a distinguisher for high rate mceliece cryptosystems the goppa code distinguishing gd problem consists in distinguishing the matrix of a goppa code from a random matrix the hardness of this problem is an assumption to prove the security of code based cryptographic primitives such as mceliece s cryptosystem up to now it is widely believed that the gd problem is a hard decision problem we present the first method allowing to distinguish alternant and goppa codes over any field our technique can solve the gd problem in polynomial time provided that the codes have sufficiently large rates the key ingredient is an algebraic characterization of the key recovery problem the idea is to consider the rank of a linear system which is obtained by linearizing a particular polynomial system describing a key recovery attack it appears that this dimension depends on the type of code considered explicit formulas derived from extensive experimentations for the rank are provided for generic random alternant and goppa codes over any field finally we give theoretical explanations of these formulas in the case of random codes alternant codes over any field of characteristic two and binary goppa codes

a new family of extended gauss quadratures with an interior interval constraint starting from two sequences ga c n and gd b n of ordinary gauss quadrature formulae with an orthogonality measure d on the open intervals a c and d b respectively we construct a new sequence ga b e n of extended gaussian quadrature formulae for d on a b which is based on some preassigned points the nodes of ga c n gd b n and the e n zeros contained in c d of a nonclassical orthogonal polynomial on a b with respect to a linear functional the principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three term recurrence formulae are related thus a new class of gaussian quadratures having some nodes contained in a given interior interval can be computed directly by standard software for ordinary gauss quadrature formulae

sort and match algorithm for soft decision decoding let a q ary linear n k code c be used over a memoryless channel we design a decoding algorithm spl psi sub n that splits the received block into two halves in n different ways first about spl radic n error patterns are found on either half then the left and right hand lists are sorted out and matched to form codewords finally the most probable codeword is chosen among at most n spl radic n codewords obtained in all n trials the algorithm can be applied to any linear code c and has complexity order of n sup 3 spl radic n for any n spl ges q sup n k the decoding error probability p sub n exceeds at most 1 q sup n k n times the probability p sub spl psi c of maximum likelihood decoding for code rates r spl ges 1 2 the complexity order q sup n k 2 grows as square root of general trellis complexity q sup min n k k when used on quantized additive white gaussian noise awgn channels the algorithm spl psi sub n can provide maximum likelihood decoding for most binary linear codes even when n has an exponential order of q sup n k

a new approach to the general minimum distance decoding problem the zero neighbors algorithm minimum distance decoding mdd for a general error correcting linear code is a hard computational problem that recently has been shown to be np hard the complexity of known decoding algorithms is determined by min 2 k 2 n k where n is the code length and k is the number of information digits two new algorithms are suggested that reduce substantially the complexity of mdd the algorithms use a new concept of zero neighbors a special set of codewords only these codewords which can be computed in advance should be stored and used in the decoding procedure the number of zero neighbors is shown to be very small compared with min 2 k 2 n k for n gg 1 and a wide range of code rates r k n for example for r approx 0 5 this number grows approximately as a square root of the number of codewords

error estimates for the implicit mac scheme for the compressible navier stokes equations we prove existence of a solution to the implicit mac scheme for the compressible navier stokes equations we derive error estimates for this scheme on two and three dimensional cartesian grids error estimates are obtained by using the discrete version of the relative energy method introduced on the continuous level in feireisl et al j math fluid mech 14 4 717 730 2012 a systematic use of the theoretical continuous analysis of the equations in combination with the numerical tools is crucial for the result this error estimate does not uses stability hypotheses on the solution of the numerical scheme

reliability based soft decision decoding with iterative information set reduction the reliability based decoding approach using the reprocessing of the most reliable information set only is extended into the iterative reprocessing of several information sets at the end of each information set reprocessing some information bits are delivered by the decoder consequently information sets with decreasing cardinality values are considered at each iteration a tight upper bound on the error performance achieved by this new method is derived compared to previously proposed competitive approaches this new method reduces the number of candidate codewords needed to achieve practically optimum decoding importantly it also preserves the very simple structured implementation of the order statistic decoding

an asymptotic preserving all speed scheme for the euler and navier stokes equations we present an asymptotic preserving all speed scheme for the simulation of compressible flows valid at all mach numbers ranging from very small to order unity the scheme is based on a semi implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly this discretization which is the key to the asymptotic preserving property provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime the divergence free condition on the velocity in the incompressible regime is respected and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations the implicit treatment of the acoustic part allows the time step to be independent of the mach number the scheme is conservative and applies to steady or unsteady flows and to general equations of state one and two dimensional numerical results provide a validation of the asymptotic preserving all speed properties

on mean convergence of extended lagrange interpolation abstract lagrange interpolation to any continuous function on 1 1 at the zeros of orthogonal polynomials is known to converge in the mean here following bellen we study mean convergence of lagrange interpolation on an extended set of nodes that includes in addition to the n zeros of the orthagonal relative to some positive weight function w polynomial n of degree n other n 1 nodes which in turn are zeros of an orthogonal polynomial gp n 1 of degree n 1 corresponding to the weight function w n 2 n w a sufficient criterion of bellen for mean convergence as n of such extended lagrange interpolation for arbitrary continuous functions is shown to fail for chebyshev weight functions of the first third and fourth kind it holds trivially for chebyshev weights of the second kind based on extensive computations it is conjectured on the other hand that the criterion is satisfied for certain jacobi weights with parameters and suitably restricted necessary conditions for mean convergence due to erd s and turan are shown to be violated for the three kinds of chebyshev weights mentioned above for smooth functions a comparison is made of the speed of convergence of simple vs extended lagrange interpolation

a new algorithm for solving ring lpn with a reducible polynomial the learning parity with noise lpn problem has recently proved to be of great importance in cryptology a special and very useful case is the ring lpn problem which typically provides improved efficiency in the constructed cryptographic primitive we present a new algorithm for solving the ring lpn problem in the case when the polynomial used is reducible it greatly outperforms the previous algorithms for solving this problem using the algorithm we can break the lapin authentication protocol for the proposed instance using a reducible polynomial in sim 2 71 bit operations

finding the permutation between equivalent linear codes the support splitting algorithm two linear codes are permutation equivalent if they are equal up to a fixed permutation on the codeword coordinates we present here an algorithm able to compute this permutation it operates by determining a set of properties invariant by permutation one for each coordinate called a signature if this signature is fully discriminant i e different for all coordinates the support of the code splits into singletons and the same signature computed for any permutation equivalent code will allow the reconstruction of the permutation a procedure is described to obtain a fully discriminant signature for most linear codes the total complexity of the support splitting algorithm is polynomial in the length of the code and exponential in the dimension of its hull i e the intersection of the code with its dual

a superlinearly convergent finite volume method for the incompressible navier stokes equations on staggered unstructured grids a method for linear reconstruction of staggered vector fields with special treatment of the divergence is presented an upwind biased finite volume scheme for solving the unsteady incompressible navier stokes equations on staggered unstructured triangular grids that uses this reconstruction is described the scheme is applied to three benchmark problems and is found to be superlinearly convergent in space

on the complexity of minimum distance decoding of long linear codes we suggest a decoding algorithm of q ary linear codes which we call supercode decoding it ensures the error probability that approaches the error probability of minimum distance decoding as the length of the code grows for n spl rarr spl infin the algorithm has the maximum likelihood performance the asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known the algorithm develops the ideas of covering set decoding and split syndrome decoding

some new results for optimal ternary linear codes let d sub 3 n k be the maximum possible minimum hamming distance of a ternary n k d code for given values of n and k we describe a package for code extension and use this to prove some new exact values of d sub 3 n k moreover we classify the ternary n k d sub 3 n k codes for some values of n and k

is code equivalence easy to decide we study the computational difficulty of deciding whether two matrices generate equivalent linear codes i e codes that consist of the same codewords up to a fixed permutation on the codeword coordinates we call this problem code equivalence using techniques from the area of interactive proofs we show on the one hand that under the assumption that the polynomial time hierarchy does not collapse code equivalence is not np complete on the other hand we present a polynomial time reduction from the graph isomorphism problem to code equivalence thus if one could find an efficient i e polynomial time algorithm for code equivalence then one could settle the long standing problem of determining whether there is an efficient algorithm for solving graph isomorphism

convergence of the mac scheme for the compressible stokes equations we prove in this paper the convergence of the marker and cell mac scheme for the discretization of the steady state compressible stokes equations on two or three dimensional cartesian grids the existence of a solution to the scheme is proven followed by estimates on approximate solutions which yield the convergence of the approximate solutions up to a subsequence and in an appropriate sense we then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations as well as the equation of state which is the main difficulty of this study

on quadrature convergence of extended lagrange interpolation quadrature convergence of the extended lagrange interpolant l 2n 1 f for any continuous function f is studied where the interpolation nodes are the n zeros i of an orthogonal polynomial of degree n and the n 1 zeros j of the corresponding induced orthogonal polynomial of degree n 1 it is found that unlike convergence in the mean quadrature convergence does hold for all four chebyshev weight functions this is shown by establishing the positivity of the underlying quadrature rule whose weights are obtained explicitly necessary and sufficient conditions for positivity are also obtained in cases where the nodes i and j interlace and the conditions are checked numerically for the jacobi weight function with parameters and it is conjectured in this case that quadrature convergence holds for 1 2 1 2

modeling bit flipping decoding based on nonorthogonal check sums with application to iterative decoding attack of mceliece cryptosystem in this correspondence iteration 1 of bit flipping decoding based on a set of nonorthogonal check sums is analyzed for both regular and irregular models in particular the tradeoff between the hamming weight and overlapping of the check sums and the number of redundant check sums required to start converging under iterative decoding is investigated the model is then applied to an iterative attack of mceliece public key cryptosystem since a successful attack of this system can be achieved by algebraic bounded distance decoding of a random code based on this model the attack can be decomposed into two phases a preprocessing phase which for one particular key kappa consists of finding a sufficiently large set s of check sums up to a certain hamming weight and a bit flipping decoding phase which uses the set s for each message encrypted with the key kappa

the hardness of solving subset sum with preprocessing the two problems subset sum and linear decoding which have given birth to the knapsack and the mceliece public key cryptosystems are known to be np complete for linear decoding it has been proved that even if one knows the linear code in advance and can preprocess it the existence of a polynomial time decoding algorithm would imply that the polynomial time hierarchy collapses at an early stage it is proved that the same holds for subset sum even if the knapsack is known in advance and can be preprocessed there is no polynomial time algorithm solving it unless the polynomial hierarchy collapses also given is the sketch of a new straightforward proof of this result for linear decoding

the leech lattice and the golay code bounded distance decoding and multilevel constructions multilevel constructions of the binary golay code and the leech lattice are described both constructions are based upon the projection of the golay code and the leech lattice onto the 6 3 4 hexacode over gf 4 however unlike the previously reported constructions the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates based upon the multilevel structure of the golay code and the leech lattice efficient bounded distance decoding algorithms are devised the bounded distance decoder for the binary golay code requires at most 431 operations as compared to 651 operations for the best known maximum likelihood decoder efficient bounded distance decoding of the leech lattice is achieved by means of partitioning it into four cosets of q sub 24 beyond the conventional partition into two h sub 24 cosets the complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case as compared to about 3600 operations for the best known in maximum likelihood decoder it is shown that the proposed algorithms decode correctly at least up to the guaranteed error correction radius of the maximum likelihood decoder thus the loss in coding gain is due primarily to an increase in the effective error coefficient which is calculated exactly for both algorithms furthermore the performance of the leech lattice decoder on the awgn channel is evaluated experimentally by means of a comprehensive computer simulation the results show a loss in coding gain of less than 0 1 db relative to the maximum likelihood decoder for ber ranging from 10 sup 1 to 10 sup 7

maximum likelihood soft decoding of binary block codes and decoders for the golay codes maximum likelihood soft decision decoding of linear block codes is addressed a binary multiple check generalization of the wagner rule is presented and two methods for its implementation one of which resembles the suboptimal forney chase algorithms are described besides efficient soft decoding of small codes the generalized rule enables utilization of subspaces of a wide variety thereby yielding maximum likelihood decoders with substantially reduced computational complexity for some larger binary codes more sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the 24 12 golay code yielding a computational gain factor of about 2 with respect to previous methods a ternary single check version of the wagner rule is applied for efficient soft decoding of the 12 6 ternary golay code

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

application of the godunov method and its second order extension to cascade flow modeling the godunov method and a new second order accurate extension of the method are used for the solution of two dimensional euler equations both numerical schemes are described in detail their performances in the subsonic transonic and supersonic flow regimes are first tested on the problem of flow in a channel with a circular arc bump the niethods are then applied to calculate the transonic flow through a supercritical com pressor cascade designed by j sanzo for this case the solution with the second order extension of the godunov method gives verygood agreement with the design distribution of parameters given by sanzo

computing automorphism groups of error correcting codes an algorithm is described for computing the automorphism group of an error correcting code the algorithm determines the order of the automorphism group and produces a set of monomial permutations which generate the group it has been implemented on a computer and has been used successfully on a great number of codes of moderate length

the complexity of information set decoding information set decoding is an algorithm for decoding any linear code expressions for the complexity of the procedure that are logarithmically exact for virtually all codes are presented the expressions cover the cases of complete minimum distance decoding and bounded hard decision decoding as well as the important case of bounded soft decision decoding it is demonstrated that these results are vastly better than those for the trivial algorithms of searching through all codewords or through all syndromes and are significantly better than those for any other general algorithm currently known for codes over large symbol fields the procedure tends towards a complexity that is subexponential in the symbol size

the existence of binary linear concatenated codes with reed solomon outer codes which asymptotically meet the gilbert varshamov bound binary linear concatenated codes with reed solomon codes as the outer code and varying nonsystematic inner codes are investigated it is shown that codes exist in this class which asymptotically meet the gilbert varshamov bound

the hardness of decoding linear codes with preprocessing the problem of maximum likelihood decoding of linear block codes is known to be hard the fact that the problem remains hard even if the code is known in advance and can be preprocessed for as long as desired in order to device a decoding algorithm is shown the hardness is based on the fact that existence of a polynomial time algorithm implies that the polynomial hierarchy collapses thus some linear block codes probably do not have an efficient decoder the proof is based on results in complexity theory that relate uniform and nonuniform complexity classes

constructions of linear codes with one dimensional hull the hull of a linear code is defined to be the intersection of the code and its dual and was originally introduced to classify finite projective planes the hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code it has been shown that these algorithms are very effective in general if the size of the hull is small the objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one dimensional hull it is shown that there are no such binary or ternary cyclic codes based on these characterizations some constructions of linear codes with one dimensional hull were given by employing quadratic number fields partial difference sets and difference sets we also construct cyclic codes with one dimensional hull some optimal codes with one dimensional hull are obtained

soft decision decoding using punctured codes let a q ary linear n k code be used over a memoryless channel we design a soft decision decoding algorithm that tries to locate a few most probable error patterns on a shorter length s spl isin k n first we take s cyclically consecutive positions starting from any initial point then we cut the subinterval of length s into two parts and examine t most plausible error patterns on either part to obtain codewords of a punctured s k code we try to match the syndromes of both parts finally the designed codewords of an s k code are re encoded to find the most probable codeword on the full length n for any long linear code the decoding error probability of this algorithm can be made arbitrarily close to the probability of its maximum likelihood ml decoding given sufficiently large t by optimizing s we prove that this near ml decoding can be achieved by using only t spl ap q sup n k k n k error patterns for most long linear codes this optimization also gives about t re encoded codewords as a result we obtain the lowest complexity order of q sup n k k n k known to date for near ml decoding for codes of rate 1 2 the new bound grows as a cubic root of the general trellis complexity q sup min n k k for short blocks of length 63 the algorithm reduces the complexity of the trellis design by a few decimal orders

defect correction based velocity reconstruction for physically consistent simulations of non newtonian flows on unstructured grids abstract a new algorithm to recover centroidal velocities from face normal data on two dimensional unstructured staggered meshes is presented the proposed approach uses iterative defect correction in conjunction with a lower order accurate gauss reconstruction to obtain second order accurate centroidal velocities we derive the conditions that guarantee the second order accuracy of the velocity reconstruction and demonstrate its efficacy on arbitrary polygonal mesh topologies the necessity of the proposed algorithm for non newtonian flow simulations is elucidated through numerical simulations of channel flow driven cavity and backward facing step problems with power law and carreau fluids numerical investigations show that second order accuracy of the reconstructed velocity field is critical to obtaining physically consistent solutions of vorticity dominated flows on non orthogonal meshes it is demonstrated that the spurious solutions are not linked to discrete conservation and arise solely due to the lower order accuracy of velocity reconstruction the importance of the proposed algorithm for hemodynamic simulations is highlighted through studies of laminar flow in an idealized stenosed artery using different blood models

beast decoding of block codes obtained via convolutional codes beast is a bidirectional efficient algorithm for searching trees in this correspondence beast is extended to maximum likelihood ml decoding of block codes obtained via convolutional codes first it is shown by simulations that the decoding complexity of beast is significantly less than that of the viterbi algorithm then asymptotic upper bounds on the beast decoding complexity for three important ensembles of codes are derived they verify beast s high efficiency compared to other algorithms for high rates the new asymptotic bound for the best ensemble is in fact better than previously known bounds

on generator matrices of mds codes corresp it is shown that the family of q ary generalized reed solomon codes is identical to the family of q ary linear codes generated by matrices of the form i a where i is the identity matrix and a is a generalized cauchy matrix using cauchy matrices a construction is shown of maximal triangular arrays over gf q which are constant along diagonals in a hankel matrix fashion and with the property that every square subarray is a nonsingular matrix by taking rectangular subarrays of the described triangles it is possible to construct generator matrices i a of maximum distance separable codes where a is a hankel matrix the parameters of the codes are n k d for 1 leq n leq q 1 1 leq k leq n and d n k 1

efficient encryption from random quasi cyclic codes we propose a framework for constructing efficient code based encryption schemes that do not hide any structure in their public matrix the framework is in the spirit of the schemes first proposed by alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight we depart somewhat from alekhnovich s approach and propose an encryption scheme based on the difficulty of decoding random quasi cyclic codes we propose two new cryptosystems instantiated within our framework the hamming quasi cyclic cryptosystem hqc based on the hamming metric and the rank quasi cyclic cryptosystem rqc based on the rank metric we give a security proof which reduces the indistinguishability under chosen plaintext attack security of our systems to a decision version of the well known problem of decoding random families of quasi cyclic codes for the hamming and rank metrics the respective mathsf qcsd and mathsf rqcsd problems we also provide an analysis of the decryption failure probability of our scheme in the hamming metric case for the rank metric there is no decryption failure our schemes benefit from a very fast decryption algorithm together with small key sizes of only a few thousand bits the cryptosystems are very efficient for low encryption rates and are very well suited to key exchange and authentication asymptotically for lambda the security parameter the public key sizes are respectively in mathcal o lambda 2 for hqc and in mathcal o left lambda frac 4 3 right for rqc practical parameter compares well to the systems based on ring learning parity with noise or the recent moderate density parity check codes system

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

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

conservation properties of unstructured staggered mesh schemes classic cartesian staggered mesh schemes have a number of attractive properties they do not display spurious pressure modes and they have been shown to locally conserve mass momentum kinetic energy and circulation to machine precision recently a number of generalizations of the staggered mesh approach have been proposed for unstructured triangular or tetrahedral meshes these unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic cartesian mesh method this work addresses the momentum kinetic energy and circulation conservation properties of unstructured staggered mesh methods it is shown that with certain choices of the velocity interpolation unstructured staggered mesh discretizations of the divergence form of the navier stokes equations can conserve kinetic energy and momentum both locally and globally in addition it is shown that unstructured staggered mesh discretizations of the rotational form of the navier stokes equations can conserve kinetic energy and circulation both locally and globally the analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation these novel conserving unstructured staggered mesh schemes have not been previously analyzed it is shown that they are first order accurate on nonuniform two dimensional unstructured meshes and second order accurate on uniform unstructured meshes numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented

optimal sectionalization of a trellis while the complexity of trellis decoding for a given block code is essentially a function of the number of states and branches in its trellis the decoding complexity may be often reduced by means of an appropriate sectionalization of the trellis notwithstanding the many examples of such sectionalizations for particular codes that appeared in the literature no systematic method for finding the best sectionalization of a given trellis is presently known we present a polynomial time algorithm which produces the optimal sectionalization of a given trellis t in time o n sup 2 where n is the length of the code generated by t the algorithm is developed in a general setting of certain operations and functions defined on the set of trellises it therefore applies to both linear and nonlinear codes and easily accommodates a broad range of optimality criteria the particular optimality criterion based on minimizing the total number of additions and comparisons required for maximum likelihood trellis decoding is investigated in detail several different methods for decoding a given trellis are discussed and compared in a number of examples finally analysis of the dynamical properties of certain optimal sectionalizations is presented

on the bcjr trellis for linear block codes in this semi tutorial paper we will investigate the computational complexity of an abstract version of the viterbi algorithm on a trellis and show that if the trellis has e edges the complexity of the viterbi algorithm is spl theta e this result suggests that the best trellis representation for a given linear block code is the one with the fewest edges we will then show that among all trellises that represent a given code the original trellis introduced by bahl cocke jelinek and raviv in 1974 and later rediscovered by wolf 1978 massey 1978 and forney 1988 uniquely minimizes the edge count as well as several other figures of merit following forney and kschischang and sorokine 1995 we will also discuss trellis oriented or minimal span generator matrices which facilitate the calculation of the size of the bcjr trellis as well as the actual construction of it

lower bounds on trellis complexity of block codes the trellis state complexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis we present a new lower bound on the state complexity of linear codes which includes most of the existing bounds as special cases the new bound is obtained by dividing the time axis for the code into several sections of varying lengths as opposed to the division into two sections the past and the future employed in the well known dlp bounds for a large number of codes this results in a considerable improvement upon the dlp bound moreover we generalize the new bound to nonlinear codes and introduce several alternative techniques for lower bounding the trellis complexity based on the distance spectrum and other combinatorial properties of the code we also show how the general ideas developed in this paper may be employed to lower bound the maximum and the total number of branches in the trellis leading to considerably tighter bounds on these quantities furthermore the asymptotic behavior of the new bounds is investigated and shown to improve upon the previously known asymptotic estimates of trellis state complexity

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

suboptimum decoding of decomposable block codes to decode a long block code with a large minimum distance by maximum likelihood decoding is practically impossible because the decoding complexity is simply enormous however if a code can be decomposed into constituent codes with smaller dimensions and simpler structure it is possible to devise a practical and yet efficient scheme to decode the code this paper investigates a class of decomposable codes their distance and structural properties it is shown that this class includes several classes of well known and efficient codes as subclasses several methods for constructing decomposable codes or decomposing codes are presented a two stage soft decision or hard decision decoding scheme for decomposable codes their translates or unions of translates is devised and its error performance is analyzed for an awgn channel the two stage soft decision decoding is suboptimum error performances of some specific decomposable codes based on the proposed two stage soft decision decoding are evaluated it is shown that the proposed two stage suboptimum decoding scheme provides an excellent trade off between the error performance and decoding complexity for codes of moderate and long block length

polynomial time attack on wild mceliece over quadratic extensions we present a polynomial time structural attack against the mceliece system based on wild goppa codes defined over a quadratic finite field extension we show that such codes can be efficiently distinguished from random codes the attack uses this property to compute a filtration that is to say a family of nested subcodes which will reveal their secret algebraic description

on the one wayness against chosen plaintext attacks of the loidreau s modified mceliece pkc mceliece public key cryptosystem pkc is one of a few alternatives for the current pkcs that are mostly based on either the integer factoring problem ifp or the discrete logarithm problem dlp that would be solved in polynomial time after the emergence of quantum computers the security of the mceliece pkc is based on the decoding problem and it is known that it satisfies with an appropriate conversion the strongest security notion i e indistinguishability of encryption against adaptively chosen ciphertext attacks ind cca2 in the random oracle model under the assumption that the underlying primitive mceliece pkc satisfies a weak security notion of one wayness against chosen plaintext attacks ow cpa ow cpa is said to be satisfied if it is infeasible for chosen plaintext attacks to recover the whole plaintext of an arbitrarily given ciphertext currently the primitive mceliece pkc satisfies ow cpa if a parameter n spl ges 2048 with optimum t and k is chosen since the binary work factor for n k t 2048 1278 70 to break it with the best cpa is around 2 sup 106 which is infeasible even if world wide computational power is used while the binary work factor for the next smaller parameter n 1024 is in a gray level of 2 sup 62 it will be improved by applying loidreau s modification that employs frobenius automorphism in goppa codes in this paper we carefully investigate the one wayness of the loidreau s modified mceliece pkc against ever known cpas and new cpas we propose and then show that it certainly improves the one wayness against ever known cpas but it is vulnerable against our new cpas thus it is rather harmful to apply the new modification to the mceliece pkc

ellipsoidal lists and maximum likelihood decoding we study an interrelation between the coverings generated by linear n k codes and complexity of their maximum likelihood ml decoding first discrete ellipsoids in the hamming spaces e sub 1 sup n are introduced these ellipsoids represent the sets of most probable error patterns that need to be tested in soft decision ml decoding we show that long linear n k codes surrounded by ellipsoids of exponential size 2 sup n k can cover the whole space e sub 2 sup n then it is proven that ml decoding of most long n k codes needs only about 2 sup n k most probable error patterns to be tested on any quantized memoryless channel finally ml decoding complexity is bounded from above by 2 sup k n k n this substantially reduces the general trellis complexity 2 sup min n k k

a cca2 secure variant of the mceliece cryptosystem the mceliece public key encryption scheme has become an interesting alternative to cryptosystems based on number theoretical problems different from rsa and elgamal mceliece pkc is not known to be broken by a quantum computer moreover even though mceliece pkc has a relatively big key size encryption and decryption operations are rather efficient in spite of all the recent results in coding theory based cryptosystems to the date there are no constructions secure against chosen ciphertext attacks in the standard model the de facto security notion for public key cryptosystems in this paper we show the first construction of a mceliece based public key cryptosystem secure against chosen ciphertext attacks in the standard model our construction is inspired by a recently proposed technique by rosen and segev

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

a superlinearly convergent mach uniform finite volume method for the euler equations on staggered unstructured grids a mach uniform finite volume scheme for solving the unsteady euler equations on staggered unstructured triangular grids that uses linear reconstruction is described the scheme is applied to three benchmark problems and is found to be considerably more accurate than a similar scheme based on piecewise constant reconstruction

weak keys in the mceliece public key cryptosystem we show that it is possible to know whether the secret goppa code of an instance of the mceliece public key cryptosystem was chosen with a binary generator polynomial furthermore whenever such a weak key is used we present an attack which can be completed for codes of length 1024 and dimension 524 with a large but feasible amount of computation

suboptimal decoding of linear codes partition technique general symmetric channels are introduced and near maximum likelihood decoding in these channels is studied first we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis it is proved that the decoding error probability of suboptimal decoding is bounded above for any q ary code of length n and code rate r by twice the error probability of its maximum likelihood decoding and tends to the latter as n grows second we design a suboptimal trellis like algorithm which reduces the known decoding complexity of the order of q sup n min r 1 r operations to that of q sup nr i r operations for all cyclic codes and virtually all long linear codes we also consider the corresponding bounds for concatenated codes an important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability

mds codes with hulls of arbitrary dimensions and their quantum error correction the hull of linear codes has promising utilization in coding theory and quantum coding theory in this paper we study the hull of generalized reed solomon codes and extended generalized reed solomon codes over finite fields with respect to the euclidean inner product several infinite families of mds codes with hulls of arbitrary dimensions are presented as an application using these mds codes with hulls of arbitrary dimensions we construct several new infinite families of entanglement assisted quantum error correcting codes with flexible parameters

a faber series approach to cardinal interpolation for a compactly supported function 9 in rd we study quasiinterpolants based on point evaluations at the integer lattice we restrict ourselves to the case where the coefficient sequence af for given data f is computed by applying a univariate polynomial q to the sequence i zd and then convolving with the data flzd such operators appear in the well known neumann series formulation of quasi interpolation a criterion for the polynomial q is given such that the corresponding operator defines a quasi interpolant since our main application is cardinal interpolation which is well defined if the symbol of p does not vanish we choose q as the partial sum of a certain faber series this series can be computed recursively by this approach we avoid the restriction that the range of the symbol of p must be contained in a disk of the complex plane excluding the origin which is necessary for convergence of the neumann series furthermore for symmetric a we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the neumann series

euclidean and hermitian hulls of mds codes and their applications to eaqeccs in this paper we construct several classes of maximum distance separable mds codes via generalized reed solomon grs codes and extended grs codes where we can determine the dimensions of their euclidean hulls or hermitian hulls it turns out that the dimensions of euclidean hulls or hermitian hulls of the codes in our constructions can take all or almost all possible values as a consequence we can apply our results to entanglement assisted quantum error correcting codes eaqeccs and obtain several new families of mds eaqeccs with flexible parameters the required number of maximally entangled states of these mds eaqeccs can take all or almost all possible values moreover several new classes of q ary mds eaqeccs of length n q 1 are also obtained