part 2 deals with the global analysis of the boundary of we develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur these methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

on vanishing criteria that control finite group structure abstract many results have been established that show how arithmetic conditions on conjugacy class sizes affect group structure a conjugacy class in g is called vanishing if there exists some irreducible character of g which evaluates to zero on the conjugacy class the aim of this paper is to show that for some classical results it is enough to consider the same arithmetic conditions on the vanishing conjugacy classes of the group

a generalisation of a theorem of wielandt abstract in 1974 helmut wielandt proved that in a finite group g a subgroup a is subnormal if and only if it is subnormal in a g for all g g in this paper we prove that the subnormality of an odd order nilpotent subgroup a of g is already guaranteed by a seemingly weaker condition a is subnormal in g if for every conjugacy class c of g there exists c c for which a is subnormal in a c we also prove the following property of finite non abelian simple groups if a is a subgroup of odd prime order p in a finite almost simple group g then there exists a cyclic p subgroup of f g which does not normalise any non trivial p subgroup of g that is generated by conjugates of a

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