on the inequivalence of generalized preparata codes if m is odd and sigma in aut gf 2 m is such that x rightarrow x sigma 2 1 is 1 1 there is a 2 m 1 1 2 m l 2m 2 nonlinear binary code p sigma having minimum distance 5 all the codes p sigma have the same distance and weight enumerators as the usual preparata codes which rise as p sigma when x sigma x 2 it is shown that p sigma and p tau are equivalent if and only if tau sigma pm 1 and aut p sigma is determined

construction of hadamard mathbb z _ 2 mathbb z _ 4 q _ 8 codes for each allowable value of the rank and dimension of the kernel this paper deals with hadamard mathbb z 2 mathbb z 4 q 8 codes which are binary codes after a gray map from a subgroup of direct products of mathbb z 2 mathbb z 4 and q 8 where q 8 is the quaternionic group in a previous work these codes were classified in five shapes in this paper we analyze the allowable range of values for the rank and dimension of the kernel which depends on the particular shape of the code we show that all these codes can be represented in a standard form from a set of generators which can help in understanding the characteristics of each shape the main results we present are the characterization of hadamard mathbb z 2 mathbb z 4 q 8 codes as a quotient of a semidirect product of mathbb z 2 mathbb z 4 linear codes and the construction of hadamard mathbb z 2 mathbb z 4 q 8 codes with each allowable pair of values for the rank and dimension of the kernel

constacyclic codes cocycles and a u v u v construction a connection between cohomology cocycles and constacyclic codes is explored it suggests an isomorphism between cyclic codes of length mn and a direct sum of m constacyclic codes of length n the isomorphism is used i to study the discrete fourier transforms and the decomposition of group ring codes ii to give a u v u v construction over gf q when q is odd the u v u v construction gives some of the best ternary cyclic codes and classes of nearly mds cyclic codes of length 2q 2 the symmetry of the construction is used to give a new proof that there are no cyclic self dual codes over gf q when q is odd

improved bounded strength decoupling schemes for local hamiltonians we address the task of switching off the hamiltonian of a system by removing all internal and system environment couplings we propose dynamical decoupling schemes that use only bounded strength controls for quantum many body systems with local system hamiltonians and local environmental couplings to do so we introduce the combinatorial concept of balanced cycle orthogonal arrays boas and show how to construct them from classical error correcting codes the derived decoupling schemes may be useful as a primitive for more complex schemes e g for hamiltonian simulation for the case of n qubits and a two local hamiltonian the length of the resulting decoupling scheme scales as o n log n improving over the previously best known schemes that scaled quadratically with n more generally using boas constructed from families of bose chaudhuri hocquenghem bch codes we show that bounded strength decoupling for any ell local hamiltonian where ell geq 2 can be achieved using decoupling schemes of length at most o n ell 1 log n

fourier transform inequalities with measure weights abstract fourier transform norm inequalities f q p are proved for measure weights on moment subspaces of lp r n density theorems are established to extend the inequalities to all of lp r n in both cases the conditions for validity are computable for n 2 and are radial and the results are applied to prove spherical restriction theorems which include power weights t t n p 1 p n p 1

nonlinear oblique projections abstract we construct nonlinear oblique projections along subalgebras of nilpotent lie algebras in terms of the baker campbell hausdorff multiplication we prove that these nonlinear projections are real analytic on every schubert cell of the grassmann manifold whose points are the subalgebras of the nilpotent lie algebra under consideration

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

a bounded distance decoding algorithm for the leech lattice with generalizations an algorithm is given that decodes the leech lattice with not much more than twice the complexity of soft decision decoding of the golay code the algorithm has the same effective minimum distance as maximum likelihood decoding and increases the effective error coefficient by less than a factor or two the algorithm can be recognized as a member of the class of multistage algorithms that are applicable to hierarchical constructions it is readily generalized to lattices that can be expressed in terms of binary code formulas and in particular to construction b lattices

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

new optimal ternary linear codes the class of quasi twisted qt codes is a generalization of the class of quasi cyclic codes similar to the way constacyclic codes are a generalization of cyclic codes in this paper rate 1 p qt codes over gf 3 are presented which have been constructed using integer linear programming and heuristic combinatorial optimization many of these attain the maximum possible minimum distance for any linear code with the given parameters and several improve the maximum known minimum distances two of these new codes namely 90 6 57 and 120 6 78 are optimal and so prove that d sub 3 90 6 57 and d sub 3 120 6 78

global growth of bandlimited local approximations abstract we obtain explicit estimates showing how the global norm of a band limited function blows up as it locally approximates a function with a jump in a derivative as an application we obtain bounds on how well a function with a jump in a derivative can be essentially time and band limited

on kirillov s lemma for nilpotent lie algebras abstract we establish a sharpening of kirillov s lemma on nilpotent lie algebras with 1 dimensional center and use it to study the structure of 3 step nilpotent lie algebras

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

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

gray isometries for finite chain rings and a nonlinear ternary 36 3 sup 12 15 code using tensor product constructions for the first order generalized reed muller codes we extend the well established concept of the gray isometry between z sub 4 spl delta sub l and z sub 2 sup 2 spl delta sub h to the context of finite chain rings our approach covers previous results by carlet see ibid vol 44 p 1543 7 1998 constantinescu see probl pered inform vol 33 no 3 p 22 8 1997 and ph d dissertation tech univ munchen munchen germany 1995 nechaev et al see proc ieee int symp information theory and its applications p 31 4 1996 and overlaps with heise et al see proc acct 6 pskov russia p 123 9 1998 and honold et al see proc acct 6 pskov russia p 135 41 1998 applying the gray isometry on z sub 9 we obtain a previously unknown nonlinear ternary 36 3 sup 12 15 code

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

z p sup k 1 linear codes we characterize codes over z sub p which are the gray images of 1 p sup k cyclic codes or cyclic codes over z p sup k l k spl ges 1 a necessary and sufficient condition for the gray image of a z p sup 2 linear 1 p cyclic code to be linear is given in many cases this yields an explicit description of the gray image of a linear 1 p cyclic code over z p sup 2 of length relatively prime to p linear cyclic codes over z p sup 2 whose gray images are linear cyclic codes over z sub p have been characterized some generalizations of these results to the case of z p sup k 1 where k spl ges 2 are also obtained

a class of optimum block codes in signal space by using convolutional coding coupled with expanded sets of channel signals ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth in this paper we construct a class of optimum signal space codes using short binary block as opposed to convolutional codes our results are compared to those obtained by ungerboeck a simplified decoding procedure that is asymptotically optimum is also presented

six dimensional nilpotent lie algebras abstract we give a full classification of six dimensional nilpotent lie algebras over an arbitrary field including fields that are not algebraically closed and fields of characteristic 2 to achieve the classification we use the action of the automorphism group on the second cohomology space as isomorphism types of nilpotent lie algebras correspond to orbits of subspaces under this action in some cases these orbits are determined using geometric invariants such as the gram determinant or the arf invariant as a byproduct we completely determine for a four dimensional vector space v the orbits of gl v on the set of two dimensional subspaces of v v

linear constructions for dna codes in this paper we translate in terms of coding theory constraints that are used in designing dna codes for use in dna computing or as bar codes in chemical libraries we propose new constructions for dna codes satisfying either a reverse complement constraint a gc content constraint or both that are derived from additive and linear codes over four letter alphabets we focus in particular on codes over gf 4 and we construct new dna codes that are in many cases better sometimes far better than previously known codes we provide updated tables up to length 20 that include these codes as well as new codes constructed using a combination of lexicographic techniques and stochastic search

binary sequences with gold like correlation but larger linear span a new construction of optimal binary sequences identical to the well known family of gold sequences in terms of maximum nontrivial correlation magnitude and family size but having larger linear span is presented the distribution of correlation values is determined for every odd integer spl tau spl ges 3 the construction provides a family that contains 2 sup spl tau 1 cyclically distinct sequences each of period 2 sup spl tau 1 the maximum nontrivial correlation magnitude equals 2 spl tau 1 sup 2 1 with one exception each of the sequences in the family has linear span at least spl tau sup 2 spl tau 2 compared to 2 spl tau for gold sequences the sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity

finding cyclic redundancy check polynomials for multilevel systems this letter describes a technique for finding cyclic redundancy check polynomials for systems for transmission over symmetric channels which encode information in multiple voltage levels so that the resulting redundancy check gives good error protection and is efficient to implement the codes which we construct have a hamming distance of 3 or 4 we discuss a way to reduce burst error in parallel transmissions and some tricks for efficient implementation of the shift register for these polynomials we illustrate our techniques by discussing a particular example where the number of levels is 9 but they are applicable in general

p adic valuation of weights in abelian codes over spl zopf p sup d counting polynomial techniques introduced by wilson are used to provide analogs of a theorem of mceliece mceliece s original theorem relates the greatest power of p dividing the hamming weights of words in cyclic codes over gf p to the length of the smallest unity product sequence of nonzeroes of the code calderbank li and poonen presented analogs for cyclic codes over zopf 2d using various weight functions hamming lee and euclidean weight as well as count of occurrences of a particular symbol some of these results were strengthened by wilson who also considered the alphabet zopf pd for p an arbitrary prime these previous results new strengthened versions and generalizations are proved here in a unified and comprehensive fashion for the larger class of abelian codes over zopf pd with p any prime for abelian codes over zopf 4 combinatorial methods for use with counting polynomials are developed these show that the analogs of mceliece s theorem obtained by wilson for hamming weight lee weight and symbol counts and the analog obtained here for euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose fourier transforms have a specified support

repeated root cyclic codes in the theory of cyclic codes it is common practice to require that n q 1 where n is the word length and f sub q is the alphabet it is shown that the even weight subcodes of the shortened binary hamming codes form a sequence of repeated root cyclic codes that are optimal in nearly all other cases one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime this statement is based on an analysis for lengths up to 100 a theorem shows why this was to be expected but it also leads to low complexity decoding methods this is an advantage especially for the codes that are not much worse than corresponding codes of odd length it is demonstrated that a binary cyclic code of length 2n n odd can be obtained from two cyclic codes of length n by the well known mod u mod u v mod construction this leads to an infinite sequence of optimal cyclic codes with distance 4 furthermore it is shown that low complexity decoding methods can be used for these codes the structure theorem generalizes to other characteristics and to other lengths some comparisons of the methods using earlier examples are given

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

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

4 phase sequences with near optimum correlation properties two families of four phase sequences are constructed using irreducible polynomials over z sub 4 family a has period l 2 sup r 1 size l 2 and maximum nontrivial correlation magnitude c sub max

binary images of cyclic codes over z sub 4 we determine all linear cyclic codes over z sub 4 of odd length whose gray images are linear codes or equivalently whose nechaev gray 1989 image are linear cyclic codes or are linear cyclic codes

a vandermonde code construction we use a vandermonde matrix to construct a code of length mn from m codes of length n a distance estimate for the construction is given when the starting codes are suitable constacyclic codes the constructed code is cyclic and from this we obtain new cyclic codes the results generalize those of hughes see ibid vol 46 p 674 80 2000 on the u spl upsi u spl upsi construction

new quasi twisted quaternary linear codes let n k d sub q codes be linear codes of length n dimension k and minimum hamming distance d over gf q in this article 31 new quaternary q 4 codes are constructed which improve the respective lower bounds on minimum distance in brouwer s tables

orbit and coset analysis of the golay and related codes let b be a code of length n over a field f with automorphism group g b sub w denotes the subset of codewords of weight w the goal is to classify the vectors of f sup n into orbits under g and to determine their distances from the various subcodes b sub w this is done for the first order reed muller nordstrom robinson and hamming codes of length 16 the golay and shortened golay codes of lengths 22 23 24 and the ternary golay code of length 12

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

new multilevel codes over gf q set partitioning is applied to multidimensional signal spaces over gf q i e gf sup n1 q n1 or d are presented these codes use reed solomon codes as component codes longer multilevel block codes are also constructed using q ary block codes with block length longer than q 1 as component codes some quaternary multilevel block codes are presented with the same length and number of information symbols as but larger distance than the best previously known quaternary one level block codes it is proved that if all the component block codes are linear the multilevel block code is also linear low rate q ary convolutional codes word error correcting convolutional codes and binary to q ary convolutional codes can also be used to construct multilevel trellis codes over gf q or binary to q ary trellis codes

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

on commutative polarizations let l be a finite dimensional lie algebra over a field k of characteristic zero and let u l be its enveloping algebra with quotient division ring d l let p be a commutative lie subalgebra of l in o2 the necessary and sufficient condition on p was given in order for d p to be a maximal commutative subfield of d l in particular this condition is satisfied if p is a commutative polarization cp with respect to any regular f l and the converse holds if l is ad algebraic the purpose of this paper is to study lie algebras admitting these cp s and to demonstrate their widespread occurrence first we have the following characterisation if l is completely solvable p is a cp of l if and only if there exists a descending chain of lie subalgebras

new ternary and quaternary linear block codes we present several new ternary and quaternary codes found with a heuristic search procedure originally proposed by the authors to find good binary codes and which is based on combinatorial optimization

negacyclic and cyclic codes over z sub 4 the negashift spl nu of z sub 4 sup n is defined as the permutation of z sub 4 sup n such that spl nu a sub 0 a sub 1 spl middot spl middot spl middot a sub i spl middot spl middot spl middot a sub n 1 a sub n 1 a sub 0 spl middot spl middot spl middot a sub i spl middot spl middot spl middot a sub n 2 and a negacyclic code of length n over z sub 4 is defined as a subset c of z sub 4 sup n such that spl nu c c we prove that the gray image of a linear negacyclic code over z sub 4 of length n is a binary distance invariant not necessary linear cyclic code we also prove that if n is odd then every binary code which is the gray image of a linear cyclic code over z sub 4 of length n is equivalent to a not necessary linear cyclic code and this equivalence is explicitely described this last result explains and generalizes the existence already known of versions of kerdock preparata and others codes as doubly extended cyclic codes furthermore we introduce a family of binary linear cyclic codes which are gray images of z sub 4 linear negacyclic codes

bounds on mixed binary ternary codes upper and lower bounds are presented for the maximal possible size of mixed binary ternary error correcting codes a table up to length 13 is included the upper bounds are obtained by applying the linear programming bound to the product of two association schemes the lower bounds arise from a number of different constructions

some weighted norm inequalities for the fourier transform of functions with vanishing moments on deduit des inegalites en norme l p ponderees entre une fonction et sa transformee de fourier dans le cas ou la fonction a des moments qui s annulent jusqu a un certain ordre

trellis oriented decomposition and trellis complexity of composite length cyclic codes the trellis complexity of composite length cyclic codes clcc s is addressed we first investigate the trellis properties of concatenated and product codes in general known factoring of clcc s into concatenated subcodes is thereby employed to derive upper bounds on the minimal trellis size and state space profile new decomposition of clcc s into product subcodes is established and utilized to derive further upper hounds on the trellis parameters the coordinate permutations that correspond to these bounds are exhibited additionally new results on the generalized hamming weights of clcc s are obtained the reduction in trellis complexity of many clcc s leads to soft decision decoders with relatively low complexity

weighted norm estimates for the fourier transform with a pair of weights we prove weighted norm inequalities of the form iifiilq ciifiihp 0 p q 0 for the fourier transform on rn for some weight functions v the hardy space hp on the right can be replaced by l the proof depends on making an atomic decomposition of f and using cancellation properties of the atoms

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

construction of fixed length insertion deletion correcting runlength limited codes an algorithm is presented for the construction of fixed length insertion deletion correcting runlength limited rll codes in this construction binary d k constrained codewords are generated by codewords of a q ary lee metric based code it is shown that this new construction always yields better codes than known constructions the use of a q ary lee 1987 metric code q odd is based on the assumption that an error insertion deletion or peak shift has maximal size q 1 2 it is shown that a decoding algorithm for the lee metric code can be extended so that it can also be applied to insertion deletion correcting rll codes furthermore such an extended algorithm can also correct some error patterns containing errors of size more than q 1 2 as a consequence if s denotes the maximal size of an error then in some cases the alphabet size of the generating code can be s 1 instead of 2 spl middot s 1

a linear construction for certain kerdock and preparata codes the nordstrom robinson kerdock and slightly modified preparata codes are shown to be linear over z4 the integers mod 4 the kerdock and preparata codes are duals over z4 and the nordstrom robinson code is self dual all these codes are just extended cyclic codes over z4 this provides a simple definition for these codes and explains why their hamming weight distributions are dual to each other firstand second order reed muller codes are also linear codes over z4 but hamming codes in general are not nor is the golay code

fast ml decoding algorithm for the nordstrom robinson code corresp the extended nordstrom robinson code is an optimum nonlinear double error correcting code of length 16 with considerable practical importance this code is also useful as a rate 1 2 vector quantizer for random waveforms such as speech linear predictive coding lpc residual a fast decoding algorithm is described for maximum likelihood decoding or nearest neighbor search in the squared error sense with 304 additions and 128 comparisons

an exponential number of generalized kerdock codes if n 1 is an odd composite integer then there are at least 2 1 2 n pairwise inequivalent binary error correcting codes of length 2 n size 2 2 n and minimum distance 2 n 1 2 1 2 n 1

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

new optimal quaternary linear codes of dimension 5 in this correspondence new optimal quaternary linear codes of dimension 5 are presented which extend previously known results these codes belong to the class of quasi twisted qt codes and have been obtained using a greedy local search algorithm other codes are also given which provide a lower bound on the maximum possible minimum distance the generator polynomials for these codes are tabulated and the minimum distances of known qt codes are given

l2 multipliers with power weights abstract necessary and sufficient conditions are found for a multiplier operator to be bounded on l 2 of the line with weight x 2 this paper is concerned primarily with the case 1 2 multiplier operators are defined on these spaces by using the usual definition on a subspace that is shown to be dense in the space the case 1 2 is treated by duality 1 2 is briefly treated using a recent result on fractional integrals the periodic case is also sketched

symbols and orbits for 3 step nilpotent lie groups abstract sufficient conditions are derived for l 2 boundedness of a convolution operator on a 3 step nilpotent lie group this is achieved by producing estimates on the kirillov symbol of the operator and is closely linked to the co adjoint orbit structure of the group a structure theorem for 3 step nilpotent lie groups with 1 dimensional center is proved

some best rate 1 p and rate p 1 p systematic quasi cyclic codes over gf 3 and gf 4 the class of quasi cyclic qc codes has been proven to contain many good codes to date the known qc codes are primarily rate 1 p and p 1 p binary codes constructed from circulant matrices these results are extended to qc codes over gf 3 and gf 4 codes are constructed using integer linear programming and heuristic combinatorial optimization many of these attain the maximum possible minimum distance for any linear code with the same parameters and several improve the maximum known distances the link between power residue pr codes and qc codes is exploited as a means of constructing new qc codes and to initialize the search algorithm previously unknown minimum distances for nonbinary quadratic residue codes are given the minimum distances for the pr codes and the maximum known distances for the qc codes are tabulated