associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

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

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

decentralized sequential detection a class of decentralized sequential detection problems is investigated under appropriate independence assumptions it is shown that at each time instant the optimal local strategies are given by threshold tests on the likelihood of ratios furthermore local decisions depend not only on the present and past observations but on the past local decision as well that is for each local processor a different threshold exists for every different sequence of past decisions examples computational techniques discussions of the difficulties at hand and suggestions for further exploration of the problem are presented

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

a note on the ordinal sum theorem and its consequence for the construction of triangular norms in this paper the well known ordinal sum theorem of semigroups is generalized and applied to construct new families of triangular subnorms and triangular norms t norms among them one can find several new families of left continuous t norms too

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

how many weights can a cyclic code have upper and lower bounds on the largest number of weights in a cyclic code of given length dimension and alphabet are given an application to irreducible cyclic codes is considered sharper upper bounds are given for the special cyclic codes called here strongly cyclic whose nonzero codewords have period equal to the length of the code asymptotics are derived on the function gamma k q that is defined as the largest number of nonzero weights a cyclic code of dimension k over mathbb f q can have and an algorithm to compute it is sketched the nonzero weights in some infinite families of reed muller codes either binary or q ary as well as in the q ary hamming code are determined two difficult results of independent interest

sequential and cooperative sensing for multi channel cognitive radios effective spectrum sensing is a critical prerequisite for multi channel cognitive radio cr networks where multiple spectrum bands are sensed to identify transmission opportunities while preventing interference to the primary users the present paper develops sequential spectrum sensing algorithms which explicitly take into account the sensing time overhead and optimize a performance metric capturing the effective average data rate of cr transmitters a constrained dynamic programming problem is formulated to obtain the policy that chooses the best time to stop taking measurements and the best set of channels to access for data transmission while adhering to hard collision constraints imposed to protect primary links given the associated lagrange multipliers the optimal access policy is obtained in closed form and the subsequent problem reduces to an optimal stopping problem a basis expansion based sub optimal strategy is employed to mitigate the prohibitive computational complexity of the optimal stopping policy a novel on line implementation based on the recursive least squares rls algorithm along with a stochastic dual update procedure is then developed to obviate the lengthy training phase of the batch scheme cooperative sequential sensing generalizations are also provided with either raw or quantized measurements collected at a central processing unit the numerical results presented verify the efficacy of the proposed algorithms

on properties of uninorms with underlying t norm and t conorm given as ordinal sums uninorms as binary operations on the unit interval have been widely applied in the fuzzy set theory this paper presents some properties of uninorm like operations for which the underlying operations are given by ordinal sums if the underlying operations of a uninorm are given by ordinal sums then the cartesian product of the union of two arbitrary intervals one in 0 e and the other in e 1 where e is the neutral element of the uninorm is a set closed under the uninorm when transposing such a cartesian product to the unit square one obtains a uninorm like operation as a result we have described the uninorm like operations for which the underlying operations are basic pseudo t norms and pseudo t conorms and one of them is idempotent

encoding of correlated observations an important class of engineering problems involves sensing an environment and making estimates based on the phenomena sensed in the traditional model of this problem the sensors observations are available to the estimator without alteration there is growing interest in em distributed sensing systems in which several observations are communicated to the estimator over channels of limited capacity the observations must be separately encoded so that the target can be estimated with minimum distortion two questions are addressed for a special case of this problem wherein there are two sensors which observe noisy data and communicate with a single estimator 1 if the encoder is unlimited in complexity what communication rates and distortions can be achieved 2 if the encoder must be a quantizer a mapping of a single observation sample into a digital output how can it be designed for good performance the first question is treated by the techniques of information theory it is proved that a given combination of rates and distortion is achievable if there exist degraded versions of the observations that satisfy certain formulas the second question is treated by two approaches in the first the outputs of the quantizers undergo a second stage of encoding which exploits their correlation to reduce the output rate algorithms which design the second stage are presented and tested the second approach is based on the em distributional distance a measure of dissimilarity between two probability distributions an algorithm to modify a quantizer for increased distributional distance is derived and tested

rotation invariant t norms the rotation invariance property revisited focusing on conjunctive left continuous increasing 0 1 2 0 1 functions t we redefine the rotation invariance property in terms of contour lines under the assumption of the existence of a neutral element email protected 0 1 this rotation invariance property requires some partial commutativity and associativity the functions that are rotation invariant w r t all of their contour lines are characterized

on locally internal monotonic operations this paper deals with monotonic binary operations f 0 1 2 0 1 with the property called locally internal property that the value at any point x y is always one of its arguments x y after stating a theorem that characterizes this kind of operations some special cases are studied in detail by considering additional properties of the operation commutativity existence of a neutral element and associativity in case of locally internal associative monotonic operations with neutral element a characterization theorem gives an improvement of a well known theorem of czogala and drewniak on idempotent associative and increasing operations with neutral element as well as an improvement of a characterization theorem for left and right continuous idempotent uninorms

rotation invariant t norms where triple rotation and rotation annihilation meet generalizing the notion of a zoom of a t norm not only allows to extend the triple rotation method and its corresponding decomposition but also allows to describe the interface between the triple rotation method and the rotation annihilation method an alternative view on the partition behind the rotation annihilation method is required to make both methods compatible besides zooms we also use contour lines and the companion of a t norm to concisely formulate the results

distributed structures sequential optimization and quantization for detection in the design of distributed quantization systems one inevitably confronts two types of constraints those imposed by a distributed system s structure and those imposed by how the distributed system is optimized structural constraints are inherent properties of any distributed quantization system and are normally summarized by functional relationships defining the inputs and outputs of their component quantizers the use of suboptimal optimization methods are often necessitated by the computational complexity encountered in distributed problems this correspondence briefly explores the impact and interplay of these two types of constraints in the context of distributed quantization for detection we introduce two structures that exploit interquantizer communications and that represent extremes in terms of their structural constraints we then develop a sequential optimization scheme that maximizes the kullback leibler divergence takes advantage of statistical dependencies in the distributed system s output variables and leads to simple parameterizations of the component quantization rules we present an illustrative example from which we draw insights into how these constraints influence the quantization boundaries and affect performance relative to lower and upper bounds

the weights in mds codes the weights in maximum distance separable mds codes of length n and dimension k over the finite field gf q are studied up to some explicit exceptional cases the mds codes with parameters given by the mds conjecture are shown to contain all k weights in the range n k 1 to n the proof uses the covering radius of the dual code

some ternary and quaternary codes and associated sphere packings tables of good ternary and quaternary codes are presented and they are used in the construction of dense sphere packings results include 1 tables of the best ternary and quaternary constacyclic codes including cyclic codes up to block length 50 2 a class of optimal n 2 codes over gf q 3 the u v w mod 2u v mod u construction a new ternary code construction technique that can be used to construct the ternary reed muller codes and others and 4 tables of linear ternary and quaternary codes obtained by modifying and combining various codes here and in the literature packings are generated in even dimensions up to 100 using these codes and a sphere packing construction in dimensions 36 and 60 new record densities appear to have been achieved

sequential scalar quantization of vectors an analysis proposes an efficient vector quantization vq technique called sequential scalar quantization ssq the scalar components of the vector are individually quantized in a sequence with the quantization of each component utilizing conditional information from the quantization of previous components unlike conventional independent scalar quantization isq ssq has the ability to exploit intercomponent correlation at the same time since quantization is performed on scalar rather than vector variables ssq offers a significant computational advantage over conventional vq techniques and is easily amenable to a hardware implementation in order to analyze the performance of ssq the authors appeal to asymptotic quantization theory where the codebook size is assumed to be large closed form expressions are derived for the quantizer mean squared error mse these expressions are used to compare the asymptotic performance of ssq with other vq techniques the authors also demonstrate the use of asymptotic theory in designing ssq for a practical application color image quantization where the codebook size is typically small theoretical and experimental results show that ssq far outperforms isq with respect to mse while offering a considerable reduction in computation over conventional vq at the expense of a moderate increase in mse

on the weights of general mds codes the weight spectra of mds codes of length n and dimension k over arbitrary alphabets are studied for all q ary mds codes of dimension k and length n ne q k 1 containing the zero codeword it is shown that all k weights from n to n k 1 are realized the weight spectrum in the remaining case n q k 1 is also determined additionally we prove that all binary mds codes are equivalent to linear mds codes the proofs are combinatorial and self contained

extremal properties of likelihood ratio quantizers m hypotheses and a random variable y with a different probability distribution under each hypothesis are considered a quantizer is applied to form a quantized random variable gamma y the extreme points of the set of possible probability distributions of gamma y as gamma ranges over all quantizers is characterized optimality properties of likelihood ratio quantizers are established for a very broad class of quantization problems including problems involving the maximization of an ali silvey 1966 distance measure and the neyman pearson variant of the decentralized detection problem

quantization for decentralized hypothesis testing under communication constraints in a decentralized hypothesis testing network several peripheral nodes observe an environment and communicate their observations to a central node for the final decision the presence of capacity constraints introduces theoretical and practical problems the following problem is addressed given that the peripheral encoders that satisfy these constraints are scalar quantizers how should they be designed in order that the central test to be performed on their output indices is most powerful the scheme is called cooperative design separate encoding since the quantizers process separate observations but have a common goal they seek to maximize a system wide performance measure the bhattacharyya distance of the joint index space as such a criterion is suggested and a design algorithm to optimize arbitrarily many quantizers cyclically is proposed a simplified version of the algorithm namely an independent design separate encoding scheme where the correlation is either absent or neglected for the sake of simplicity is outlined performances are compared through worked examples

remarks on uninorm aggregation operators it is proved that continuous uninorm aggregation operators are only t norms or t conorms some general constructions about uninorm aggregation operators are given