anti collusion fingerprinting for multimedia digital fingerprinting is a technique for identifying users who use multimedia content for unintended purposes such as redistribution these fingerprints are typically embedded into the content using watermarking techniques that are designed to be robust to a variety of attacks a cost effective attack against such digital fingerprints is collusion where several differently marked copies of the same content are combined to disrupt the underlying fingerprints we investigate the problem of designing fingerprints that can withstand collusion and allow for the identification of colluders we begin by introducing the collusion problem for additive embedding we then study the effect that averaging collusion has on orthogonal modulation we introduce a tree structured detection algorithm for identifying the fingerprints associated with k colluders that requires o klog n k correlations for a group of n users we next develop a fingerprinting scheme based on code modulation that does not require as many basis signals as orthogonal modulation we propose a new class of codes called anti collusion codes accs which have the property that the composition of any subset of k or fewer codevectors is unique using this property we can therefore identify groups of k or fewer colluders we present a construction of binary valued acc under the logical and operation that uses the theory of combinatorial designs and is suitable for both the on off keying and antipodal form of binary code modulation in order to accommodate n users our code construction requires only o spl radic n orthogonal signals for a given number of colluders we introduce three different detection strategies that can be used with our acc for identifying a suspect set of colluders we demonstrate the performance of our acc for fingerprinting multimedia and identifying colluders through experiments using gaussian signals and real images

new families of optimal frequency hopping sequence sets frequency hopping sequences fhss are employed to mitigate the interferences caused by the hits of frequencies in frequency hopping spread spectrum systems in this paper we present combinatorial constructions for fhs sets including direct constructions by using cyclotomic classes recursive constructions based on cyclic difference matrices merging blocks and discrete logarithm by these constructions a number of series of new fhs sets are then produced these fhs sets are optimal with respect to the peng fan bounds

generalised incomplete trojan designs summary generalised incomplete m x n k trojan designs for m replicates of nk treatments based on sets of k cyclic generators are discussed normal equations for plots within columns plots within blocks and blocks within columns treatment effects are developed the nk treatments are divided into k subsets each of size n and the conditional plots within blocks and blocks within columns information matrix for each subset is defined efficient conditional treatment estimates are discussed and efficient generators for the various strata are discussed balanced m x n k incomplete trojan designs based on youden generators are constructed and designs based on multiples of a single generator are discussed some ideas for constructing efficient general m x n 2 designs are outlined and some advantages of generalised incomplete trojan designs are discussed

a family of highly symmetric codes we define a class of codes admitting a large automorphism group this family contains the binary extended hamming code the hexacode the golay codes the pless symmetry codes as well as the 16 4 12 sub 8 codes constructed by marcugini et al a computer search resulted in the construction of codes with new parameters 28 7 18 sub 8 32 8 20 sub 8 and 39 13 17 sub 4 belonging to this family

new constructions of quasi cyclic ldpc codes based on special classes of bibd s for the awgn and binary erasure channels this paper presents new methods for efficiently constructing encodable quasi cyclic low density parity check ldpc codes based on special balanced incomplete block designs bibds codes constructed perform well over both the additive white gaussian noise awgn and binary erasure channels with iterative decoding

characterization of m sequences of lengths 2 2k 1 and 2 k 1 with three valued cross correlation considered is the distribution of the cross correlation between in sequences of length 22k 1 where m 2k and m sequences of shorter length 2k 1 new pairs of m sequences with three valued cross correlation are found and the complete correlation distribution is determined finally we conjecture that there are no more cases with a three valued cross correlation apart from the ones proven here

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

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

construction of low density parity check codes based on balanced incomplete block designs this correspondence presents a method for constructing structured regular low density parity check ldpc codes based on a special type of combinatoric designs known as balance incomplete block designs codes constructed by this method have girths at least 6 and they perform well with iterative decoding furthermore several classes of these codes are quasi cyclic and hence their encoding can be implemented with simple feedback shift registers

a new three valued cross correlation between m sequences of different lengths the distribution of the cross correlation between m sequences of period 2m 1 where m 2k and m sequences of shorter period 2 k 1 is considered a new pair of m sequences with three valued cross correlation is found and the complete correlation distribution is determined

margulis spacetimes via the arc complex we study strip deformations of convex cocompact hyperbolic surfaces defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface we prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation in an essentially unique way the infinitesimal version of this result gives a parameterization by the arc complex of the moduli space of margulis spacetimes with fixed convex cocompact linear holonomy as an application we provide a new proof of the tameness of such margulis spacetimes m by establishing the crooked plane conjecture which states that m admits a fundamental domain bounded by piecewise linear surfaces called crooked planes the noninfinitesimal version gives an analogous theory for noncompact complete anti de sitter 3 manifolds

achieving the welch bound with difference sets consider a codebook containing n unit norm complex vectors in a k dimensional space in a number of applications the codebook that minimizes the maximal cross correlation amplitude i sub max is often desirable relying on tools from combinatorial number theory we construct analytically optimal codebooks meeting in certain cases the welch lower bound when analytical constructions are not available we develop an efficient numerical search method based on a generalized lloyd algorithm which leads to considerable improvement on the achieved i sub max over existing alternatives we also derive a composite lower bound on the minimum achievable i sub max that is effective for any codebook size n

cross correlation of m sequences of different lengths we consider the distribution of the cross correlation between m sequences of period 2 sup m 1 where m 2k and m sequences of shorter period 2 sup k 1 we present some general properties of the cross correlation between these m sequences and we find some decimations with 3 valued cross correlation we further present some numerical results and open problems

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

propagation algorithms for lexicographic ordering constraints finite domain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia to apply finite domain constraint programming to a problem it is modelled by a set of constraints on a set of decision variables a common modelling pattern is the use of matrices of decision variables the rows and or columns of these matrices are often symmetric leading to redundancy in a systematic search for solutions an effective method of breaking this symmetry is to constrain the assignments of the affected rows and columns to be ordered lexicographically this paper develops an incremental propagation algorithm gaclexleq that establishes generalised arc consistency on this constraint in o n operations where n is the length of the vectors furthermore this paper shows that decomposing gaclexleq into primitive constraints available in current finite domain constraint toolkits reduces the strength or increases the cost of constraint propagation also presented are extensions and modifications to the algorithm to handle strict lexicographic ordering detection of entailment and vectors of unequal length experimental results on a number of domains demonstrate the value of gaclexleq

on z_ 2 k dual binary codes a new generalization of the gray map is introduced the new generalization phi z2 kn rarr z2 2k 1n is connected with the known generalized gray map phi in the following way if we take two dual linear z2 k codes and construct binary codes from them using the generalizations phi and phi of the gray map then the weight enumerators of the binary codes obtained will satisfy the macwilliams identity the classes of z2 k linear hadamard codes and co z2 k linear extended 1 perfect codes are described where co z2 k linearity means that the code can be obtained from a linear z2 k code with the help of the new generalized gray map

a new family of four valued cross correlation between m sequences of different lengths the cross correlation function between m sequences of period 2m 1 where m 6k and m sequences of shorter period 2m 2 1 is investigated the first infinite family of pairs of m sequences with four valued cross correlation is constructed and the complete correlation distribution of this family is determined

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 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

construction of bbz _4 linear reed muller codes in this paper new quaternary plotkin constructions are given and are used to obtain new families of quaternary codes the parameters of the obtained codes such as the length the dimension and the minimum distance are studied using these constructions new families of quaternary reed muller rm codes are built with the peculiarity that after using the gray map the obtained z4 linear codes have the same parameters and fundamental properties as the codes in the usual binary linear rm family to make the duality relationships in the constructed families more evident the concept of kronecker inner product is introduced

parity check matrices separating erasures from errors most decoding algorithms of linear codes in general are designed to correct or detect errors however many channels cause erasures in addition to errors in principle decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code for any given linear code and any given maximum number of correctable erasures parity check matrices are introduced that yield parity check equations which do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures these matrices allow for the separation of erasures from errors to facilitate decoding several constructions of such separating parity check matrices are presented to reduce decoding complexity separating parity check matrices with small number of rows are preferred the minimum number of rows in a parity check matrix separating a given maximum number of erasures is called the separating redundancy upper and lower bounds on the separating redundancies are derived in particular it is shown that the separating redundancies tend to grow linearly with the number of rows in full rank parity check matrices of codes the separating redundancies of some classes of codes are determined for some maximum numbers of erasures

some new results on the cross correlation of m sequences the determination of the cross correlation between an m sequence and its decimated sequence has been a longstanding research problem considering a ternary m sequence of period 33r 1 we determine the cross correlation distribution for decimations d 3r 2 and d 32r 2 where gcd r 3 1 meanwhile for a binary m sequence of period 22lm 1 we make an initial investigation for the decimation d 22lm 1 2m 1 2s where l 2 is even and 0 s 2m 1 it is shown that the cross correlation takes at least four values furthermore we confirm the validity of two famous conjectures due to sarwate et al and helleseth in this case

traffic grooming on the path in a wdm network routing a request consists in assigning it a route in the physical network and a wavelength if each request uses at most 1 c of the bandwidth of the wavelength we will say that the grooming factor is c that means that on a given edge of the network we can groom group at most c requests on the same wavelength with this constraint the objective can be either to minimize the number of wavelengths related to the transmission cost or minimize the number of add drop multiplexers shortly adm used in the network related to the cost of the nodes we consider here the case where the network is a path on n nodes pn thus the routing is unique for a given grooming factor c minimizing the number of wavelengths is an easy problem well known and related to the load problem but minimizing the number of adms is np complete for a general set of requests and no results are known here we show how to model the problem as a graph partition problem and using tools of design theory we completely solve the case where c 2 and where we have a static uniform all to all traffic one request for each pair of vertices

niho type cross correlation functions via dickson polynomials and kloosterman sums suppose that n 2k is even we study the cross correlation function between two m sequences for niho type decimations d 2 sup k 1 s 1 we develop a new technique to study the value distribution of these cross correlation functions which makes use of dickson polynomials as a first application we derive here the distribution of the six valued cross correlation function for s 3 and odd k up to a term which depends on kloosterman sums in addition applying simpler methods we prove a theorem providing niho type decimations with four valued cross correlation functions and their distribution we conjecture that the latter result actually covers all such decimations

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 intersection of bbz _ 2 bbz _ 4 additive hadamard codes the intersection structure for z 2 z 4 additive hadamard codes is investigated for any two of these codes c 1 and c 2 the abelian group structure of the intersection c 1 capc 2 is characterized the parameters of this abelian group structure corresponding to the intersection codes are computed establishing lower and upper bounds for them constructions are given of codes whose intersection has any parameters between these bounds

construction of regular and irregular ldpc codes geometry decomposition and masking two algebraic methods for systematic construction of structured regular and irregular low density parity check ldpc codes with girth of at least six and good minimum distances are presented these two methods are based on geometry decomposition and a masking technique numerical results show that the codes constructed by these methods perform close to the shannon limit and as well as random like ldpc codes furthermore they have low error floors and their iterative decoding converges very fast the masking technique greatly simplifies the random like construction of irregular ldpc codes designed on the basis of the degree distributions of their code graphs

a lattice based systematic recursive construction of quasi cyclic ldpc codes this paper presents a low complexity recursive and systematic method to construct good well structured low density parity check ldpc codes the method is based on a recursive application of a partial kronecker product operation on a given gamma x q q ges 3 a prime integer lattice l gamma x q the n 1 fold product of l gamma x q by itself denoted ln gamma x q represents a regular quasi cyclic qc ldpc code denoted see pdf of high rate and girth 6 the minimum distance of see pdf is equal to that of the core code see pdf introduced by l gamma x q the support of the minimum weight codewords in see pdf are characterized by the support of the same type of codewords in see pdf from performance perspective the constructed codes compete with the pseudorandom ldpc codes

the tournament scheduling problem with absences abstract we study time scheduling problems with allowed absences as a new kind of graph coloring problem one may think of a sport tournament where each player each team is permitted a certain number t of absences we then examine how many rounds are needed to schedule the whole tournament in the worst case this upper limit depends on t and on the structure of the graph g whose edges represent the games that have to be played but also on whether or not the absences are announced before the tournament starts therefore we actually have two upper limits for the number of required rounds we have t g for pre scheduling if all absences are pre fixed and we have ol t g for on line scheduling if we have to stay flexible and deal with absences when they occur we conjecture that t g g 2 t and that ol t g g 2 t the first conjecture is stronger than the total coloring conjecture while the second is weaker than the on line list edge coloring conjecture our conjectures hold for all bipartite graphs for complete graphs we prove them partially lower and upper bounds to t g and ol t g for general multigraphs g are established too

on the intersection of bbz _ 2 bbz _ 4 additive perfect codes the intersection problem for z2z4 additive extended and nonextended perfect codes i e which are the possibilities for the number of codewords in the intersection of two z2z4 additive codes c1 and c2 of the same length is investigated lower and upper bounds for the intersection number are computed and for any value between these bounds codes which have this given intersection value are constructed for all these z2z4 additive codes c1 and c2 the abelian group structure of the intersection codes c1 cap c2 is characterized the parameters of this abelian group structure corresponding to the intersection codes are computed and lower and upper bounds for these parameters are established finally for all possible parameters between these bounds constructions of codes with these parameters for their intersections are given

the terwilliger algebras of johnson graphs abstract let j n m denote the johnson graph with vertex set x fix a vertex x x let t t x denote the terwilliger algebra of j n m corresponding to x in this paper we study the structure of t under the assumption that m 3 and n 4 m let u sl 2 be the universal enveloping algebra of sl 2 we display a c algebra homomorphism u sl 2 t we show that t is generated by the image of and some central elements of t we obtain these results using the theory of leonard pairs

constructions of permutation arrays a permutation array pa of length n and minimum distance d is a set of permutations of n elements such that any two permutations coincide in at most n d positions some constructions of pas are given

families of hadamard bbz_ 2 bbz_ 4 q_ 8 codes a z2z4q8 code is the binary image after a gray map of a subgroup of z2k1 z4k2 q8k3 where q8 is the quaternion group on eight elements such z2z4q8 codes are translation invariant propelinear codes as are the well known z4 linear or z2z4 linear codes in this paper we show that there exist pure z2z4q8 codes that is codes that do not admit any abelian translation invariant propelinear structure we study the dimension of the kernel and rank of the z2z4q8 codes and we give upper and lower bounds for these parameters we give tools to construct a new class of hadamard codes formed by several families of z2z4q8 codes we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are hadamard

optimal complete diallel crosses summary optimality of complete diallel crosses in incomplete blocks is investigated optimal complete diallel crosses are characterized and such binary plans are derived using nested balanced incomplete block designs a table of optimal complete diallel crosses for up to 15 lines is also provided

nonexistence of completely transitive codes with error correcting capability e 3 the class of completely transitive codes was introduced by sole 1990 as a proper subclass of binary linear completely regular codes there exist completely transitive codes with error correcting capabilities e 1 2 and 3 in a previous correspondence borges and rifa see ibid vol 46 no 1 p 279 80 jan 2000 proved the nonexistence of completely transitive codes with more than two codewords and error correcting capability e 4 in this correspondence we prove the nonexistence for the remaining case namely e 4 therefore the question of the existence of such codes depending on their error correcting capability is completely solved

combinatorial constructions for optimal two dimensional optical orthogonal codes optical orthogonal codes oocs have been designed for ocdma a one dimensional 1 d optical orthogonal code 1 d ooc is a set of one dimensional binary sequences having good auto and cross correlations one limitation of 1 d ooc is that the length of the sequence increases rapidly when the number of users or the weight of the code is increased which means large bandwidth expansion is required if a big number of codewords is needed to lessen this problem two dimensional 2 d coding also called multiwavelength oocs was invested a two dimensional 2 d optical orthogonal code 2 d ooc is a set of utimesv matrices with 0 1 elements having good auto and cross correlations recently many researchers are working on constructions and designs of 2 d oocs in this paper we shall reveal the combinatorial properties of 2 d oocs and give an equivalent combinatorial description of a 2 d ooc based on this we are able to use combinatorial methods to obtain many optimal 2 d oocs

crc handbook of combinatorial designs balanced incomplete block designs and t designs2 v k l designs of small orderbibds with small block sizet designs t 3steiner systemssymmetric designsresolvable and near resolvable designslatin squares mols and orthogonal arrayslatin squaresmutually orthogonal latin squares mols incomplete molsorthogonal arrays of index more than oneorthogonal arrays of strength more than twopairwise balanced designspbds and gdds the basicspbds recursive constructionspbd closurepairwise balanced designs as linear spacespbds and gdds of higher indexpbds frames and resolvabilityother combinatorial designsassociation schemesbalanced part ternary designsbalanced tournament designsbhaskar rao designscomplete mappings and sequencings of finite groupsconfigurationscostas arrayscoveringscycle systemsdifference familiesdifference matricesdifference sets abeliandifference sets nonabeliandifference triangle setsdirected designsd optimal matricesembedding partial quasigroupsequidistant permutation arraysfactorial designsfrequency squaresgeneralized quadranglesgraph decompositions and designsgraphical designshadamard matrices and designshall triple systemshowell designsmaximal sets of molsmendelsohn designsthe oberwolfach problemordered designs and perpendicular arraysorthogonal designsorthogonal main effect planspackingspartial geometriespartially balanced incomplete block designsquasigroupsquasi symmetric designs r l designsroom squaresself orthogonal latin squares sols sols with a symmetric orthogonal mate solssom sequences with zero autocorrelationskolem sequencesspherical t designsstarterstrades and defining sets t m s netstuscan squarest wise balanced designsuniformly resolvable designsvector space designsweighing matrices and conference matriceswhist tournamentsyouden designs generalizedyouden squaresapplicationscodescomputer science selected applicationsapplications of designs to cryptographyderandomizationoptimality and efficiency comparing block designsgroup testingscheduling a tournamentwinning the lotteryrelated mathematics and computational methodsfinite groups and designsnumber theory and finite fieldsgraphs and multigraphsfactorizations of graphsstrongly regular graphstwo graphsclassical geometriesprojective planes nondesarguesiancomputational methods in design theoryindex

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 combinatorial construction for strictly optimal frequency hopping sequences frequency hopping sequences fhss with favorable partial hamming correlation properties have important applications in many synchronization and multiple access systems strictly optimal fhss are those fhss with optimal partial hamming autocorrelation irrespective of the correlation window length in this paper strictly optimal fhss are investigated from a combinatorial approach a generic connection between strictly optimal fhss and disjoint cyclic perfect mendelsohn difference families is established by virtue of this connection new strictly optimal fhss are generated from some disjoint cpmdfs these strictly optimal fhss have new parameters not covered in the literature

a sensitive algorithm for detecting the inequivalence of hadamard matrices a hadamard matrix of side n is an n n matrix with every entry either 1 or 1 which satisfies hh t ni two hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations to identify the equivalence of two hadamard matrices by a complete search is known to be an np hard problem when n increases in this paper a new algorithm for detecting inequivalence of two hadamard matrices is proposed which is more sensitive than those known in the literature and which has a close relation with several measures of uniformity as an application we apply the new algorithm to verify the inequivalence of the known 60 inequivalent hadamard matrices of order 24 furthermore we show that there are at least 382 pairwise inequivalent hadamard matrices of order 36 the latter is a new discovery

binary sequences with three valued cross correlations of different lengths in general it is hard to find the cross correlation distribution between two sequences in particular there are only few cases that the cross correlation is three valued in this paper new pairs of binary sequences with three cross correlation values are presented these two sequences have different lengths 2 2k 1 and 3 2 k 1 respectively we determine the cross correlation distribution between these two sequences we also give the possible correlation values for the shift decimated sequences the magnitude of cross correlation values is shown to be low the techniques presented in this paper may be useful to handle other similar cases

on spl zopf sub 4 linear preparata like and kerdock like codes we say that a binary code of length n is additive if it is isomorphic to a subgroup of spl zopf sub 2 sup spl alpha spl times spl zopf sub 4 sup spl beta where the quaternary coordinates are transformed to binary by means of the usual gray map and hence spl alpha 2 spl beta n in this paper we prove that any additive extended preparata 1968 like code always verifies spl alpha 0 i e it is always a spl zopf sub 4 linear code moreover we compute the rank and the dimension of the kernel of such preparata like codes and also the rank and the kernel of the spl zopf sub 4 dual of these codes i e the spl zopf sub 4 linear kerdock like codes

rank inequalities and separation algorithms for packing designs and sparse triple systems combinatorial designs find numerous applications in computer science and are closely related to problems in coding theory packing designs correspond to codes with constant weight 4 sparse partial steiner triple systems 4 sparse pstss correspond to erasure resilient codes that are useful in handling failures in large disk arrays chee colbourn ling discrete appl math to appear hellerstein gibson karp katz paterson algorithmica 12 1994 182 208 the study of polytopes associated with combinatorial problems has proven to be important for both algorithms and theory but only recently the study of design polytopes has been pursued moura math appl 368 1996 227 254 moura ph d thesis university of toronto 1999 moura proc seventh annu european symp prague 1999 lecture notes in computer science vol 1643 springer berlin 1999 pp 462 475 wengrzik master s thesis universitat berlin 1995 zehendner doctoral thesis universitat augsburg 1986 in this article we study polytopes associated with t v k packing designs and with m sparse pstss subpacking and l sparseness inequalities are introduced and studied they can be regarded as rank inequalities for the independence systems associated with these designs conditions under which subpacking inequalities define facets are derived in particular those which define facets for pstss are determined for m 4 the l sparseness inequalities with 2 l m are proven to induce facets for the m sparse psts polytope this proof uses extremal families of pstss known as erdos configurations separation algorithms for these inequalities are proposed we incorporate some of the sparseness inequalities in a polyhedral algorithm and determine maximal 4 sparse psts v v 16 an upper bound on the size of m sparse pstss is presented

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

the maximal determinant of cocyclic 1 1 matrices over d2t abstract cocyclic construction has been successfully used for hadamard matrices of order n these 1 1 matrices satisfy that hh t h t h ni and give the solution to the maximal determinant problem when n 1 2 or a multiple of 4 in this paper we approach the maximal determinant problem using cocyclic matrices when n 2 mod 4 more concretely we give a reformulation of the criterion to decide whether or not the 2 t 2 t determinant with entries 1 attains the ehlich wojtas bound in the d 2 t cocyclic framework we also provide some algorithms for constructing d 2 t cocyclic matrices with large determinants and some explicit calculations up to t 19

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

low density parity check codes from transversal designs with improved stopping set distributions this paper examines the construction of low density parity check ldpc codes from transversal designs based on sets of mutually orthogonal latin squares mols by transferring the concept of configurations in combinatorial designs to the level of latin squares we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes based on large sets of simple structured mols we derive powerful constraints for the choice of suitable subsets leading to improved stopping set distributions for the corresponding codes we focus on ldpc codes with column weight 4 but the results are also applicable for the construction of codes with higher column weights finally we show that a subclass of the presented codes has quasi cyclic structure which allows low complexity encoding

trace minimal graphs and d optimal weighing designs let g v be the set of all regular graphs on v vertices certain graphs from among those in g v with maximum girth have a special property called trace minimality in particular all strongly regular graphs with no triangles and some cages are trace minimal these graphs play an important role in the statistical theory of d optimal weighing designs each weighing design can be associated with a 0 1 matrix let mm n 0 1 denote the set of all m n 0 1 matrices and let g m n maxdetxtx x mm n 0 1 a matrix x mm n 0 1 is a d optimal design matrix if det xtx g m n in this paper we exhibit some new formulas for g m n where n 1 mod 4 and m is sufficiently large these formulas depend on the congruence class of n mod n more precisely let m nt r where 0 r n for each pair n r there is a polynomial p n r t of degree n in t which depends only on n r such that g nt r n p n r t for all sufficiently large t the polynomial p n r t is computed from the characteristic polynomial of the adjacency matrix of a trace regular graph whose degree of regularity and number of vertices depend only on n and r we obtain explicit expressions for the polynomial p n r t for many pairs n r in particular we obtain formulas for g nt r n for n 19 23 and 27 all 0 r n and all sufficiently large t and we obtain families of formulas for p n r t from families of trace minimal graphs including bipartite graphs obtained from finite projective planes generalized quadrilaterals and generalized hexagons

a construction method for optimally universal hash families and its consequences for the existence of rbibds we introduce a method for constructing optimally universal hash families and equivalently rbibds as a consequence of our construction we obtain minimal optimally universal hash families if the cardinalities of the universe and the range are powers of the same prime a corollary of this result is that the necessary conditions for the existence of an rbibd with parameters v k namely v 0 mod k and v 1 0 mod k 1 are sufficient if v and k are powers of the same prime as an application of our construction we show that the k maxcut algorithm of hofmeister and lefmann a combinatorial design approach to maxcut random struct algorithms 9 1996 163 173 can be implemented such that it has a polynomial running time in the case that the number of vertices and k are powers of the same prime