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

two constructions of asymptotically optimal codebooks according to the welch bound abstract in a number of applications the codebook that minimizes the maximal cross correlation amplitude imax is often desirable there are two major ingredients in this paper these two contents are to propose two constructions of codebooks the codebooks generated by these constructions asymptotically meet the welch bound what we find is the codebook constructed in section 2 can also asymptotically meet the levenshtein bound the parameters of these codebooks are new

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

discrete breathers in a mass in mass chain with hertzian local resonators we report on the existence of discrete breathers in a one dimensional mass in mass chain with linear intersite coupling and nonlinear precompressed hertzian local resonators which is motivated by recent studies of the dynamics of microspheres adhered to elastic substrates after predicting theoretically the existence of discrete breathers in the continuum and anticontinuum limits of intersite coupling we use numerical continuation to compute a family of breathers interpolating between the two regimes in a finite chain where the displacement profiles of the breathers are localized around one lattice site we then analyze the frequency amplitude dependence of the breathers by performing numerical continuation on a linear eigenmode vanishing amplitude solution of the system near the upper band gap edge finally we use direct numerical integration of the equations of motion to demonstrate the formation and evolution of the identified localized modes in energy conserving and dissipative scenarios including within settings that may be relevant to future experimental studies

commutative semifields and symplectic spreads abstract commutative semifields are constructed by using their relationship with symplectic spreads the number of pairwise nonisomorphic commutative semifield planes of even order n obtained in this manner is not bounded above by any polynomial in n the number previously known for any n was less than log n

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

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

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

a unified approach to whiteman s and ding helleseth s generalized cyclotomy over residue class rings the theory of cyclotomy dates back to gauss and has a number of applications in combinatorics coding theory and cryptography cyclotomy over a residue class ring bbzv can be divided into classical cyclotomy or generalized cyclotomy depending on v prime or composite in this paper we introduce a generalized cyclotomy of order d over bbzp1e1p2e2 pnen which includes whiteman s and ding helleseth s generalized cyclotomy as special cases here p1 p2 pn are pairwise distinct odd primes satisfying d pi 1 for all 1 i n and e1 e2 en are positive integers we derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers as applications we completely solve an open problem and a conjecture on whiteman s generalized cyclotomy of order four over bbzp1p2 besides we also construct an infinite series of near optimal codebooks over bbzp1p2 as well as some infinite series of asymptotically optimal difference systems of sets over bbzp1e1p2e2 pnen

generalized discrete fourier transform with nonlinear phase constant modulus transforms like discrete fourier transform dft walsh transform and gold codes have been successfully used over several decades in several engineering applications including discrete multi tone dmt orthogonal frequency division multiplexing ofdm and code division multiple access cdma communications systems in this paper we present a generalized framework for dft called generalized dft gdft with nonlinear phase by exploiting the phase space we show that gdft offers sizable correlation improvements over dft walsh oppermann and gold codes leading to better performance in all multi carrier communications scenarios investigated we also highlight how known constant modulus orthogonal transforms are special solutions of the proposed gdft framework moreover we introduce practical design methods offering computationally efficient implementations of gdft as enhancements to dft we conclude the paper with examples of communications applications where gdft is shown to outperform dft and other known constant modulus bases

new constructions of asymptotically optimal codebooks with multiplicative characters in practical applications such as direct spread code division multiple access communications space time codes and compressed sensing and codebooks with small inner product correlation are required it is extremely difficult to construct codebooks achieving the levenshtein bound in this paper two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented these constructions produce complex codebooks asymptotically achieving the levenshtein bound and codebooks asymptotically achieving the welch bound the codebooks presented in this paper have new parameters

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

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

two classes of codebooks nearly meeting the welch bound in this paper the value of imax c for codebooks c c d constructed by certain almost difference sets d in fqx is determined and expressed in terms of jacobi sums from which it shows that such codebooks nearly meet the welch bound this result is an answer of a question raised by c ding and t feng in 2 we also present another series of codebooks which nearly meet the welch bound where the codebook is constructed by a subset r fq1 fq2 when q1 q2 de is an almost difference set of r

a family of alltop functions that are ea inequivalent to the cubic function sequences with optimal correlation properties are much sought after for applications in communication systems in 1980 alltop ieee trans inf theory 26 3 350 354 1980 described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation subsequently these sequences were used to construct mutually unbiased bases mubs a structure of importance in quantum information theory the key feature of this cubic function is that its difference function is a planar function functions with planar difference functions have been called alltop functions this paper provides a new family of alltop functions and establishes the use of alltop functions for construction of sequence sets and mubs

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

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

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

new families of codebooks achieving the levenstein bound in this paper a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced it includes some earlier constructions of codebooks meeting the levenstein bound as special cases with this construction two new families of codebooks achieving the levenstein bound are obtained the codebooks constructed in this paper could have a very small alphabet size

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

signal sets from functions with optimum nonlinearity signal sets with the best correlation property are desirable in code division multiple access cdma systems in this paper the construction of wootters and fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions then specific classes of planar functions and almost bent functions are employed to obtain q2 q q signal sets the signal sets derived from planar functions are optimal with respect to the levenstein bound and those obtained from almost bent functions nearly meet the levenstein bound the signal sets constructed in this paper could have a very small alphabet size and have applications in synchronous ds cdma systems where the number of users is greater than the signal space dimension or the spreading factor

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

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

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 construction of codebooks associated with binary sequences an n k codebook is a set of n unit norm code vectors in a k dimensional vector space for its applications it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible meeting the welch bound equality strictly or asymptotically in this paper an n k codebook is constructed from a k n partial matrix with k n where each code vector is equivalent to a column of the matrix to obtain the k n matrix k rows are selected from a j n matrix associated with a binary sequence of length j and hamming weight k where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence it is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of transform of the binary sequence thus constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its transform is as small as possible from the discovery new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from fourier and hadamard matrices associated with binary sequences

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

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

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

optimal codebooks from binary codes meeting the levenshtein bound in this paper a generic construction of codebooks based on binary codes is introduced with this generic construction a few previous constructions of optimal codebooks are extended and a new class of codebooks almost meeting the levenshtein bound is presented exponentially many codebooks meeting or almost meeting the levenshtein bound from binary codes are obtained in this paper the codebooks constructed in this paper have alphabet size 4 as a byproduct three bounds on the parameters of binary codes are derived

coherence optimization and best complex antipodal spherical codes vector sets with optimal coherence according to the welch bound cannot exist for all pairs of dimension and cardinality if such an optimal vector set exists it is an equiangular tight frame and represents the solution to a grassmannian line packing problem best complex antipodal spherical codes bcascs are the best vector sets with respect to the coherence by extending methods used to find best spherical codes in the real valued euclidean space the proposed approach aims to find bcascs and thereby a complex valued vector set with minimal coherence there are many applications demanding vector sets with low coherence examples are not limited to several techniques in wireless communication or to the field of compressed sensing within this contribution existing analytical and numerical approaches for coherence optimization of complex valued vector spaces are summarized and compared to the proposed approach the numerically obtained coherence values improve previously reported results the drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases

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

orthogonal complex valued spreading sequences with a wide range of correlation properties a new family of complex valued spreading sequences was proposed for code division multiple access cdma systems by oppermann and vucetic 1994 1997 which offer the system designer the choice of a wide range of correlation properties this letter identifies a subfamily of spreading sequences for use in cdma systems in which the sequence elements in any set are orthogonal the subfamily also offers a very wide range of values for both aperiodic auto correlation ac and aperiodic cross correlation cc functions with the added property of having orthogonal elements

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

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

a new approach to constructing quadratic pseudo planar functions over mathbb f _ 2 n planar functions over finite fields give rise to finite projective planes they were also used in the constructions of des like iterated ciphers error correcting codes and codebooks they were originally defined only in finite fields with odd characteristic but recently zhou introduced pesudo planar functions in even characteristic which yields similar applications all known pesudo planar functions are quadratic and hence they give presemifields in this paper a new approach to constructing quadratic pseudo planar functions is given then five explicit families of pseudo planar functions are constructed one of which is a binomial two of which are trinomials and the other two are quadrinomials all known pesudo planar functions are revisited some of which are generalized these functions not only lead to projective planes relative difference sets and presemifields but also give optimal codebooks meeting the levenstein bound complete sets of mutually unbiased bases and compressed sensing matrices with low coherence

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

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

near optimal partial hadamard codebook construction using binary sequences obtained from quadratic residue mapping in this paper a new class of n k near optimal partial hadamard codebooks is proposed the construction of the proposed codebooks from hadamard matrices is based on binary row selection sequences which are generated by quadratic residue mapping of p ary m sequences the proposed codebooks have parameters n p n and k p 1 2p n sqrt n 1 for an odd prime p and an even positive integer n we prove that the maximum magnitude of inner products between the code vectors of the proposed codebooks asymptotically achieves the welch bound equality for sufficiently large p and derive their inner product distribution

on meeting the peak correlation bounds in this paper we study the problem of meeting peak periodic or aperiodic correlation bounds for complex valued sets of sequences to this end the welch levenstein and exponential bounds on the peak inner product of sequence sets are considered and used to provide compound peak correlation bounds in both periodic and aperiodic cases the peak aperiodic correlation bound is further improved by using the intrinsic dimension deficiencies associated with its formulation in comparison to the compound bound the new aperiodic bound contributes an improvement of more than 35 for some specific values of the sequence length n and set cardinality m we study the tightness of the provided bounds by using both analytical and computational tools in particular novel algorithms based on alternating projections are devised to approach a given peak periodic or aperiodic correlation bound several numerical examples are presented to assess the tightness of the provided correlation bounds as well as to illustrate the effectiveness of the proposed methods for meeting these bounds

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

linear codes from perfect nonlinear mappings and their secret sharing schemes in this paper error correcting codes from perfect nonlinear mappings are constructed and then employed to construct secret sharing schemes the error correcting codes obtained in this paper are very good in general and many of them are optimal or almost optimal the secret sharing schemes obtained in this paper have two types of access structures the first type is democratic in the sense that every participant is involved in the same number of minimal access sets in the second type of access structures there are a few dictators who are in every minimal access set while each of the remaining participants is in the same number of minimal access sets

a generic construction of complex codebooks meeting the welch bound codebooks also called signal sets meeting the welch bound on the maximum correlation amplitude are called mwbe codebooks and are desirable in code division multiple access systems two different but related constructions of mwbe codebooks from difference sets were developed by xia and ding recently the objectives of this correspondence are to present a generic construction of mwbe codebooks that contains the previous two constructions as special cases and describe new mwbe codebooks that cannot be produced by the earlier two constructions

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

new constructions of codebooks nearly meeting the welch bound with equality an n k codebook c is a set of n unit norm complex vectors in bbck optimal codebooks meeting the welch bound with equality are desirable in a number of areas however it is very difficult to construct such optimal codebooks there have been a number of attempts to construct codebooks nearly meeting the welch bound with equality i e the maximal cross correlation amplitude imax c is slightly higher than the welch bound equality but asymptotically achieves it for large enough n in this paper using difference sets and the product of abelian groups we propose new constructions of codebooks nearly meeting the welch bound with equality our methods yield many codebooks with new parameters in some cases our constructions are comparable to known constructions

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

traveling waves and their tails in locally resonant granular systems in the present study we revisit the theme of wave propagation in locally resonant granular crystal systems also referred to as mass in mass systems we use three distinct approaches to identify relevant traveling waves the first consists of a direct solution of the traveling wave problem the second one consists of the solution of the fourier tranformed variant of the problem or more precisely of its convolution reformulation upon an inverse fourier transform in real space finally our third approach will restrict considerations to a finite domain utilizing the notion of fourier series for important technical reasons namely the avoidance of resonances which will be discussed in detail all three approaches can be utilized in either the displacement or the strain formulation typical resulting computations in finite domains result in the solitary waves bearing symmetric non vanishing tails at both ends of the computational domain importantly however a countably infinite set of anti resonance conditions is identified for which solutions with genuinely rapidly decaying tails arise

complex spreading sequences with a wide range of correlation properties this paper presents a new family of complex valued pseudo random sequences for use in coded multiple access communication systems the family offers a very wide range of values for both auto correlation ac and cross correlation cc functions allowing great flexibility in the selection of characteristics of sequence sets based on the measure described in this paper for the mean square aperiodic ac and cc values of a set of sequences the correlation properties of sets of these sequences are compared to well known sequence sets and it is shown that sets from the new family of sequences have superior qualities tables of parameters for various sequence sets are presented to enable the construction and comparison of sets from this new family

low degree planar polynomials over finite fields of characteristic two abstract planar functions are mappings from a finite field f q to itself with an extremal differential property such functions give rise to finite projective planes and other combinatorial objects there is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even we classify polynomials of degree at most q 1 4 that induce planar functions on f q by showing that such polynomials are precisely those in which the degree of every monomial is a power of two as a corollary we obtain a complete classification of exceptional planar polynomials namely polynomials over f q that induce planar functions on infinitely many extensions of f q the proof strategy is to study the number of f q rational points of an algebraic curve attached to a putative planar function our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions