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

a class of quasi cyclic ldpc codes over gf 2 m low density parity check ldpc codes over gf 2m are an extension of binary ldpc codes performances of gf 2m ldpc codes have been shown to be higher than binary ldpc codes but the complexity of the encoders decoders increases hence there is a substantial lack of implementations for ldpc codes over gf 2m codes this paper presents a class of quasi cyclic ldpc codes over gf 2m these codes can alleviate the encoding decoding complexity without excessive loss of performances it is shown how from a performance point of view such codes are better than m times bigger binary codes and as good as 2m longer binary codes

on the algebraic structure of quasi cyclic codes iii generator theory following parts i and ii quasi cyclic codes of given index are studied as codes over a finite polynomial ring these latter codes are decomposed by the chinese remainder theorem crt or equivalently the mattson solomon transform into products of shorter codes over larger alphabets we characterize and enumerate self dual one generator quasi cyclic codes in that context we give an algorithm to remove some equivalent codes from that enumeration a generalization to multigenerator codes is sketched

sklyanin algebras and hilbert schemes of points we construct projective moduli spaces for torsion free sheaves on noncommutative projective planes these moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual commutative projective plane p2 the generic noncommutative plane corresponds to the sklyanin algebra s skl e constructed from an automorphism of infinite order on an elliptic curve e p2 in this case the fine moduli space of line bundles over s with first chern class zero and euler characteristic 1 n provides a symplectic variety that is a deformation of the hilbert scheme of n points on p2 e

a class of linear codes with good parameters a construction of linear codes with good parameters is given based on brouwer s table more than 100 new codes are obtained from our construction

existence of lipschitz and semiconcave control lyapunov functions given a locally lipschitz control system which is globally asymptotically controllable at the origin we construct a control lyapunov function for the system which is lipschitz on bounded sets and we deduce the existence of another one which is semiconcave and so locally lipschitz outside the origin the proof relies on value functions and nonsmooth calculus

a lyapunov like characterization of asymptotic controllability it is shown that a control system in bf r n is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls

stochastic versions of the lasalle theorem the main aim of this paper is to establish stochastic versions of the well known lasalle stability theorem from these stochastic versions follow many classical results on stochastic stability this shows clearly the power of our new results

stability of dynamical systems with competitive controls the degenerate case abstract we study the stability of a dynamical system with two competitive controls to a general closed set the stability is proven to be equivalent to the continuity on such set of the value function of a differential game with maximum cost functional the existence of a suitably defined and continuous lyapunov function is proven to imply the stability our lyapunov function satisfies a differential inequality in the sense of the theory of viscosity solutions

a stochastic representation for mean curvature type geometric flows a smooth solution t t 0 t rd of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem in this control problem the controller tries to steer the state process into a given deterministic set tc with probability one the reachability set v t for the target problem is the set of all initial data x from which the state process xx t tc for some control process this representation is proved by studying the squared distance function to t for the codimension k mean curvature flow the state process is dx t 2 pdw t where w t is a d dimensional brownian motion and the control p is any projection matrix onto a d k dimensional plane smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given

lyapunov functions for infinite dimensional systems abstract we study lyapunov functions for infinite dimensional dynamical systems governed by general maximal monotone operators we obtain a characterization of lyapunov pairs by means of the associated hamilton jacobi partial differential equations whose solutions are meant in the viscosity sense as evolved in works of tataru and crandall lions our approach also leads to a new sufficient condition for lyapunov pairs generalizing a classical result of pazy

multidimensional quasi cyclic and convolutional codes we introduce multidimensional analogues of quasi cyclic qc codes and study their algebraic structure we demonstrate a concatenated structure for multidimensional qc codes and use this to prove that this class of codes is asymptotically good we also relate the new family of codes to convolutional codes it is known that the minimum distance of qc codes provides a natural lower bound on the free distance of convolutional codes we show that the same relation also holds between certain rank one 2 d convolutional codes and the related multidimensional qc codes we provide examples which show that our bound is sharp in some cases we also present some optimal 2 d qc codes along the way we provide a condition on the encoders of rank one convolutional codes which are equivalent to noncatastrophicity for 1 d convolutional codes in the n text d case n 1 our condition is sufficient for the noncatastrophicity of the encoder

provably good codes for hash function design a new technique to lower bound the minimum distance of certain types of quasi cyclic codes with large dimension by reducing the problem to lower bounding the minimum distance of a few significantly smaller codes has been developed these codes have the property that they have extremely efficient software encoders using this technique it is proved that a code which is similar to the sha 1 secure hash algorithm to be explained shortly message expansion code has minimum distance 82 and that too in just the last 64 of the 80 expanded words in fact the proposed code has much greater distance than that of sha 1 code which makes our proposed hashing scheme robust against cryptographic attacks the technique is further used to find the minimum weight of the sha 1 code itself 25 in last 60 words which was an open problem estimating minimum distance of a code given by its parity check matrix is well known to be a hard problem our technique is expected to be helpful in estimating minimum distance of similar codes as well as in designing future practical cryptographic hash functions

a matrix theoretic approach for analyzing quasi cyclic low density parity check codes a matrix theoretic approach for studying quasi cyclic codes based on matrix transformations via fourier transforms and row and column permutations is developed these transformations put a parity check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field the approach is amicable to the analysis and construction of quasi cyclic low density parity check codes since it takes into account the specific parity check matrix used for decoding with iterative message passing algorithms based on this approach the dimension of the codes and parity check matrices for the dual codes can be determined several algebraic and geometric constructions of quasi cyclic codes are presented as applications along with simulation results showing their performance over additive white gaussian noise channels decoded with iterative message passing algorithms

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

on the existence of optimal controls the optimal control problem where the state is governed by an ito stochastic differential equation possibly just an ordinary differential equation is formulated in martingale terms under a coercivity condition which is weaker than compactness of the control set a convexity condition and mild continuity hypotheses on the data it is shown by the direct method that optimal controls exist hard and soft constraints are allowed in the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state i e the synthesis problem has a solution the main tool here is krylov s markovian selection theorem

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

almost sure stabilizability of controlled degenerate diffusions we develop a direct lyapunov method for the almost sure open loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes the infinitesimal decrease condition for a lyapunov function is a new form of hamilton jacobi bellman partial differential inequality of second order we give local and global versions of the first and second lyapunov theorems assuming the existence of a lower semicontinuous lyapunov function satisfying such an inequality in the viscosity sense an explicit formula for a stabilizing feedback is provided for affine systems with smooth lyapunov function several examples illustrate the theory

on existence of good self dual quasicyclic codes for a long time asymptotically good self dual codes have been known to exist asymptotically good 2 quasicyclic codes of rate 1 2 have also been known to exist for a long time recently it was proved that there are binary self dual n 3 quasicyclic codes of length n asymptotically meeting the gilbert varshamov bound unlike 2 quasicyclic codes which are defined to have a cyclic group of order n 2 as a subgroup of their permutation group the n 3 quasicyclic c codes are defined with a permutation group of fixed order of 3 so from the decoding point of view 2 quasicyclic c codes are preferable to n 3 quasicyclic c codes in this correspondence with the assumption that there are infinite primes p with respect to w r t which 2 is primitive we prove that there exist classes of self dual 2p quasicyclic c codes and type ii 8p quasicyclic c codes of length respectively 2p sup 2 and 8p sup 2 which asymptotically meet the gilbert varshamov bound when compared with the order of the defining permutation groups these classes of codes lie between the 2 quasicyclic c codes and the n 3 quasicyclic c codes of length n considered in previous works

a class of 1 generator quasi cyclic codes if r f sub q x spl rceil x sup m 1 s f sub qn x x sup m 1 we define the mapping a x spl rarr a x spl sigma sub 0 sup n 1 a sub i x spl alpha sub i from r sup n onto s where spl alpha sub 0 spl alpha sub i spl alpha sub n 1 is a basis for f sub qn over f sub q this carries the q ray 1 generator quasicyclic qc code r a x onto the code ra x in s whose parity check polynomial p c p is defined as the monic polynomial h x over f sub q of least degree such that h x a x 0 in the special case where gcd q m 1 and where the prime factorizations of x sub m 1 over f sub q and f sub qn are the same we show that there exists a one to one correspondence between the q ary 1 generator quasis cyclic codes with p c p h x and the elements of the factor group j i where j is the ideal in s with p c p h x and i the corresponding quantity in r we then describe an algorithm for generating the elements of j i next we show that if we choose a normal basis for f sub qn over f sub q then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes thereby rending the algorithm more attractive from a computational point of view finally in section iv we show how to modify the above algorithm in order to generate all the binary self dual 1 generator qc codes

generic noncommutative surfaces abstract we study a class of noncommutative surfaces and their higher dimensional analogs which come from generic subalgebras of twisted homogeneous coordinate rings of projective space such rings provide answers to several open questions in noncommutative projective geometry specifically these rings r are the first known graded algebras over a field k which are noetherian but not strongly noetherian in other words r k b is not noetherian for some choice of commutative noetherian extension ring b this answers a question of artin small and zhang the rings r are also maximal orders but they do not satisfy all of the conditions of artin and zhang in particular they satisfy the 1 condition but not i for i 2 answering a question of stafford and zhang and a question of stafford and van den bergh finally we show that the noncommutative scheme r proj has finite global dimension

higher order cis codes we introduce complementary information set codes of higher order a binary linear code of length tk and dimension k is called a complementary information set code of order t t cis code for short if it has t pairwise disjoint information sets the duals of such codes permit to reduce the cost of masking cryptographic algorithms against side channel attacks as in the case of codes for error correction given the length and the dimension of a t cis code we look for the highest possible minimum distance in this paper this new class of codes is investigated the existence of good long cis codes of order 3 is derived by a counting argument general constructions based on cyclic and quasi cyclic codes and on the building up construction are given a formula similar to a mass formula is given a classification of 3 cis codes of length 12 is given nonlinear codes better than linear codes are derived by taking binary images of z 4 codes a general algorithm based on edmonds basis packing algorithm from matroid theory is developed with the following property given a binary linear code of rate 1 t it either provides t disjoint information sets or proves that the code is not t cis using this algorithm all optimal or best known tk k codes where t 3 4 256 and 1 k 256 t are shown to be t cis for all such k and t except for t 3 with k 44 and t 4 with k 37

constacyclic codes of length 2 s over galois extension rings of bbf _ 2 u bbf _2 we study all constacyclic codes of length 2s over gr rfr m the galois extension ring of dimension m of the ring rfr f2 uf2 the units of the ring gr rfr m are of the forms alpha and alpha ubeta where alpha beta are nonzero elements of f2m which correspond to 2 m 2m 1 such constacyclic codes first the structure and hamming distances of 1 ugamma constacyclic codes are established we then classify all cyclic codes of length 2s over gr rfr m and obtain a formula for the number of those cyclic codes as well as the number of codewords in each code finally one to one correspondences between cyclic and alpha constacyclic codes as well as 1 ugamma constacyclic and alpha ubeta constacyclic codes are provided via ring isomorphisms that allow us to carry over the results about cyclic and 1 ugamma constacyclic accordingly to all constacyclic codes of length 2s over gr rfr m

stochastic lyapunov functions without differentiability at supposed equilibria abstract stochastic calculus indicates that allowing a lyapunov candidate function to be non differentiable at the origin helps its expected value to decrease to utilize this observation this paper investigates stability of stochastic systems supposing that the origin of the state space is an equilibrium before influenced by gaussian white noises for identifying such stability properties notions of instantaneous points and almost sure equilibria which are mutually exclusive are introduced this paper clarifies the relationship between the stability properties and develops their lyapunov type characterizations without differentiability at the origin discussions are given in relation to the notion of noise to state stability and this paper proposes a method to address both transient and global in time properties simultaneously with a single positive definite function even when stochastic noises prevent the origin from being an equilibrium

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

stochastic target problems dynamic programming and viscosity solutions in this paper we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward sdes and forward backward sdes the controlled process x nu y nu takes values in mathbb r d times mathbb r and a given initial data for x nu 0 then the control problem is to find the minimal initial data for y nu so that it reaches a stochastic target at a specified terminal time t the main application is from financial mathematics in which the process x nu is related to stock price y nu is the wealth process and nu is the portfolio we introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation the boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense this provides a unique characterization of the value function which is the minimal initial data for y nu

cohomology of schematic algebras letrbe a positively graded noetheriank algebra and let proj r r grbe the quotient category of the category of gradedr modulesr grmodulo those of finite length ifris schematic cf f van oystaeyen and l willaert j pure appl algebra104 1995 109 122 and has finite global dimension we show that the cohomology groups ofr as defined in m artin and j j zhang adv in math 109 1994 228 287 are finite dimensional furthermore we introduce cech cohomology groups for schematic algebras and prove that they coincide with the usual graded cohomology groups these generalized cech cohomology groups are still manageable we illustrate this with an example

stabilization of stochastic nonlinear systems driven by noise of unknown covariance this paper poses and solves a new problem of stochastic nonlinear disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise this is a natural stochastic counterpart of the problem of input to state stabilization in the sense of sontag 1989 our development starts with a set of new global stochastic lyapunov theorems for an exemplary class of stochastic strict feedback systems with vanishing nonlinearities where the equilibrium is preserved in the presence of noise we develop an adaptive stabilization scheme based on tuning functions that requires no a priori knowledge of a bound on the covariance next we introduce a control lyapunov function formula for stochastic disturbance attenuation finally we address optimality and solve a differential game problem with the control and the noise covariance as opposing players for strict feedback systems the resulting isaacs equation has a closed form solution

lyapunov s second method for random dynamical systems abstract the method of lyapunov functions lyapunov s second or direct method was originally developed for studying the stability of a fixed point of an autonomous or non autonomous differential equation it was then extended from fixed points to sets from differential equations to dynamical systems and to stochastic differential equations we go one step further and develop lyapunov s second method for random dynamical systems and random sets together with matching notions of attraction and stability as a consequence lyapunov functions will also be random our test is that the extension be coherent in the sense that it reduces to the deterministic theory in case the noise is absent and that we can prove that a random set is asymptotically stable if and only if it has a lyapunov function several examples are treated including the stochastic lorenz system