cyclic self dual codes it is shown that if the automorphism group of a binary self dual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4 in particular no cyclic binary self dual code can have all its weights divisible by four the number of cyclic binary self dual codes of length n is determined and the shortest nontrivial code in this class is shown to have length 14

the automorphism group of a self dual 72 36 16 code does not contain z_ 4 it has been proven in a series of works that the order of the automorphism group of a self dual binary 72 36 16 code does not exceed five up to equivalence we obtain a parametrization of all self dual binary codes of length 72 with automorphism of order four which can be extremal we use extensive computations in magma and on a supercomputer to show that the automorphism group of an extremal self dual binary code of length 72 does not have an element of order four

on the 24 12 10 quaternary code and binary codes with an automorphism having two cycles a general decomposition theorem is given for codes over finite fields which have an automorphism of a given type such codes can be decomposed as direct sums of subcodes which may be viewed as shorter length codes over extension fields if such a code is self dual sometimes the subcodes are also this decomposition is applied to prove that the self dual 24 12 10 quaternary code has no automorphism of order 3 this decomposition is also applied to count the number of equivalent 2r r and 2r 2r 1 self dual binary codes with an automorphism of prime order r

on the algebraic structure of quasi cyclic codes i finite fields a new algebraic approach to quasi cyclic codes is introduced the key idea is to regard a quasi cyclic code over a field as a linear code over an auxiliary ring by the use of the chinese remainder theorem crt or of the discrete fourier transform dft that ring can be decomposed into a direct product of fields that ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the u spl upsi u spl upsi and vandermonde constructions all binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction quinting and septing constructions are introduced other results made possible by the ring decomposition are a characterization of self dual quasi cyclic codes and a trace representation that generalizes that of cyclic codes

how many weights can a quasi cyclic code have we investigate the largest number of nonzero weights of quasi cyclic codes in particular we focus on the function gamma q n ell k q that is defined to be the largest number of nonzero weights a quasi cyclic code of index gcd ell n length n and dimension k over mathbb f q can have and connect it to similar functions related to linear and cyclic codes we provide several upper and lower bounds on this function using different techniques and studying its asymptotic behavior moreover we determine the smallest index for which a q ary reed muller code is quasi cyclic a result of independent interest

a strengthening of the assmus mattson theorem let w sub 1 d w sub 2 w sub s be the weights of the nonzero codewords in a binary linear n k d code c and let w sub 1 w sub 2 w sub 3 be the nonzero weights in the dual code c1 let t be an integer in the range 0 or d 4 then either the words of any nonzero weight w sub i form a t 1 design or else the codewords of minimal weight d form a 1 2 t t 2 design if in addition c is self dual with all weights divisible by 4 then the codewords of any given weight w sub i form either a t 1 design or a 1 2 t t 2 design the proof avoids the use of modular forms

nonbinary quantum codes we present several results on quantum codes over general alphabets that is in which the fundamental units may have more than two states in particular we consider codes derived from finite symplectic geometry assumed to have additional global symmetries from this standpoint the analogs of calderbank shor steane codes and of gf 4 linear codes turn out to be special cases of the same construction this allows us to construct families of quantum codes from certain codes over number fields in particular we get analogs of quadratic residue codes including a single error correcting code encoding one letter in five for any alphabet size we also consider the problem of fault tolerant computation through such codes generalizing ideas of gottesman see phys rev a vol 57 no 1 p127 37 1998

reverse time wave field extrapolation imaging and inversion the scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background which are expressed as a scattering potential in general there is no closed form expression for the kernel of this integral representation although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array when the receiver array completely surrounds the scatterer the kernel reduces to the imaginary part of the green s function for the background medium this integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential two such relations are given 1 for the migrated image obtained by deconvolving the extrapolated field with the incident field and 2 for the reconstructed image obtained by applying a one way wave operator to the extrapolated field and then deconvolving by the incident field the migrated image highlig

extremal self dual codes with an automorphism of order 2 a method to design binary self dual codes with an automorphism of order two without fixed points is presented extremal self dual codes with lengths 40 42 44 54 58 68 are constructed many of them have weight enumerators for which extremal codes were previously not known to exist

type ii codes over f sub 2 uf sub 2 the alphabet f sub 2 uf sub 2 is viewed here as a quotient of the gaussian integers by the ideal 2 self dual f sub 2 uf sub 2 codes with lee weights a multiple of 4 are called type ii they give even unimodular gaussian lattices by construction a while type i codes yield unimodular gaussian lattices construction b makes it possible to realize the leech lattice as a gaussian lattice there is a gray map which maps type ii codes into type ii binary codes with a fixed point free involution in their automorphism group combinatorial constructions use weighing matrices and strongly regular graphs gleason type theorems for the symmetrized weight enumerators of type ii codes are derived all self dual codes are classified for length up to 8 the shadow of the type i codes yields bounds on the highest minimum hamming and lee weights

multidimensional born velocity inversion single wideband point source the inverse scattering problem for an acoustic medium is considered within the approximate direct inversion framework as opposed to iterative methods the direct inversion approach gives an estimate of the medium velocities by operating on the observed scattered field without repeated solutions of the forward problem previous solutions to the multidimensional born velocity inversion problem require either coincident source and receiver arrays or plane wave sources in both cases an array of point sources is required to collect the proper data for inversion in this paper the solution to the single point source problem is derived it is shown that by extrapolating and imaging the observed scattered field appropriately the projections of the velocity function at all angles can be obtained the velocities are then reconstructed by the inverse radon transform method of tomography

cyclic codes and self dual codes over f sub 2 uf sub 2 we introduce linear cyclic codes over the ring f sub 2 uf sub 2 0 1 u u u 1 where u sup 2 0 and study them by analogy with the z sub 4 case we give the structure of these codes on this new alphabet self dual codes of odd length exist as in the case of z sub 4 codes unlike the z sub 4 case here free codes are not interesting some nonfree codes give rise to optimal binary linear codes and extremal self dual codes through a linear gray map

the automorphism group of a binary self dual doubly even 72 36 16 code is solvable in this correspondence we prove that the automorphism group of a putative binary self dual doubly even 72 36 16 code is solvable moreover its order is 5 7 10 14 56 or a divisor of 72

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

on the automorphism group of a doubly even 72 36 16 code we prove that a putative doubly even 72 36 16 code has no automorphism of order 3 with fixed points moreover such a code has no automorphism of an odd order greater than 7 eventually we prove that 25 and 49 do not divide the order of its automorphism group

construction of an extremal self dual code of length 62 we construct an extremal singly even self dual 62 31 12 code the existence of such a self dual code was not previously known combined with known results the above code also determines all lengths for which there are self dual codes with minimum weight 12

on the automorphism group of a binary self dual doubly even 72 36 16 code we prove that the automorphism group of a binary self dual doubly even 72 36 16 code has order 5 7 10 14 or d where d divides 18 or 24 or it is a4 c3

on repeated root cyclic codes a parity check matrix for a q ary repeated root cyclic code is derived using the hasse derivative then the minimum distance of a q ary repeated root cyclic code is expressed in terms of the minimum distance of a certain simple root cyclic code with the help of this result several binary repeated root cyclic codes of lengths up to n 62 are shown to contain the largest known number of codewords for their given length and minimum distance the relative minimum distance d sub min n of q ary repeated root cyclic codes of rate r or r is proven to tend to zero as the largest multiplicity of a root of the generator g x increases to infinity it is further shown that repeated root cycle codes cannot be asymptotically better than simple root cyclic codes

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

wave field extrapolation techniques in seismic migration a tutorial the objective of this paper is to provide a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration i e the kirchhoff summation approach the plane wave method k f method and the finite difference technique particular emphasis is given to the relationship between the different methods by formulating the problem in the space frequency domain x y omega domain a systems approach can be adopted which results in simple and concise expressions these expressions clearly show that forward extrapolation is described by a spatial convolution procedure and inverse extrapolation is described by a spatial deconvolution procedure in the situation of lateral velocity variations the de convolution procedure becomes space variant the space frequency domain is most suitable for recursive depth migration in addition frequency dependent properties such as absorption dispersion and spatial bandwidth can be handled easily it is shown that all extrapolation methods are based on two equations taylor series and wave equation in the kirchhoff summation approach all terms of the taylor series are summed to an exact analytical expression the kirchhoff integral for plane surfaces it formulates the extrapolation procedure in terms of a spatial convolution integral which must be discretized in practical applications the fourier transformed version of the kirchhoff integral is used in the plane wave method k f method this actually means that spatial de convolution in the x y omega domain is translated into multiplication in the k x k y omega domain of course this is not allowed if the extrapolation operators are space variant in explicit finite difference techniques a truncated version of the taylor series is used with some optimum adjustments of the coefficients for only one or two terms in the taylor series a spatial low pass filter must be applied to compensate for the amplitude errors at high tilt angles explicit methods are simple and most suitable for three dimensional 3 d applications in implicit finite difference schemes the wave field extrapolator is written in terms of an explicit forward extrapolator and an explicit inverse extrapolator properly designed implicit schemes do not show amplitude errors and therefore amplitude correction filters need not be applied in comparison with explicit schemes implicit schemes are more sensitive to improper boundary conditions at both ends of the data file it is shown that the forward seismic model can be elegantly described by a matrix equation using separate operators for downward and upward traveling waves using this model inverse extrapolation involves one matrix inversion procedure to compensate for the downward propagation effects and one matrix inversion procedure to compensate for the upward propagation effects

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

extremal binary self dual codes in this correspondence we investigate binary extremal self dual codes numerous extremal self dual codes and interesting self dual codes with minimum weight d 14 and 16 are constructed in particular the first extremal type i 86 43 16 code and new extremal self dual codes with weight enumerators which were not previously known to exist for lengths 40 50 52 and 54 are constructed we also determine the possible weight enumerators for extremal type i codes of lengths 66 100

type iv self dual codes over rings we study type iv self dual codes over the commutative rings of order 4 gleason type theorems of type iv codes and their shadow codes are investigated a mass formula of type iv codes over these rings are given we give a classification of type tv codes over z sub 4 and f2 uf sub 2 for reasonable lengths we also construct a number of optimal type tv codes

mass formulas for self dual codes over z sub 4 and f sub q uf sub q rings we give a mass formula for quaternary self dual and type ii codes and for self dual codes over f sub q uf sub q rings we define type ii codes and compute the number of distinct type ii codes we also give formulas for the number of doubly even binary codes necessary to compute n sub d n the number of distinct codes of length n for any dimension n and conclude that the classification of quaternary self dual codes could be reasonably extended to length 14

a survey of the near field far field inverse scattering inverse source integral equation the kirchhoff direct integration of the scalar wave equation is reviewed and some properties of the kirchhoff surface integral are discussed from the perspective of the inverse scattering inverse source problem a modified kirchhoff surface integral is introduced leading to a fredholm integral equation of the first kind for the unknown sources induced by the incident field inside a volume in terms of the scattered fields on the surface enclosing this volume the properties and physical meaning of this integral equation are discussed a generalization of this integral equation for the vector electromagnetic wave equations is presented

the asymptotic behavior of e entropy of a compact positive operator abstract the purpose of this paper is to apply the methods of real analysis to the investigation of the asymptotic behavior of e entropy of a compact positive operator for sufficiently small e 0 the lower growth order about the e entropy of a compact positive operator is characterized by the proper values associated with this operator moreover about the e entropy per unit time of convolution operators generated by a continuous positive definite and lebesgue integrable function both the upper and the lower growth orders are characterized by the spectral density associated with this function

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

compactification of the stationary channel space topologization of the set consisting of all channels is given this result is applied to the proof that the entropy rates and the transmission rate are jointly upper semi continuous on the product topological space constructed from the stationary regular borel probability measures and the stationary channels finally it is proved that there exists a combination of an input stationary measure and a stationary channel having finite memory which can make the transmission rate maximal

variable background born inversion by wavefield backpropagation the inverse scattering problem for an acoustic medium is formulated by using the variable background born approximation a constant density acoustic medium is probed by a wide band point source and the scattered field is observed along a curved receiver array located outside the region where the medium velocity is different from the assumed background velocity function the solution proposed relies on the introduction of a backpropagated field this field is obtained by using a finite difference scheme backward in time to backpropagate into the medium the scattered field observed along the receiver array the backpropagated field is imaged at the source travel times giving an image of the same type as obtained by reverse time finite difference migration techniques the gradient of this image is then taken along rays linking the source to points in the medium and after scaling this gives the reconstructed potential to relate the reconstructed potential to the true scattering potential high frequency

upper bounds for the lengths of s extremal codes over bbf _ 2 bbf _ 4 and bbf _ 2 u bbf _ 2 our purpose is to find an upper bound for the length of an s extremal code over f2 resp f4 when d equiv 2 mod 4 resp d odd this question is left open in a bound for certain s extremal lattices and codes archiv der mathematik vol 89 no 2 pp 143 151 2007 resp s extremal additive f4 codes advances in mathematics of communications vol 1 no 1 pp 111 130 2007 more precisely we show that if there is an n n 2 d s extremal type i binary self dual code with d 6 and d equiv 2 mod 4 then n 1 and d equiv 1 mod 2 then n 13d 26 we also define s extremal self dual codes over f2 uf2 and derive an upper bound for the length of an s extremal self dual code over f2 uf2 using the information on binary s extremal codes

continuity of entropy and mutual entropy in c dynamical systems the lower semicontinuities of the entropy and the mutual entropy of a state for c dynamical systems are proved with respect to the set of all kms states and the set of all invariant states

a method for constructing inequivalent self dual codes with applications to length 56 necessary and sufficient conditions are presented for two doubly even codes with an automorphism of odd prime order to be equivalent under certain assumptions for the codes as an application we obtain all extremal doubly even codes of length 56 with an automorphism of order 13 sixteen such codes are inequivalent

monomials of orders 7 and 11 cannot be in the group of a 24 12 10 self dual quaternary code corresp it is an interesting open question whether a self dual quaternary 24 12 10 code c exists it was shown by conway and pless that the only primes which can be orders of permutations in the group of c are 11 7 and 3 in this correspondence we eliminate 11 and 7 not only as permutations but also as orders of monomials in the group of c this is done by reducing the problems to the consideration of several codes and finding low weight vectors in these codes

some results on cyclic codes over f _ 2 v f _ 2 in this paper we investigate the structure and properties of cyclic codes over the ring f 2 vf 2 we first study the relationship between cyclic codes over f 2 vf 2 and binary cyclic codes then we prove that cyclic codes over the ring are principally generated and give the generator polynomial of cyclic codes over the ring finally we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given length n over f 2 vf 2

construction of optimal type iv self dual codes over f sub 2 uf sub 2 previously type iv self dual codes over the ring f sub 2 uf sub 2 of order 4 have been introduced in this paper we construct five optimal type iv self dual codes which are the first examples of such codes for that length we also give several new optimal type iv self dual codes and type iv self dual codes which improve the upper bounds for minimum weights

quaternary quadratic residue codes and unimodular lattices we construct new self dual and isodual codes over the integers module 4 the binary images of these codes under the gray map are nonlinear but formally self dual the construction involves hensel lifting of binary cyclic codes quaternary quadratic residue codes are obtained by hensel lifting of the classical binary quadratic residue codes repeated hensel lifting produces a universal code defined over the 2 adic integers we investigate the connections between this universal code and the codes defined over z sub 4 the composition of the automorphism group and the structure of idempotents over z sub 4 we also derive a square root bound on the minimum lee weight and explore the connections with the finite fourier transform certain self dual codes over z sub 4 are shown to determine even unimodular lattices including the extended quadratic residue code of length q 1 where q spl equiv 1 mod8 is a prime power when q 23 the quaternary golay code determines the leech lattice in this way this is perhaps the simplest construction for this remarkable lattice that is known

imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized radon transform this paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers the linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids this integral equation is connected with the causal generalized radon transform grt and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the grt the first term of this asymptotic expansion is interpreted as a migration algorithm as a result this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium also a partial reconstruction operator is explicitly derived for a limited aperture when specialized to a constant background velocity and specific source receiver geometries our results are directly related to some known migration algorithms

on the automorphism group of a putative code a long standing open question in coding theory is whether a binary self dual 72 36 16 code exists it was shown recently that the automorphism group order of such code is equal to 2imiddot3 jmiddot5middot7 2imiddot3jmiddot7 2imiddot3jmiddot5 or 2imiddot3 j for some nonnegative integers i and j we show that the group order is not a multiple of 35 and find that only 22 orders are possible

binary self dual codes with automorphisms of composite order in this paper we present some results concerning a decomposition of binary self dual codes having an automorphism of an order which is the product of two odd prime numbers these results are applied to construct self dual 72 36 12 codes with an automorphism of order 15 furthermore it is proved that the automorphism group of a putative binary extremal self dual 72 36 16 code contains at most 28 nontrivial types of automorphisms of odd order a complete list of these possible types of automorphisms is presented

constacyclic codes of length 2 s over galois extension rings of bbf _ 2 u bbf _2 we study all constacyclic codes of length 2s over gr rfr m the galois extension ring of dimension m of the ring rfr f2 uf2 the units of the ring gr rfr m are of the forms alpha and alpha ubeta where alpha beta are nonzero elements of f2m which correspond to 2 m 2m 1 such constacyclic codes first the structure and hamming distances of 1 ugamma constacyclic codes are established we then classify all cyclic codes of length 2s over gr rfr m and obtain a formula for the number of those cyclic codes as well as the number of codewords in each code finally one to one correspondences between cyclic and alpha constacyclic codes as well as 1 ugamma constacyclic and alpha ubeta constacyclic codes are provided via ring isomorphisms that allow us to carry over the results about cyclic and 1 ugamma constacyclic accordingly to all constacyclic codes of length 2s over gr rfr m

on ternary self dual codes of length 24 a partial classification is given of the self dual codes of length 24 over gf 3 the main results are as follows there are exactly two codes with minimum hamming distance d 9 most of the codes have d 6 and are indecomposable one code with d 6 has a trivial automorphism group this is the first such self dual code that has been found the codes generated by the 59 inequivalent 24 times 24 hadamard matrices have been investigated and there appear to be only nine inequivalent codes two with d 9 and seven with d 6 and in all there are 27 decomposable codes at least 96 indecomposable codes with d 6 and the total number of inequivalent codes is at least 140

quantum ergodic channels in operator algebras abstract the quantum ergodic channel is studied by operator algebraic methods the ergodic and kms channels are introduced and their dynamical properties are discussed

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

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

23 does not divide the order of the group of a 72 36 16 doubly even code corresp an interesting open question is whether a 72 36 16 doubly even code c exists in 3 the odd prime numbers which can divide the order of the group of c were determined and 23 is the largest of these twenty three is eliminated by reducing the problem to the consideration of 348 codes each of which is shown to have minimum weight 12 or less one of these codes denoted by c arises from the a x b x a b x construction where a and b are in one quadratic residue code and x is in the other the weight distribution of c is given

a characterization of a 56 28 extremal self dual code corresp a 54 27 code is characterized in terms of a 27 times 27 matrix made up of circulant 3 times 3 matrices the minimum distance of this code is found to be 11 and its weight distribution is computed the 54 27 d 11 code is extended to a 56 28 d 12 external self dual code a 54 27 d 10 self dual code is also described in terms of the incidence matrix of a 27 13 6 group difference set defined on gf 3 3

maximum distance codes over rings of order 4 in this correspondence we study bounds on the euclidean hamming lee and bachoc weights of codes over rings of order 4 similar to the singleton bound and investigate the relationship between these bounds moreover we give some characterizations of the codes meeting these bounds

a 72 36 16 doubly even code does not have an automorphism of order 11 corresp we prove that there does not exist a 72 36 16 doubly even code with an automorphism of order 11 if such a code exists it can be decomposed as a direct sum of two codes one can be viewed as a self dual 12 6 4 binary code and the other a self dual 6 3 4 code over f 1024 this is shown to be impossible

the extended quadratic residue code is the only 48 24 12 self dual doubly even code an extremal self dual doubly even binary n k d code has a minimum weight d 4 spl lfloor n 24 spl rfloor 4 of such codes with length divisible by 24 the golay code is the only 24 12 8 code the extended quadratic residue code is the only known 48 24 12 code and there is no known 72 36 16 code one may partition the search for a 48 24 12 self dual doubly even code into three cases a previous search assuming one of the cases found only the extended quadratic residue code we examine the remaining two cases separate searches assuming each of the remaining cases found no codes and thus the extended quadratic residue code is the only doubly even self dual 48 24 12 code

mathbb z _ 2 mathbb z _ 2 u cyclic and constacyclic codes following the very recent studies on mathbb z 2 mathbb z 4 additive codes mathbb z 2 mathbb z 2 u linear codes have been introduced by aydogdu et al in this paper we introduce and study the algebraic structure of cyclic constacyclic codes and their duals over the r module mathbb z 2 alpha r beta where r mathbb z 2 u mathbb z 2 left 0 1 u u 1 right is the ring with four elements and u 2 0 we determine the generating independent sets and the types and sizes of both such codes and their duals finally we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in mathbb z 2 alpha r beta

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

automorphisms of codes with applications to extremal doubly even codes of length 48 general results on automorphisms of self dual binary codes are given these results are applied to the study of extremal self dual doubly even binary codes of length 48 the main theorem proved is that an extremal self dual doubly even code of length 48 with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code interesting constructions of the binary extended golay code as well as a conjecture about a possible connection between an extremal self dual doubly even code of length 72 and an extremal quaternary code of length 24 arc yielded by techniques used in the proof

cyclic codes over r f p uf p u k 1 f p with length p s n in this paper all cyclic codes with length p sn n prime to p over the ring r f p uf p u k 1f p are classified it is first proved that tor j c is an ideal of s f p m w so that the structure of ideals over extension ring s u k m w gr u k m w is determined then an isomorphism between r x and a direct sum h is u k m h w can be obtained using discrete fourier transform the generator polynomial representation of the corresponding ideals over f p uf p u k 1f p is calculated via the inverse isomorphism moreover torsion codes ms polynomial and inversion formula are described

a putative doubly even 72 36 16 code does not have an automorphism of order 9 in this paper we prove that there does not exist a binary self dual doubly even code with an automorphism of order 9 to do so we apply a method for constructing binary self dual codes possessing an automorphism of order for an odd prime

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen