on mds codes via cauchy matrices a special form of cauchy matrix is used to obtain a tighter bound for the validity region of the maximum distance separable mds conjecture and a new compact characterization of generalized reed solomon codes the latter is further used to obtain constructions and some existence results for long 2k k double circulant mds codes

additive rank metric codes we give an infinite family of maximum rank distance mrd codes which covers properly the largest known linear mrd code family our family contains infinite families of non linear mrd codes which are the first non linear examples for most of the parameters we also give explicit examples and a table that demonstrates the proportion of linear and non linear families for some small parameters

transitions and kinematics of reaction convection fronts in a cell population model abstract here we consider cell population dynamics in which there is a simultaneous proliferation and maturation the mathematical model of this process is formulated as a nonlinear first order partial differential equation for the cell density u t x in which there is retardation delay in the temporal t variable thus we consider a transient reaction convection equation in which the cell density is convected with maturation velocity r for localizedd initial perturbations the equation has positive and negative traveling front solutions positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution and negative fronts correspond to the reversed case three classes of fronts are found according to the strength of the convection velocity i for strong convection r 1 the fronts are simple translations of the initial data regardless of delay strength ii for weak convection r 1 two types of fronts exist a reaction convection fronts arise if the localized initial perturbation acts at a non zero maturation on the zero amplitude state and b convection fronts arise if the localized initial perturbation acts at zero maturation on a finite amplitude state for weak convection cases ii a and ii b a further classification arises according to whether the magnitude of the temporal delay is larger or smaller than a critical value b it is found that the critical delay b corresponds to the hopf bifircation of the reaction equation that is obtained in the absence of convection r 0 for delays larger than b the convective and reaction convection fronts are oscillatory in addition all the reaction convection fronts reverse their direction of motion undergo a positive to negative transition and display non uniform kinematics simulation results are validated and interpreted using solutions for the unretarded equations and by local stability analysis

isometries for rank distance and permutation group of gabidulin codes the rank distance was introduced by e m gabidulin see probl pered inform vol 21 p 1 12 1985 he determined an upper bound for the minimum rank distance of a code moreover he constructed a class of codes which meet this bound the so called gabidulin codes we first characterize the linear and semilinear isometries for the rank distance then we determine the isometry group and the permutation group of gabidulin codes of any length we give a characterization of equivalent gabidulin codes finally we prove that the number of equivalence classes of gabidulin codes is exactly the number of equivalence classes of vector spaces of dimension n contained in gf p sup m under some particular relations

maximum rank array codes and their application to crisscross error correction a mu n n k array code c over a field f is a k dimensional linear space of n n matrices over f such that every nonzero matrix in c has rank or mu it is first shown that the dimension of such array codes must satisfy the singleton like bound k

systematic encoders for generalized gabidulin codes and the q analogue of cauchy matrices abstract we characterize the generator matrix in standard form of generalized gabidulin codes the parametrization we get for the non systematic part of this matrix coincides with the q analogue of generalized cauchy matrices leading to the definition of q cauchy matrices these matrices can be represented very conveniently and their representation allows to define new interesting subfamilies of generalized gabidulin codes whose generator matrix is a structured matrix in particular as an application we construct gabidulin codes whose generator matrix is the concatenation of an identity block and a toeplitz hankel matrix in addition our results allow to give a new efficient criterion to verify whether a rank metric code of dimension k and length n is a generalized gabidulin code this criterion is only based on the computation of the rank of one matrix and on the verification of the linear independence of two sets of elements and it requires o m f k n field operations where f k n is the cost of computing the reduced row echelon form of a k n matrix moreover we also provide a characterization of the generator matrix in standard form of general mrd codes

a nonlinear age and maturity structured model of population dynamics ii chaos instability hypercyclicity and chaos are investigated in a nonlinear model of age and maturity structured population dynamics it is demonstrated that the behavior of solutions depends on the viewpoint of the observer viewed in the direction of age structure the population stabilizes to a regular distribution viewed in the direction of maturity structure there is the possibility of chaotic behavior

maximum rank distance codes as space time codes the critical design criterion for space time codes in asymptotically good channels is the minimum rank between codeword pairs rank codes are a two dimensional matrix code construction where by the rank is the metric of merit we look at the application of rank codes to space time code design in particular we provide construction methods of full rank codes over different complex signal constellations for arbitrary numbers of antennas and codeword periods we also derive a singleton type bound on the rate of a code for the rank metric and we show that rank codes satisfy this bound with equality

a model of the dynamics of intramolluscan trematode populations some problems concerning oscillatory behavior abstract a model of the development of a helminth population in a snail host is considered the model simulates the conditions of a real biological experiment it is based on the equations of nonlinear age dependent population dynamics and is investigated numerically using a difference scheme method an interesting oscillatory behavior dependence on certain parameters is established if the nonlinearity incorporates the population number of a given age group

error correcting regenerating and locally repairable codes via rank metric codes this paper presents and analyzes a novel concatenated coding scheme for enabling error resilience in two distributed storage settings one being storage using existing regenerating codes and the second being storage using locally repairable codes the concatenated coding scheme brings together a maximum rank distance code as an outer code and either a globally regenerating or a locally repairable code as an inner code in addition error resilience for combination of locally repairable codes with regenerating codes is considered this concatenated coding system is designed to handle two different types of adversarial errors the first type includes an adversary that can replace the content of an affected node only once while the second type studies an adversary that is capable of polluting data an unbounded number of times the paper establishes an upper bound on the resilience capacity for a locally repairable code this paper also proves that the proposed concatenated coding approach attains the upper bound on the resilience capacity in the presence of the first type of adversary for both minimum storage regenerating codes and locally repairable codes further this paper presents mechanisms that combine the presented concatenated coding scheme with subspace signatures to achieve error resilience for the second type of errors

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

a new family of mrd codes abstract we introduce a family of linear sets of pg 1 q 2 n arising from maximum scattered linear sets of pseudoregulus type of pg 3 q n for n 3 4 and for certain values of the parameters we show that these linear sets of pg 1 q 2 n are maximum scattered and they yield new mrd codes with parameters 6 6 q 5 for q 2 and with parameters 8 8 q 7 for q odd

global stability of a partial differential equation with distributed delay due to cellular replication in this paper we investigate a nonlinear partial differential equation arising from a model of cellular proliferation this model describes the production of blood cells in the bone marrow it is represented by a partial differential equation with a retardation of the maturation variable and a distributed temporal delay our aim is to prove that the behaviour of primitive cells influences the global behaviour of the population

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

a nonlinear age and maturity structured model of population dynamics i basic theory a nonlinear model of age and maturity structured population dynamics is analyzed the population is structured by age and maturity of the individuals and the nonlinearity in the equations corresponds to density dependent limitation of population growth the existence and asymptotic behavior of solutions are studied

numerical integration of a mathematical model of hematopoietic stem cell dynamics a mathematical model of hematopoiesis describing the dynamics of stem cell population is investigated this model is represented by a system of two nonlinear age structured partial differential equations describing the dynamics of resting and proliferating hematopoietic stem cells it differs from previous attempts to model the hematopoietic system dynamics by taking into account cell age dependence of coefficients that prevents a usual reduction of this system to an unstructured delay differential system we prove the existence and uniqueness of a solution to our problem and we investigate the existence of stationary solutions a numerical scheme adapted to the problem is presented we show the effectiveness of this numerical technique in the simulation of the dynamics of the solution numerical simulations show that long period oscillations can be obtained in this model corresponding to a destabilization of the system these oscillations can be related to observations of some periodical hematological diseases such as chronic myelogenous leukemia

a rank metric approach to error control in random network coding the problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of rotter and kschischang a large class of constant dimension subspace codes is investigated it is shown that codes in this class can be easily constructed from rank metric codes while preserving their distance properties moreover it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank metric codes where partial information about the error is available this partial information may be in the form of erasures knowledge of an error location but not its value and deviations knowledge of an error value but not its location taking erasures and deviations into account when they occur strictly increases the error correction capability of a code if mu erasures and delta deviations occur then errors of rank t can always be corrected provided that 2t les d 1 mu delta where d is the minimum rank distance of the code for gabidulin codes an important family of maximum rank distance codes an efficient decoding algorithm is proposed that can properly exploit erasures and deviations in a network coding application where n packets of length m over f q are transmitted the complexity of the decoding algorithm is given by o dm operations in an extension field f qn

equivalence and characterizations of linear rank metric codes based on invariants abstract we show that the sequence of dimensions of the linear spaces generated by a given rank metric code together with itself under several applications of a field automorphism is an invariant for the whole equivalence class of the code the same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism these invariants give rise to easily computable criteria to check if two codes are inequivalent we derive some concrete values and bounds for these dimension sequences for some known families of rank metric codes namely gabidulin and generalized twisted gabidulin codes we then derive conditions on the length of the codes with respect to the field extension degree such that codes from different families cannot be equivalent furthermore we derive upper and lower bounds on the number of equivalence classes of gabidulin codes and twisted gabidulin codes improving a result of schmidt and zhou for a wider range of parameters in the end we use the aforementioned sequences to determine a characterization result for gabidulin codes

propagation of population pulses and fronts in a cell replication problem non locality and dependence on the initial function abstract here we study the solution types in a simplified equation arising from a cell population dynamics model formulated as a non linear first order partial differential equation for the cell density u t x in the presence of non local maturational x interactions this can be considered as a reaction convection equation in which the cell density is convected with maturation velocity r the non local action affects only the reaction birth rate an initial function representing a pulse of small amplitude on an almost extinguished population may be dissipated may produce multiple pulses or may remain intact as it travels driven by reaction and convection depending on the magnitude of the non local action alternately for localized initial perturbations the equation also has positive traveling front solutions positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution for weak convection r 1 which is the only case treated here reaction driven fronts arise if the localized initial perturbation acts at a non zero maturation on the zero amplitude state a classification arises according to whether the magnitude of the maturation delay is larger or smaller than a critical value and scaling arguments are used to determine the critical delay for delays larger than critical the reaction driven fronts are oscillatory and correspond to the case for which the correlation length for non local action is of the same order as the characteristic length of maturation population gradients that yield oscillatory birth rates simulation results are validated and interpreted using scaling arguments and solutions for the local equations

a mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia this paper is devoted to the analysis of a mathematical model of blood cell production in the bone marrow hematopoiesis the model is a system of two age structured partial differential equations integrating these equations over the age we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration this system describes the evolution of the total cell populations by constructing a lyapunov functional it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium it is also shown that the nontrivial equilibrium the most biologically meaningful one can become unstable via a hopf bifurcation numerical simulations are carried out to illustrate the analytical results the study may be helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases

mrd codes arising from the trinomial xq xq3 cxq5 fq6 x abstract in 10 the existence of f q linear mrd codes of f q 6 6 with dimension 12 minimum distance 5 and left idealiser isomorphic to f q 6 defined by a trinomial of f q 6 x when q is odd and q 0 1 mod 5 has been proved in this paper we show that this family produces f q linear mrd codes of f q 6 6 with the same properties also in the remaining q odd cases but not in the q even case these mrd codes are not equivalent to the previously known mrd codes we also prove that the corresponding maximum scattered f q linear sets of pg 1 q 6 are not p l 2 q 6 equivalent to any previously known linear set