on bounded weight codes the maximum size of a binary code is studied as a function of its length n minimum distance d and minimum codeword weight ssi w this function b n d w is first characterized in terms of its exponential growth rate in the limit n for fixed d n and w n the exponential growth rate of b n d w is shown to be equal to the exponential growth rate of a n d for 0 1 2 and equal to the exponential growth rate of a n d w for 1 2 1 second analytic and numerical upper bounds on b n d w are derived using the semidefinite programming sdp method these bounds yield a nonasymptotic improvement of the second johnson bound and are tight for certain values of the parameters

association schemes and coding theory this paper contains a survey of association scheme theory with its algebraic and analytical aspects and of its applications to coding theory in a wide sense it is mainly concerned with a class of subjects that involve the central notion of the distance distribution of a code special emphasis is put on the linear programming method inspired by the macwilliams transform this produces upper bounds for the size of a code with a given minimum distance and lower bounds for the size of a design with a given strength the most specific results are obtained in the case where the underlying association scheme satisfies certain well defined polynomial properties this leads one into the realm of orthogonal polynomial theory in particular some universal bounds are derived for codes and designs in polynomial type association schemes throughout the paper the main concepts methods and results are illustrated by two examples that are of major significance in classical coding theory namely the hamming scheme and the johnson scheme other topics that receive special attention are spherical codes and designs and additive codes in translation schemes including z sub 4 additive binary codes

new code upper bounds from the terwilliger algebra and semidefinite programming we give a new upper bound on the maximum size a n d of a binary code of word length n and minimum distance at least d it is based on block diagonalizing the terwilliger algebra of the hamming cube the bound strengthens the delsarte bound and can be calculated with semidefinite programming in time bounded by a polynomial in n we show that it improves a number of known upper bounds for concrete values of n and d from this we also derive a new upper bound on the maximum size a n d w of a binary code of word length n minimum distance at least d and constant weight w again strengthening the delsarte bound and yielding several improved upper bounds for concrete values of n d and w

a characterization of 1 perfect additive codes the characterization of perfect single error correcting codes or 1 perfect codes has been an open question for a long time recently rifa has proved that a binary 1 perfect code can be viewed as a distance compatible structure in f sup n and a homomorphism spl theta f sup n spl rarr spl omega where spl omega is a loop a quasi group with identity element in this correspondence we consider 1 perfect codes that are subgroups of f sup n with a distance compatible abelian structure we compute the set of admissible parameters and give a construction for each case we also prove that two such codes are different if they have different parameters the resulting codes are always systematic and we prove their unicity therefore we give a full characterization easy coding and decoding algorithms are also presented

classification of binary constant weight codes a binary code c f2n with minimum distance at least d and codewords of hamming weight w is called an n d w constant weight code the maximum size of an n d w constant weight code is denoted by a n d w and codes of this size are said to be optimal in a computer aided approach optimal n d w constant weight codes are here classified up to equivalence for d 4 n 12 d 6 n 14 d 8 n 17 d 10 n 20 with one exception d 12 n 23 d 14 n 26 d 16 n 28 and d 18 n 28 moreover several new upper bounds on a n d w are obtained leading among other things to the exact values a 12 4 5 80 a 15 6 7 69 a 18 8 7 33 a 19 8 7 52 a 19 8 8 78 and a 20 8 8 130 since a 15 6 6 70 this gives the first known example of parameters for which a n d w 1 a n d w with w n 2 a scheme based on double counting is developed for validating the classification results

some nonregular designs from the nordstrom robinson code and their statistical properties the nordstrom robinson code is a well known nonlinear code in coding theory this paper explores the statistical properties of this nonlinear code many nonregular designs with 32 64 128 and 256 runs and 7 16 factors are derived from it it is shown that these nonregular designs are better than regular designs of the same size in terms of resolution aberration and projectivity furthermore many of these nonregular designs are shown to have generalised minimum aberration among all possible designs seven orthogonal arrays are shown to have unique word length pattern and four of them are shown to be unique up to isomorphism

the perfect binary one error correcting codes of length 15 part ii properties a complete classification of the perfect binary one error correcting codes of length 15 as well as their extensions of length 16 was recently carried out in p r j o stergard and o pottonen the perfect binary one error correcting codes of length 15 part i classification ieee trans inf theory vol 55 pp 4657 4660 2009 in the current accompanying work the classified codes are studied in great detail and their main properties are tabulated the results include the fact that 33 of the 80 steiner triple systems of order 15 occur in such codes further understanding is gained on full rank codes via switching as it turns out that all but two full rank codes can be obtained through a series of such transformations from the hamming code other topics studied include non systematic codes embedded one error correcting codes and defining sets of codes a classification of certain mixed perfect codes is also obtained

intersection of hadamard codes for two binary codes c1 c2 define i c1 c2 c1capc2 to be their intersection number this correspondence establishes that there exist hadamard codes of length 2t for all tges3 with intersection number i if and only if iisin 0 2 4 2t 1 12 2t 1 8 2t 1 also it is proved that for all tges4 if there exists a hadamard matrix of order 4s then there exist hadamard codes of length 2t 2 s with intersection number i if and only if iisin 0 2 4 2 t 3s 12 2t 3s 8 2t 3s

nonparametric methods for survival sacrifice experiments in many carcinogenicity studies the time to disease occurrence is not clinically observable a survival sacrifice experiment is considered for nonparametric inference about the rate of disease occurrence a multistate model for disease development and death is considered and an algorithm of the em type for maximum likelihood estimation is obtained questions of identifiability and estimability are addressed under the model interval hazards for disease occurrence are identifiable for intervals defined by the sacrifice times a score test is developed appropriate for the comparison of two groups with respect to disease development without need of any assumption concerning lethality of the disease concerned

on the additive spl zopf sub 4 linear and non spl zopf sub 4 linear hadamard codes rank and kernel all the possible nonisomorphic additive spl zopf sub 4 linear and non spl zopf sub 4 linear hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one

involutions in additive 1 perfect codes given a 1 perfect code c the group of symmetries of c sym c sn c c is a subgroup of the group of automorphisms of c in this paper we focus on symmetries of order 2 i e involutions let inv c sym c be the set of involutions of c an additive code c of type is a subgroup of bbz2 bbz4 for additive 1 perfect codes c of length 2t 1 we establish that any involution inv c must have at least 2k 1 fixed points where k t 2

perfect binary codes constructions properties and enumeration properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties an upper bound on the cardinality of the intersection of two perfect codes of length n is presented and perfect codes whose intersection attains the upper bound are constructed for all n as an immediate consequence of the proof of the upper bound the authors obtain a simple closed form expression for the weight distribution of a perfect code furthermore they prove that the characters of a perfect code satisfy certain constraints and provide a sufficient condition for a binary code to be perfect the latter result is employed to derive a generalization of the construction of phelps 1983 which is shown to give rise to some perfect codes that are nonequivalent to the perfect codes obtained from the known constructions moreover for any m spl ges 4 the authors construct full rank perfect binary codes of length 2 sup m 1 these codes are obviously nonequivalent to any of the previously known perfect codes furthermore the latter construction exhibits the existence of full rank perfect tilings finally they construct a set of 2 2 sup cn nonequivalent perfect codes of length n for sufficiently large n and a constant c 0 5 spl epsiv precise enumeration of the number of codes in this set provides a slight improvement over the results reported by phelps

improving the em algorithm summary a modification of the em algorithm is proposed for situations in which the maximization of the complete data likelihood function does not have a closed form solution self consistency of the modified algorithm is established application to carcinogenicity experiments is illustrated and the results of a simulation study comparing the original and modified versions indicate that use of the proposed modification can lead to significant computational savings

modular forms which behave like theta series in this paper we determine all modular forms of weights 36 k 56 4 k for the full modular group sl 2 z which behave like theta series i e which have in their fourier expansions the constant term 1 and all other fourier coefficients are non negative rational integers in fact we give convex regions in r 3 resp in r 4 for the cases k 36 40 and 44 resp for the cases k 48 52 and 56 corresponding to each lattice point in these regions we get a modular form with the above property as an application we determine the possible exceptions of quadratic forms in the respective dimensions

propelinear 1 perfect codes from quadratic functions perfect codes obtained by the vasil ev schonheim construction from a linear base code and quadratic switching functions are transitive and moreover propelinear this gives at least exp cn2 propelinear 1 perfect codes of length n over an arbitrary finite field while an upper bound on the number of transitive codes is exp c nlnn 2 vphantom

improved semidefinite programming bound on sizes of codes let a n d respectively a n d w be the maximum possible number of codewords in a binary code respectively binary constant weight w code of length n and minimum hamming distance at least d by adding new linear constraints to schrijver s semidefinite programming bound which is obtained from block diagonalizing the terwilliger algebra of the hamming cube we obtain two new upper bounds on a n d namely a 18 8 71 and a 19 8 131 twenty three new upper bounds on a n d w for n 28 are also obtained by a similar way

on optimal binary one error correcting codes of lengths 2 m 4 and 2 m 3 best and brouwer proved that triply shortened and doubly shortened binary hamming codes which have length 2m 4 and 2m 3 respectively are optimal properties of such codes are here studied determining among other things parameters of certain subcodes a utilization of these properties makes a computer aided classification of the optimal binary one error correcting codes of lengths 12 and 13 possible there are 237 610 and 117 823 such codes respectively with 27 375 and 17 513 inequivalent extensions this completes the classification of optimal binary one error correcting codes for all lengths up to 15 some properties of the classified codes are further investigated finally it is proved that for any m 4 there are optimal binary one error correcting codes of length 2m 4 and 2m 3 that cannot be lengthened to perfect codes of length 2m 1

crossed random effect models for multiple outcomes in a study of teratogenesis human teratogens often manifest themselves through a broad spectrum of adverse effects although often not serious when considered individually such outcomes taken together may represent a syndrome that can lead to serious developmental problems accordingly studies that investigate the effect of human teratogens on fetal development typically record the presence or absence of a multitude of abnormalities resulting in the data of multivariate binary form for each infant such studies typically have three objectives 1 estimate an overall effect of exposure across outcomes 2 identify subjects having the syndrome and 3 identify those outcomes that constitute the syndrome so that doctors know what to look for when diagnosing the syndrome in other exposed newborns this article proposes the use of a logistic regression model with crossed random effect structure to address all three questions simultaneously we use the proposed models to analyze data from a study investigating the effects of in uter

translation invariant propelinear codes a class of binary group codes is investigated these codes are the propelinear codes defined over the hamming metric space f sup m f 0 1 with a group structure generally they are neither abelian nor translation invariant codes but they have good algebraic and combinatorial properties linear codes and z sub 4 linear codes can be seen as a subclass of propelinear codes it is shown here that the subclass of translation invariant propelinear codes is of type z sub 2 sup k1 spl oplus z sub 4 sup k2 spl oplus q sub 8 k3 where q sub 8 is the non abelian quaternion group of eight elements exactly every translation invariant propelinear code of length n can be seen as a subgroup of z sub 2 sup k1 spl oplus z sub 4 sup k2 spl oplus q sub 8 sup k3 with k sub 1 2k sub 2 4k sub 3 n for k sub 2 k sub 3 0 we obtain linear binary codes and for k sub 1 k sub 3 0 we obtain z sub 4 linear codes the class of additive propelinear codes the abelian subclass of the translation invariant propelinear codes is studied and a family of nonlinear binary perfect codes with a very simply construction and a very simply decoding algorithm is presented

the perfect binary one error correcting codes of length 15 part i classification a complete classification of the perfect binary one error correcting codes of length 15 as well as their extensions of length 16 is presented there are 5983 such inequivalent perfect codes and 2165 extended perfect codes efficient generation of these codes relies on the recent classification of steiner quadruple systems of order 16 utilizing a result of blackmore the optimal binary one error correcting codes of length 14 and the 15 1024 4 codes are also classified there are 38 408 and 5983 such codes respectively

delsarte s linear programming bound for constant weight codes we give an alternative proof of delsarte s linear programming bound for binary codes and its improvements applying the technique which is used in the proof to binary constant weight codes we obtain new upper bounds on sizes of binary constant weight codes

the steiner triple systems of order 19 using an orderly algorithm the steiner triple systems of order 19 are classied there are 11 084 874 829 pairwise nonisomorphic such designs for each design the order of its automorphism group and the number of pasch congurations it contains are recorded 2 591 of the designs are anti pasch there are three main parts of the classication constructing an initial set of blocks the seeds completing the seeds to triple systems with an algorithm for exact cover and carrying out isomorph rejection of the nal triple systems isomorph rejection is based on the graph canonical labeling software nauty supplemented with a vertex invariant based on pasch congurations the possibility of using the strongly regular block graphs of these designs in the isomorphism tests is utilized the aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters 57 24 11 9

on binary 1 perfect additive codes some structural properties the rank and kernel of 1 perfect additive codes is determined additive codes could be seen as translation invariant propelinear codes and in this correspondence a characterization of propelinear codes as codes having a regular subgroup of the full group of isometrics of the code is established a characterization of the automorphism group of a 1 perfect additive code is given and also the cardinality of this group is computed finally an efficiently computable characterization of the steiner triple systems associated with a 1 perfect binary additive code is also established

on the number of 1 perfect binary codes a lower bound we present a construction of 1 perfect binary codes which gives a new lower bound on the number of such codes we conjecture that this lower bound is asymptotically tight

estimating survival curves with left truncated and interval censored data under monotone hazards we show that the nonparametric maximum likelihood estimator npmle of a survival function may severely underestimate the survival probabilities at very early times for left truncated and interval censored data as an alternative we propose to compute the nonparametric mle under a nondecreasing hazard assumption the monotone mle by a gradient projection algorithm when the assumption holds the projection step is accomplished via an isotonic regression algorithm the pool adjacent violators algorithm this gradient projection algorithm is computationally efficient and converges globally monte carlo simulations show superior performance of the monotone mle over that of the npmle in terms of either bias or variance even for large samples the methodology is illustrated with the application to the wisconsin epidemiological study of diabetic retinopathy data to estimate the probability of incidence of retinopathy

every binary 2 sup m 2 2 sup 2 m 2 m 3 code can be lengthened to form a perfect code of length 2 sup m 1 we answer a problem posed by etzion and vardy 1998 by showing that a binary code of length n 2 m 2 with 2 n m codewords and minimum distance three can always be lengthened to form a perfect code of length 2 m 1

lower bounds on trellis complexity of block codes the trellis state complexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis we present a new lower bound on the state complexity of linear codes which includes most of the existing bounds as special cases the new bound is obtained by dividing the time axis for the code into several sections of varying lengths as opposed to the division into two sections the past and the future employed in the well known dlp bounds for a large number of codes this results in a considerable improvement upon the dlp bound moreover we generalize the new bound to nonlinear codes and introduce several alternative techniques for lower bounding the trellis complexity based on the distance spectrum and other combinatorial properties of the code we also show how the general ideas developed in this paper may be employed to lower bound the maximum and the total number of branches in the trellis leading to considerably tighter bounds on these quantities furthermore the asymptotic behavior of the new bounds is investigated and shown to improve upon the previously known asymptotic estimates of trellis state complexity

optimal binary one error correcting codes of length 10 have 72 codewords the maximum number of codewords in a binary code with length n and minimum distance d is denoted by a n d by construction it is known that a 10 3 spl ges 72 and a 11 3 spl ges 144 these bounds have long been conjectured to be the exact values this is here proved by classifying various codes of smaller length and lengthening these using backtracking and isomorphism rejection there are 562 inequivalent codes attaining a 10 3 72 and 7398 inequivalent codes attaining a 11 3 144

upper bounds for constant weight codes let a n d w denote the maximum possible number of codewords in an n d w constant weight binary code we improve upon the best known upper bounds on a n d w in numerous instances for n spl les 24 and d spl les 12 which is the parameter range of existing tables most improvements occur for d 8 10 where we reduce the upper bounds in more than half of the unresolved cases we also extend the existing tables up to n spl les 28 and d spl les 14 to obtain these results we develop new techniques and introduce new classes of codes we derive a number of general bounds on a n d w by means of mapping constant weight codes into euclidean space this approach produces among other results a bound on a n d w that is tighter than the johnson bound a similar improvement over the best known bounds for doubly constant weight codes studied by johnson and levenshtein is obtained in the same way furthermore we introduce the concept of doubly bounded weight codes which may be thought of as a generalization of the doubly constant weight codes subsequently a class of euclidean space codes called zonal codes is introduced and a bound on the size of such codes is established this is used to derive bounds for doubly bounded weight codes which are in turn used to derive bounds on a n d w we also develop a universal method to establish constraints that augment the delsarte inequalities for constant weight codes used in the linear programming bound in addition we present a detailed survey of known upper bounds for constant weight codes and sharpen these bounds in several cases all these bounds along with all known dependencies among them are then combined in a coherent framework that is amenable to analysis by computer this improves the bounds on a n d w even further for a large number of instances of n d and w

bounds on the size of permutation codes with the kendall tau metric the rank modulation scheme has been proposed for efficient writing and storing data in nonvolatile memory storage error correction in the rank modulation scheme is done by considering permutation codes in this paper we consider codes in the set of all permutations on n elements s n using the kendall tau metric the main goal of this paper is to derive new bounds on the size of such codes for this purpose we also consider perfect codes diameter perfect codes and the size of optimal anticodes in the kendall tau metric structures which have their own considerable interest we prove that there are no perfect single error correcting codes in s n where n 4 is a prime or 4 leq n leq 10 we present lower bounds on the size of optimal anticodes with odd diameter as a consequence we obtain a new upper bound on the size of codes in s n with even minimum kendall tau distance we present larger single error correcting codes than the known ones in s 5 and s 7

maximizing generalized linear mixed model likelihoods with an automated monte carlo em algorithm summary two new implementations of the em algorithm are proposed for maximum likelihood fitting of generalized linear mixed models both methods use random independent and identically distributed sampling to construct monte carlo approximations at the e step one approach involves generating random samples from the exact conditional distribution of the random effects given the data by rejection sampling using the marginal distribution as a candidate the second method uses a multivariate t importance sampling approximation in many applications the two methods are complementary rejection sampling is more efficient when sample sizes are small whereas importance sampling is better with larger sample sizes monte carlo approximation using random samples allows the monte carlo error at each iteration to be assessed by using standard central limit theory combined with taylor series methods specifically we construct a sandwich variance estimate for the maximizer at each approximate e step this suggests a rule for automatically increasing the monte carlo sample size after iterations in which the true em step is swamped by monte carlo error in contrast techniques for assessing monte carlo error have not been developed for use with alternative implementations of monte carlo em algorithms utilizing markov chain monte carlo e step approximations three different data sets including the infamous salamander data of mccullagh and nelder are used to illustrate the techniques and to compare them with the alternatives the results show that the methods proposed can be considerably more efficient than those based on markov chain monte carlo algorithms however the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension

the uniqueness of the best code proves that the 10 40 4 code found by best 1980 is unique the authors then employ this fact to show that a 10 3 a 11 4 spl les 78 and a 11 3 a 12 4 spl les 156

involutions in binary perfect codes given a 1 perfect code c the group of symmetries of c sym c sn c c is a subgroup of the group of automorphisms of c in this paper we focus on symmetries of order two i e involutions let invf c sym c be the set of involutions that stabilize f pointwise for linear 1 perfect codes the possibilities for the number of fixed points f are given establishing lower and upper bounds for any m 2 and any value k between these bounds m 2 k m 1 linear 1 perfect codes of length n 2m 1 which have an involution that fixes f 2k 1 coordinates are constructed moreover for any m 4 1 r m 1 and m 2 k m 1 nonlinear 1 perfect codes of length n 2m 1 having rank n m r and an involution that fixes 2k 1 coordinates are also constructed except one case when m 6 is even r m 1 and k m 2

a new binary code of length 10 and covering radius 1 a mixed code of covering radius 1 that has 60 codewords is constructed this code is then used to show that k 10 1

two new binary codes with minimum distance three the author obtains a binary code of length 18 which has 164 2 sup 6 10496 codewords the minimum distance of this code is three therefore a 18 3 or 10496 shortening the author obtains find a binary code of length 17 and minimum distance 3 which has 5248 codewords showing that a 17 3 or 5248

reconstructing extended perfect binary one error correcting codes from their minimum distance graphs the minimum distance graph of a code has the codewords as vertices and edges exactly when the hamming distance between two codewords equals the minimum distance of the code a constructive proof for reconstructibility of an extended perfect binary one error correcting code from its minimum distance graph is presented consequently inequivalent such codes have nonisomorphic minimum distance graphs moreover it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code

splittings of cyclic groups and perfect shift codes a splitting m s of an additive abelian group g consists of a set of integers m and a subset s spl sub g such that every nonzero element g spl isin g can be uniquely written as m spl middot h for some m spl isin m and h spl isin s splittings m spl plusmn 1 spl middot spl middot spl middot spl plusmn k correspond to perfect k shift codes used in the analysis of run length limited codes correcting single peak shifts we shall determine the set s for splittings of cyclic groups z sub p p prime by m 1 a spl middot spl middot spl middot a sup r b spl middot spl middot spl middot b sup s and m spl plusmn 1 spl plusmn a spl middot spl middot spl middot spl plusmn a sup r spl plusmn b spl middot spl middot spl middot spl plusmn b sup s this yields new conditions on the existence of perfect 3 and 4 shift codes further it can be shown that splittings of z sub p by spl plusmn 1 spl plusmn 2 spl plusmn 3 exist exactly if z sub p is also split by 1 2 3

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

the rank and kernel of extended 1 perfect spl zopf sub 4 linear and additive non spl zopf sub 4 linear codes a binary extended 1 perfect code of length n 1 2 sup t is additive if it is a subgroup of spl zopf sub 2 sup spl alpha spl times spl zopf sub 4 sup spl beta the punctured code by deleting a spl zopf sub 2 coordinate if there is one gives a perfect additive code 1 perfect additive codes were completely characterized and by using that characterization we compute the possible parameters spl alpha spl beta rank and dimension of the kernel for extended 1 perfect additive codes a very special case is that of extended 1 perfect spl zopf sub 4 linear codes

representations of integral quadratic forms by sums of squares let g n be the minimum number of squares whose sum represents all positive definite integral quadratic forms of rank n which are represented by sums of squares in this article we first discuss representations of integers by unimodular lattices we then estimate the order of magnitude of the diameter of the 2 graph of unimodular lattices of rank n combining these results we prove g n o 3 n 2n log n we also provide a lower bound for g n finally we discuss s integrable lattices as an application of our method

product constructions for perfect lee codes a well known conjecture of golomb and welch is that the only nontrivial perfect codes in the lee and manhattan metrics have length two or minimum distance three this problem and related topics were subject for extensive research in the last 40 years in this paper two product constructions for perfect lee codes and diameter perfect lee codes are presented these constructions yield a large number of nonlinear perfect codes and nonlinear diameter perfect codes in the lee and manhattan metrics a short survey and other related problems on perfect codes in the lee and manhattan metrics are also discussed