the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

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

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

krawtchouk polynomials and universal bounds for codes and designs in hamming spaces universal bounds for the cardinality of codes in the hamming space f sub r sup n with a given minimum distance d and or dual distance d are stated a self contained proof of optimality of these bounds in the framework of the linear programming method is given the necessary and sufficient conditions for attainability of the bounds are found the parameters of codes satisfying these conditions are presented in a table a new upper bound for the minimum distance of self dual codes and a new lower bound for the crosscorrelation of half linear codes are obtained

duadic codes a new family of binary cyclic n n 1 2 and n n 1 2 codes are introduced which include quadratic residue qr codes when n is prime these codes are defined in terms of their idempotent generators and they exist for all odd n p 1 a 1 p 2 a 2 cdots p r a r where each p i is a prime equiv pm 1 pmod 8 dual codes are identified the minimum odd weight of a duadic n n 1 2 code satisfies a square root bound when equality holds in the sharper form of this bound vectors of minimum weight hold a projective plane the unique projective plane of order 8 is held by the minimum weight vectors in two inequivalent 73 37 9 duadic codes all duadic codes of length less than 127 are identified and the minimum weights of their extensions are given one of the duadic codes of length 113 has greater minimum weight than the qr code of that length

linear programming bounds for doubly even self dual codes using a variant of the linear programming method we derive a new upper bound on the minimum distance d of doubly even self dual codes of length n asymptotically for n growing it gives d n spl les 0 166315 spl middot spl middot spl middot o 1 thus improving on the mallows odlyzko sloane bound of 1 6 to establish this we prove that in any doubly even self dual code the distance distribution is asymptotically upper bounded by the corresponding normalized binomial distribution in a certain interval

a generalized minkowski problem with dirichlet boundary condition we prove the existence of hypersurfaces with prescribed boundary whose weingarten curvature equals a given function that depends on the normal of the hypersurface

some self dual codes invariant under the hall janko group abstract we construct self dual codes of length 100 invariant under the hall janko group j 2 we also construct a 2 100 22 168 design with the automorphism group j 2 2 which is the extension of j 2 by its outer automorphism

parameters of goppa codes revisited we discuss parameters of goppa 1970 codes such as minimum distance covering radius distance distribution and generalized hamming weights by a variation on the exponential sums method and combinatorial arguments we sharpen known bounds on these parameters

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

more on the distance distribution of bch codes we derive a new estimate for the error term in the binomial approximation to the distance distribution of bch codes this is an improvement on the earlier bounds by kasami fujiwara lin 1985 vladuts skorobogatov 1991 and krasikov litsyn 1995

bounds for the weight distribution of weakly self dual codes upper bounds are given for the weight distribution of binary weakly self dual codes to get these new bounds we introduce a novel method of utilizing unitary operations on hilbert spaces this method is motivated by recent progress on quantum computing this new approach leads to much simpler proofs for such genre of bounds on the weight distributions of certain classes of codes moreover in some cases our bounds are improvements on the earlier bounds these improvements are achieved either by extending the range of the weights over which the bounds apply or by extending the class of codes subjected to these bounds

a bound for divisible codes a divisible code is a linear code whose word weights have a common divisor larger than one if the divisor is a power of the field characteristic there is a simple bound on the dimension of the code in terms of its weight range when this bound is applied to the subcode of words with weight divisible by four in a type i binary self dual code it yields an asymptotic improvement of the conway sloane bound for self dual codes

optimal self dual codes over f sub 2 spl times f sub 2 with respect to the hamming weight in this paper we study optimal self dual codes and type iv self dual codes over the ring f sub 2 spl times f sub 2 of order 4 we give improved upper bounds on minimum hamming and lee weights for such codes using the bounds we determine the highest minimum hamming and lee weights for such codes of lengths up to 30 we also construct optimal self dual codes and type iv self dual codes

shadow codes and weight enumerators the technique of using shadow codes to build larger self dual codes is extended to codes over arbitrary fields it is shown how to build the codes and how to determine the new weight enumerator as well for codes over fields equipped with a square root of 1 and not of characteristic 2 a self dual code of length n 2 can be built from a self dual code of length n for codes over a field without a square root of 1 and not of characteristic 2 a self dual code of length n 4 is built from a self dual code of length n and for codes over fields of characteristic 2 the length of the new self dual code depends on the presence of the all one vector in the subcode chosen in certain cases using the subcode of vectors orthogonal to the all one vector the new weight enumerator can be calculated directly from the original weight enumerator specific examples of the technique are illustrated for codes over f sub 3 f sub 4 and f sub 5

new extremal self dual codes of lengths 36 38 and 58 we construct new extremal self dual binary codes of lengths 36 38 and 58 we show that there are at least 14 inequivalent extremal self dual 38 18 8 codes and that there are at least 368 inequivalent extremal self dual 38 19 8 codes for length 58 we construct 11 extremal self dual 58 29 10 codes whose weight enumerators were previously unknown

on spectra of bch codes derives an estimate for the error term in the binomial approximation of spectra of bch codes this estimate asymptotically improves on the bounds by sidelnikov 1971 kasami et al 1985 and sole 1990

the dirichlet problem for monge ampere equations in non convex domains and spacelike hypersurfaces of constant gauss curvature in this paper we extend the well known results on the existence and regularity of solutions of the dirichlet problem for monge ampere equations in a strictly convex domain to an arbitrary smooth bounded domain in r as well as in a general riemannian manifold we prove for the nondegenerate case that a sufficient and necessary condition for the classical solvability is the existence of a subsolution for the totally degenerate case we show that the solution is in c11 q if the given boundary data extends to a locally strictly convex c2 function ori q as an application we prove some existence results for spacelike hypersurfaces of constant gauss kronecker curvature in minkowski space spanning a prescribed boundary

the dirichlet problem for weingarten hypersurfaces in lorentz manifolds we solve the dirichlet problem for strictly convex spacelike hypersurfaces of prescribed weingarten curvature under the main assumption that there exists an upper barrier we consider curvature functions that correspond in general to fully nonlinear elliptic partial differential equations which are not uniformly elliptic

packing radius covering radius and dual distance tietaivainen 1991 derived an upper bound on the covering radius of codes as a function of the dual distance this was generalized to the minimum distance and to q polynomial association schemes by levenshtein and fazekas both proofs use a linear programming approach in particular levenshtein and fazekas 1990 use linear programming bounds for codes and designs in this article proofs relying solely on the orthogonality relations of krawtchouk 1929 lloyd and more generally krawtchouk adjacent orthogonal polynomials are derived as a by product upper bounds on the minimum distance of formally self dual binary codes are derived

on the minimal weight of some singly even codes it is shown that the minimal distance d of a singly even self dual 24t 8 12t 4 code is at most 4t 2 if its shadow contains a weight 4 vector t is even and sub t sup 5t is odd it is proved particularly that there does not exist a 56 28 12 singly even self dual code with 4862 words of weight 12 this answers a question raised by conway and sloane see ibid vol 36 p 1319 33 1990

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

a new upper bound for the minimum of an integral lattice of determinant 1 let be an n dimensional integral lattice of determinant 1 we show that for all sufficiently large n the minimal nonzero squared length in does not exceed n 6 10 this bound is a consequence of some new conditions on the theta series of these lattices these conditions also enable us to find the greatest possible minimal squared length in all dimensions n 33 in particular we settle the no roots problem there is a determinant 1 lattice containing no vectors of squared length 1 or 2 precisely when n 23 n 25 there are also analogues of all these results for codes

an approximation to the weight distribution of binary linear codes binary primitive bch codes form a large class of powerful error correcting codes the weight distributions of primitive bch codes are unknown except for some special classes such as the single double triple error correcting codes and some very low rate primitive bch codes however asymptotic results for the weight distribution of a large subclass of primitive bch codes have been derived by sidel nikov these results provide some insight into the weight structure of primitive bch codes sidel nikov s approach is improved and applied to the weight distribution of any binary linear block code then sidel nikov s results on the weight distributions of binary primitive bch codes are improved and it is shown that the weights of a binary primitive code have approximate binomial distribution

cubic self dual binary codes we study binary self dual codes with a fixed point free automorphism of order three all binary codes of that type can be obtained by a cubic construction that generalizes turyn s we regard such cubic codes of length 3 spl lscr as codes of length spl lscr over the ring f sub 2 spl times f sub 4 classical notions of type ii codes shadow codes and weight enumerators are adapted to that ring two infinite families of cubic codes are introduced new extremal binary codes in lengths spl les 66 are constructed by a randomized algorithm necessary conditions for the existence of a cubic 72 36 16 type ii code are derived

on the weight enumerators of duadic and quadratic residue codes in this correspondence we compute the weight enumerators of various quadratic residue codes over f sub 2 and f sub 3 together with certain codes of related families like the duadic and the quadratic double circulant codes we use a parallel algorithm to find the number of codewords of a given not too high weight from which we deduce by usual classical methods for self dual and formally self dual codes over f sub 2 and f sub 3 their associated previously unknown weight enumerators we compute weight enumerators for lengths as high as 152 for binary codes and 96 for ternary codes

an improved upper bound on the minimum distance of doubly even self dual codes we derive a new upper bound on the minimum distance d of doubly even self dual codes of length n asymptotically for n growing it gives lim sub n spl rarr spl infin sup d n spl les 5 5 sup 3 4 10 0 165630 thus improving on the mallows odlyzko sloane bound of 1 6 and our recent bound of 0 166315

new binary singly even self dual codes in this paper we construct new binary singly even self dual codes with larger minimum weights than the previously known singly even self dual codes for several lengths several known construction methods are used to construct the new self dual codes

arithmetic of formal groups and applications i universal norm subgroups the theory of the norm map for the formal multiplicative group is well known as local class field theory in 1972 mazur 14 in his generalization of iwasawa theory to abelian varieties over number fields made quite clear that it is absolutely crucial for such a global theory first to understand the norm map for the formal groups attached to abelian varieties he achieved this understanding in case of formal groups of multiplicative type by using the theory of proalgebraic groups later on lubin and rosen 12 gave a completely elementary approach to mazur s results in the meantime hazewinkel 8 had settled the case of one dimensional commutative formal lie groups his method consists in an explicit and very complicated study of the properties of the power series coefficients of the logarithm of a formal group law by means of higher ramification theory in the same spirit vvedenskij 18 and konovalov 10 reproved and slightly extended hazewinkel s results let k q be a finite extension with ring of integers r and residue class field and koo k be a ramified zv extension with ring of integers r and f gal koo k let i r denote a smooth connected commutative formal r group of finite dimension d i e a commutative formal lie group if k k is the intermediate layer of degree p in koo k r its ring of integers and g gal k k its galois group then we know from the description of ff as a formal lie group that

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

classification of extremal and s extremal binary self dual codes of length 38 in this paper we classify all extremal and s extremal binary self dual codes of length 38 there are exactly 2744 extremal self dual codes two s extremal codes and 1730 s extremal codes we obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self dual codes of length 36 and from a generalization of this recursive algorithm for the shadow the classification of extremal codes permits to achieve the classification of all extremal codes with

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

a new extremal self dual code of length 64 in this correspondence we construct an extremal singly even self dual 64 32 12 code with a weight enumerator which was not previously known to exist

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

spectra of long primitive binary bch codes cannot approach the binomial distribution usually spectra weight distributions of primitive binary bch codes are supposed to approximate binomial weight distributions well for a wide range of code rates and code lengths it is shown that for any fixed code rate r 1 spectra of long n spl rarr spl infin primitive binary bch codes cannot approximate the binomial distribution at all

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

a generalization of a theorem of imai and its applications to iwasawa theory it is proved that if k is a complete discrete valuation field of mixed characteristic 0 p with residue field satisfying a mild condition then any abelian variety over k with potentially good reduction has finite k k 1 p infty rational torsion subgroup this can be used to remove certain conditions assumed in some theorems in iwasawa theory

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

quantum shadow enumerators in a previous paper shor and laflamme see phys rev lett vol 78 p 1600 02 1997 define two weight enumerators for quantum error correcting codes connected by a macwilliams 1977 transform and use them to give a linear programming bound for quantum codes we extend their work by introducing another enumerator based on the classical theory of shadow codes that tightens their bounds significantly in particular nearly all of the codes known to be optimal among additive quantum codes codes derived from orthogonal geometry can be shown to be optimal among all quantum codes we also use the shadow machinery to extend a bound on additive codes to general codes obtaining as a consequence that any code of length can correct at most n 1 6 errors

on the accuracy of the binomial approximation to the distance distribution of codes the binomial distribution is a well known approximation to the distance spectra of many classes of codes we derive a lower estimate for the deviation from the binomial approximation

estimates for the range of binomiality in codes spectra we derive new estimates for the range of binomiality in a code s spectra where the distance distribution of a code is upperbounded by the corresponding normalized binomial distribution the estimates depend on the code s dual distance

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

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

the covering radius of extremal self dual code d11 and its application d11 constructed by conway and sloane is an extremal self dual 56 28 12 code with weight enumerator 1 8190y sup 12 622314y sup 16 11699688y sup 20 we find that the covering radius of d11 is 10 this shows that the inequality r c sub 56 spl les 10 cannot be improved any more where r c sub 56 is the covering radius of any extremal doubly even 56 28 12 code c sub 56 and we use it to get ten new 58 29 10 codes

a limit law on the distance distribution of binary codes an approximation is given of the distance distribution of a binary code by the binomial distribution with an exponentially decreasing error term specifically the upper bound of the relative error term between the normalized distance distribution of a binary code and the binomial distribution has been asymptotically improved in particular the bound becomes exponentially small for large distances in families of codes with small sigma and rate 0 5 the approach used was based on an integral representation of krawtchouk polynomials examples of interest are bch codes of primitive length duals of irreducible cyclic codes and preparata codes

new extremal binary 44 22 8 codes one of the problems in coding theory is constructing self dual codes whose weight enumerators are not yet known to exist we use the concept of neighbors and construct extremal binary self dual 44 22 8 codes whose weight enumerators are not yet known to exist

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