a characterization of certain griesmer codes mmd codes in a more general sense let c be an n k d sub q linear code the defect of c is the parameter s s c n k 1 d if k spl ges m 1 spl ges 2 then by the griesmer bound d spl les q sup m q 1 q sub m 1 s m the author s interest is in those linear codes having the maximum minimum distance i e d q sup m q 1 q sup m 1 s m for m 1 we have d q s 1 and the codes are maximum minimum distance mmd codes in the sense of faldum and willems see ibid vol 44 p 1555 58 1998 thus we consider mmd codes in a more general sense we refer to them simply as mmd codes all mmd codes with m 1 are known up to formal equivalence note that two codes are formally equivalent if they have the same weight distribution the author classifies up to formal equivalence the mmd codes with m spl ges 2

generalized hamming weights of linear codes the generalized hamming weight d sub r c of a binary linear code c is the size of the smallest support of any r dimensional subcode of c the parameter d sub r c determines the code s performance on the wire tap channel of type ii bounds on d sub r c and in some cases exact expressions are derived in particular a generalized griesmer bound for d sub r c is presented and examples are given of codes meeting this bound with equality

on list decodability of random rank metric codes and subspace codes codes in rank metric have a wide range of applications to construct such codes with better list decoding performance explicitly it is of interest to investigate the listdecodability of random rank metric codes it is shown that if n m b is a constant then for every rank metric code in fm n q with rate r and list decoding radius must obey the gilbert varshamov bound that is r 1 1 b otherwise the list size can be exponential and hence no polynomial time list decoding is possible on the other hand for arbitrary 0 0 with e and being independent of each other with high probability a random rank metric code with rate r 1 1 b can be efficiently list decoded up to a fraction of rank errors with constant list size o 1 e we establish similar results for constant dimension subspace codes moreover we show that with high probability the list decoding radius of random fq linear rank metric codes also achieve the gilbert varshamov bound with constant list size o exp 1 e

the weight hierarchy of some reducible cyclic codes the generalized hamming weights ghws of linear codes are fundamental parameters the knowledge of which is of great interest in many applications however to determine the ghws of linear codes is difficult in general in this paper we study the ghws for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases this is achieved by extending the idea of yang et al into higher dimension and by employing some interesting combinatorial arguments it shall be noted that these cyclic codes may have arbitrary number of nonzeros

erasure list decodable codes from random and algebraic geometry codes erasure list decoding was introduced to correct a larger number of erasures by outputting a list of possible candidates in this paper we consider both random linear codes and algebraic geometry codes for list decoding from erasures the contributions of this paper are twofold first for arbitrary 0 and epsilon 0 r and epsilon are independent we show that with high probability a q ary random linear code of rate r is an erasure list decodable code with constant list size q o 1 epsilon that can correct a fraction 1 r epsilon of erasures i e a random linear code achieves the information theoretic optimal tradeoff between information rate and fraction of erasures second we show that algebraic geometry codes are good erasure list decodable codes precisely speaking a q ary algebraic geometry code of rate r from the garcia stichtenoth tower can correct 1 r 1 sqrt q 1 1 q epsilon fraction of erasures with list size o 1 epsilon this improves the johnson bound for erasures applied to algebraic geometry codes furthermore list decoding of these algebraic geometry codes can be implemented in polynomial time note that the code alphabet size q in this paper is constant and independent of epsilon

list decoding from erasures bounds and code constructions we consider the problem of list decoding from erasures we establish lower and upper bounds on the rate of a binary linear code that can be list decoded with list size l when up to a fraction p of its symbols are adversarially erased such bounds already exist in the literature albeit under the label of generalized hamming weights and we make their connection to list decoding from erasures explicit our bounds show that in the limit of large l the rate of such a code approaches the capacity 1 p of the erasure channel such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate spl omega spl epsiv sup 2 log 1 spl epsiv that can be efficiently list decoded using lists of size o 1 spl epsiv when an adversarially chosen 1 spl epsiv fraction of symbols are erased for arbitrary spl epsiv 0 this improves previous results in this vein which achieved a rate of spl omega spl epsiv sup 3 log 1 spl epsiv

on the generalized hamming weights of product codes the rth generalized hamming weight of a linear code is the minimum support size of any r dimensional subcode it has been found useful in the studies of cryptography and trellis coding the authors derive several results on expressing the generalized hamming weights of a product code in terms of those of its component codes they also formulate a general conjecture

generalized bezout s theorem and its applications in coding theory this paper presents a generalized bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more plane curves a new approach to determine a lower bound on the minimum distance for algebraic geometric codes defined from a class of plane curves is introduced based on the generalized bezout theorem examples of more efficient linear codes are constructed using the generalized bezout theorem and the new approach for d 4 the linear codes constructed by the new construction are better than or equal to the known linear codes for d spl ges 5 these new codes are better than the known ag codes defined from whole spaces the klein codes 22 16 5 and 22 15 6 over gf 2 sup 3 and the improved hermitian code 64 56 6 over gf 2 sup 4 are also constructed

on the generalized hamming weights of cyclic codes we prove several results on the generalized hamming weights ghw s of linear codes particularly for cyclic codes based on these and previously known results we give some efficient algorithms for computing ghw hierarchy of cyclic codes we give complete weight hierarchy for each of the binary cyclic codes of odd lengths spl les 31 a table of second and third ghw s of binary cyclic codes of odd lengths spl les 57 is also presented we have also computed the second ghw of all binary cyclic codes of length 63 and the third ghw of one code from each dimension

the weight hierarchy of higher dimensional hermitian codes studies a class of projective systems and linear codes corresponding to hermitian varieties over finite fields the weight hierarchy also known as the set of generalized hamming weights of the code is calculated the higher weight distribution is also found

brief paper adaptive inverse dynamics control of robots with uncertain kinematics and dynamics it has been about two decades since the first globally convergent adaptive tracking controller was derived for robots with dynamic uncertainties however not until recently has the problem of concurrent adaptation to both the kinematic and dynamic uncertainties found its solution this adaptive controller belongs to passivity based control though passivity based controllers have many attractive properties in general they are not able to guarantee the uniform performance of the robot over the entire workspace even in the ideal case of perfect knowledge of the manipulator parameters the closed loop system remains nonlinear and coupled thus the closed loop tracking performance is difficult to quantify while the inverse dynamics controllers can overcome these deficiencies therefore in this work we will develop a new adaptive jacobian tracking controller based on the inverse manipulator dynamics using the lyapunov approach we have proved that the end effector motion tracking errors converge asymptotically to zero simulation results are presented to show the performance of the proposed controller

bounds on the minimum support weights the minimum support weight d sub r c of a linear code c over gf q is the minimal size of the support of an r dimensional subcode of c a number of bounds on d sub r c are derived generalizing the plotkin bound and the griesmer bound as well as giving two new existential bounds as the main result it is shown that there exist codes of any given rate r whose ratio d sub rd sub 1 is lower bounded by a number ranging from q sup r 1 q sup r q sup r 1 to r depending on r

generalized column distances the notion of generalized hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography in this paper we extend this concept to the context of convolutional codes in particular we focus on column distances and introduce the novel notion of generalized column distances gcd we first show that the hierarchy of gcd is strictly increasing we then provide characterizations of such distances in terms of the truncated parity check matrix of the code that will allow us to determine their values finally the case in which the parity check matrix is in systematic form is treated

on binary constructions of quantum codes recently steane 1998 proposed an enlargement of the calderbank shor steane construction here we further improve on the parameters obtained from his construction present examples of new codes and analyze asymptotical non constructive bounds

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

on generalized hamming weights of binary primitive bch codes with minimum distance one less than a power of two the generalized hamming weights introduced by wei 1991 have been shown to be fundamental descriptive parameters of a linear block code they have been found to be useful in certain cryptographic applications and in the studies of minimal trellis diagrams of linear block codes in this correspondence we determine the first few and the last few generalized hamming weights of binary primitive bch codes with minimum distance one less than a power of two of their extensions and of the duals of both

compact embeddings in the generalized sobolev space w01 p g and existence of solutions for nonlinear elliptic problems abstract our aim in this paper is to deal with the compact embedding in the generalized sobolev space w 0 1 p g with a variable exponent satisfying the log holder condition as an application we find a nontrivial weak solution of the nonlinear elliptic problem div u x p x 2 u x f x u x in g u x 0 on g which is an extension of fan zhang x fan g h zhang existence of solutions for p x laplacian dirichlet problem nonlinear anal 52 2003 pp 1843 1852 theorem 4 7

codes satisfying the chain condition the authors considered weight hierarchies of codes satisfying the chain condition they called these chain good first they gave a set of simple necessary conditions for a sequence to be chain good they proved that given one chain good sequence there is an infinite set of chain good sequences that can be constructed from this one sequence finally they used this result to completely describe the sets of chain good sequences of dimensions up to 5

minimum support weights of binary codes some relations between the minimum support weights are discussed in particular the possible weight hierarchies of codes of dimension 4 are determined

matrix characterization of linear codes with arbitrary hamming weight hierarchy the support of an n k linear code c over a finite field f q is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position the rth generalized hamming weight d r c 1 leq r leq k of c is defined as the minimum of the cardinalities of the supports of all n r subcodes of c the sequence d 1 c d 2 c d k c is called the hamming weight hierarchy hwh of c the hwh d r c n k r r 1 2 k characterizes maximum distance separable mds codes therefore the matrix characterization of mds codes is also the characterization of codes with the hwh dr c n k r r 1 2 k a linear code c with systematic check matrix i p where i is the n k x n k identity matrix and p is a n k x k matrix is mds iff every square submatrix of p is nonsingular in this paper we extend this characterization to linear codes with arbitrary hwh using this result we characterize near mds codes near near mds n 2 mds codes and a mu mds codes the mds rank of c is the smallest integer n such that d n 1 n k n 1 and the defect vector of c with mds rank n is defined as the ordered set mu1 c mu2 c mu3 c mu n c mu n 1 c where mu i c n k i di c we call c a dually defective code if the defect vector of the code and its dual are the same we also discuss matrix characterization of dually defective codes further the codes meeting the generalized greismer bound are characterized in terms of their generator matrix the hwh of dually defective codes meeting the generalized greismer bound are also reported