linear authentication codes bounds and constructions in this paper we consider a new class of unconditionally secure authentication codes called linear authentication codes or linear a codes we show that a linear a code can be characterized by a family of subspaces of a vector space over a finite field we then derive an upper bound on the size of the source space when other parameters of the system that is the sizes of the key space and the authenticator space and the deception probability are fixed we give constructions that are asymptotically close to the bound and show applications of these codes in constructing distributed authentication systems

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

integrality of a ratio of petersson norms and level lowering congruences we prove integrality of the ratiof f g g outside an explicit finite set of primes where g is an arithmetically normalized holomorphic newform on a shimura curve f is a normalized hecke eigenform on gl 2 with the same hecke eigenvalues as g and denotes the petersson inner product the primes dividing this ratio are shown to be closely related to certain level lowering con gruences satisfied by f and to the central values of a family of rankin selberg l functions finally we give two applications the first to proving the integral ity of a certain triple product l value and the second to the computation of the faltings height of jacobians of shimura curves

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

new constructions of subspace codes using subsets of mrd codes in several blocks a basic problem for the constant dimension subspace coding is to determine the maximal possible size mathrm a q n d k of a set of k dimensional subspaces in mathrm f q n such that the subspace distance satisfies d u v 2k 2 dim u cap v geq d for any two different subspaces u and v in this set we present two new constructions of constant dimension subspace codes using subsets of maximum rank distance mrd codes in several blocks this method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting mrd codes in these two constructions subsets of mrd codes with bounded ranks play an essential role the delsarte theorem about the rank distribution of mrd codes is an important ingredient to count codewords in our constructed constant dimension subspace codes we give many new lower bounds for mathrm a q n d k more than 110 new constant dimension subspace codes better than previously best known codes are constructed

on metrics for error correction in network coding the problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model for coherent network coding where knowledge of the network topology and network code is assumed at the source and destination nodes the error correction capability of an outer code is succinctly described by the rank metric as a consequence it is shown that universal network error correcting codes achieving the singleton bound can be easily constructed and efficiently decoded for noncoherent network coding where knowledge of the network topology and network code is not assumed the error correction capability of a subspace code is given exactly by a new metric called the injection metric which is closely related to but different than the subspace metric of koumltter and kschischang in particular in the case of a non constant dimension code the decoder associated with the injection metric is shown to correct more errors then a minimum subspace distance decoder all of these results are based on a general approach to adversarial error correction which could be useful for other adversarial channels beyond network coding

packing and covering properties of subspace codes for error control in random linear network coding codes in the projective space and codes in the grassmannian over a finite field referred to as subspace codes and constant dimension codes cdcs respectively have been proposed for error control in random linear network coding for subspace codes and cdcs a subspace metric was introduced to correct both errors and erasures and an injection metric was proposed to correct adversarial errors in this paper we investigate the packing and covering properties of subspace codes with both metrics we first determine some fundamental geometric properties of the projective space with both metrics using these properties we then derive bounds on the cardinalities of packing and covering subspace codes and determine the asymptotic rates of optimal packing and optimal covering subspace codes with both metrics our results not only provide guiding principles for the code design for error control in random linear network coding but also illustrate the difference between the two metrics from a geometric perspective in particular our results show that optimal packing cdcs are optimal packing subspace codes up to a scalar for both metrics if and only if their dimension is half of their length up to rounding in this case cdcs suffer from only limited rate loss as opposed to subspace codes with the same minimum distance we also show that optimal covering cdcs can be used to construct asymptotically optimal covering subspace codes with the injection metric only

congruences between modular forms cyclic isogenies of modular elliptic curves and integrality of p adic l functions let f be a congruence subgroup of type ni n2 and of level n we study congruences between weight 2 normalized newforms f and eisenstein series e on f modulo a prime p above a rational prime p assume that p t 6n e is a common eigenfunction for all hecke operators and f is ordinary at p we show that the abelian variety associated to f and the cuspidal subgroup associated to e intersect non trivially in their p torsion points let a be a modular elliptic curve over q with good ordinary reduction at p we apply the above result to show that an isogeny of degree divisible by p from the optimal curve a1 in the q isogeny class of elliptic curves containing a to a extends to an etale morphism of neron models over 7p if p 7 we use this to show that p adic distributions associated to the p adic l functions of a are 7p valued

rank minimization over finite fields fundamental limits and coding theoretic interpretations this paper establishes information theoretic limits for estimating a finite field low rank matrix given random linear measurements of it these linear measurements are obtained by taking inner products of the low rank matrix with random sensing matrices necessary and sufficient conditions on the number of measurements required are provided it is shown that these conditions are sharp and the minimum rank decoder is asymptotically optimal the reliability function of this decoder is also derived by appealing to de caen s lower bound on the probability of a union the sufficient condition also holds when the sensing matrices are sparse a scenario that may be amenable to efficient decoding more precisely it is shown that if the n n sensing matrices contain on average nlog n entries the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field analogies are drawn between the aforementioned results and rank metric codes in the coding theory literature in fact we are also strongly motivated by understanding when minimum rank distance decoding of random rank metric codes succeeds to this end we derive minimum distance properties of equiprobable and sparse rank metric codes these distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one

on congruence modules related to hilbert eisenstein series we generalize the work of ohta on the congruence modules attached to elliptic eisenstein series to the setting of hilbert modular forms our work involves three parts in the first part we construct eisenstein series adelically and compute their constant terms by computing local integrals in the second part we prove a control theorem for one variable ordinary lambda adic hilbert modular forms following hida s work on the space of multivariable ordinary lambda adic hilbert cusp forms in part three we compute congruence modules related to hilbert eisenstein series through an analog of ohta s methods

on randomized linear network codes and their error correction capabilities randomized linear network code for single source multicast was introduced and analyzed in ho et al ieee transactions on information theory october 2006 where the main results are upper bounds for the failure probability of the code in this paper these bounds are improved and tightness of the new bounds is studied by analyzing the limiting behavior of the failure probability as the field size goes to infinity in the linear random coding setting for single source multicast the minimum distance of the code defined in zhang ieee transactions on information theory january 2008 is a random variable taking nonnegative integer values that satisfy the inequality in the singleton bound recently established in yeung and cai communications in information and systems 2006 for network error correction codes we derive a bound on the probability mass function of the minimum distance of the random linear network code based on our improved upper bounds for the failure probability codes having the highest possible minimum distance in the singleton bound are called maximum distance separable mds the bound on the field size required for the existence of mds codes reported in zhang ieee transactions on information theory january 2008 and matsumoto arxiv cs it 0610121 oct 2006 suggests that such codes exist only when field size is large define the degradation of a code as the difference between the highest possible minimum distance in the singleton bound and the actual minimum distance of the code the bound for the probability mass function of the minimum distance leads to a bound on the field size required for the existence of network error correction codes with a given maximum degradation the result shows that allowing minor degradation reduces the field size required dramatically

convolutional codes with maximum column sum rank for network streaming the column hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels we introduce a metric known as the column sum rank that parallels the column hamming distance when streaming over a network with link failures we prove the rank analogues of several known column hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory our construction involves finding a class of super regular matrices that preserve this property after multiplication with non singular block diagonal matrices in the ground field

new lmrd code bounds for constant dimension codes and improved constructions we prove upper bounds for the cardinality of constant dimension codes cdc which contain a lifted maximum rank distance lmrd code as a subset thereby we cover all parameters fulfilling k lt 3d 2 where k is the codeword dimension and d is the minimum subspace distance the proofs of the bounds additionally show that an lmrd code l can be unioned with a cdc c of fitting parameters without violating the subspace distance condition iff each codeword of c intersects the special subspace of l in at least dimension d 2 this connection is used to find the new largest and sometimes bound achieving cdcs for small parameters

optimal ferrers diagram rank metric codes optimal rank metric codes in ferrers diagrams are considered such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space first we consider rank metric anticodes and prove a code anticode bound for ferrers diagram rank metric codes the size of optimal linear anticodes is given four techniques and constructions of ferrers diagram rank metric codes are presented each providing optimal codes for different diagrams and parameters for which no optimal solution was known before the first construction uses maximum distance separable codes on the diagonals of the matrices the second one takes a subcode of a maximum rank distance code and the last two combine codes in small diagrams to a code in a larger diagram the constructions are analyzed and compared and unsolved diagrams are identified

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

eigenform product identities for hilbert modular forms we prove that amongst all real quadratic fields and all spaces of hilbert modular forms of full level and of weight 2 or greater the product of two hecke eigenforms is not a hecke eigenform except for finitely many real quadratic fields and finitely many weights we show that for mathbb q sqrt 5 there are exactly two such identities

error correcting codes in projective spaces via rank metric codes and ferrers diagrams coding in the projective space has received recently a lot of attention due to its application in network coding reduced row echelon form of the linear subspaces and ferrers diagram can play a key role for solving coding problems in the projective space in this paper we propose a method to design error correcting codes in the projective space we use a multilevel approach to design our codes first we select a constant weight code each codeword defines a skeleton of a basis for a subspace in reduced row echelon form this skeleton contains a ferrers diagram on which we design a rank metric code each such rank metric code is lifted to a constant dimension code the union of these codes is our final constant dimension code in particular the codes constructed recently by koetter and kschischang are a subset of our codes the rank metric codes used for this construction form a new class of rank metric codes we present a decoding algorithm to the constructed codes in the projective space the efficiency of the decoding depends on the efficiency of the decoding for the constant weight codes and the rank metric codes finally we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension

compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas in this paper for any d d dilation matrix m we demonstrate in a constructive way that we can construct compactly supported tight m wavelet frames and orthonormal m wavelet bases in l2 rd of exponential decay which are derived from compactly supported m refinable functions such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments this paper improves several results in battle comm math phys 110 1987 601 bownik j fourier anal appl 7 2001 489 grochenig and ron proc amer math soc 126 1998 1101 lemarie j math pures appl 67 1988 227 and strichartz constr approx 9 1993 327

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

linearized polynomial interpolation and its applications in this paper we first propose an interpolation algorithm in a well ordered free module of a linearized polynomial ring and then use this algorithm to decode several important families of codes gabidulin codes kotter and kschischang kk codes and mahdavifar and vardy mv codes our decoding algorithm for gabidulin codes is different from the polynomial reconstruction algorithm by loidreau when applied to decode kk codes our interpolation algorithm is equivalent to the sudan style list 1 decoding algorithm proposed by kotter and kschischang for kk codes the interpolation approach is also capable of solving the interpolation problem for the list decoding of mv codes proposed by mahdavifar and vardy and has a lower complexity than gaussian elimination an interpolator for list decoding of mv codes has also been implemented in hardware and the synthesis results show that it leads to better throughput and efficiency than gaussian elimination

modularity of fibres in rigid local systems k as is explained for example in w1 and w2 fontaine and mazur fm conjectured that this is always the case signi cantprogress on this conjecture was achieved w3 by proving particular instancesof the following lifting conjecture conjecture 1 1 suppose that l is odd and that the residual represen tation attached to is modular then itself is modular conjecture 1 1 is proved in w3 and tw when k q and the restriction of to the decomposition groups at the primes above l are semistable in the senseof ddt 2 4 this is enough using the primes l 3 and 5 to establish theshimura taniyama conjecture for semistable elliptic curves thanks to a keyresult of langlands and tunnell progressively stronger cases of conjecture 1 1were subsequently proved by di cdt fu and sw1 in sw1 skinnerand wiles obtain quite general results in the context where k is any totallyreal eld the principal assumption being that is ordinary at the primesabove l in this note we consider galois representations which occur in rigid fam ilies and establish their modularity under conjecture 1 1 this implies themodularity over suitable real abelian extensions of the galois representationsoccurring in the cohomology of the curvesy

universal secure network coding via rank metric codes the problem of securing a network coding communication system against an eavesdropper is considered the network implements linear network coding to deliver n packets from source to each receiver and the adversary can eavesdrop on arbitrarily chosen links the objective is to provide reliable communication to all receivers while guaranteeing that the source information remains information theoretically secure from the adversary a coding scheme is proposed that can achieve the maximum possible rate of n packets the scheme which is based on rank metric codes has the distinctive property of being universal it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying linear network code the only requirement of the scheme is that the packet length be at least n which is shown to be strictly necessary for universal communication at the maximum rate a further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network and the network may suffer from a rank deficiency of at most in this case the proposed scheme can be extended to achieve the rate of n 2t packets this rate is shown to be optimal under the assumption of zero error communication

hilbert modular forms and the gross stark conjecture let f be a totally real eld and an abelian totally odd character of f in 1988 gross stated a p adic analogue of stark s conjecture that relates the value of the derivative of the p adic l function associated to and the p adic logarithm of a p unit in the extension of f cut out by in this paper we prove gross s conjecture when f is a real quadratic eld and is a narrow ring class character the main result also applies to general totally real elds for which leopoldt s conjecture holds assuming that either there are at least two primes above p in f or that a certain condition relating the l invariants of and 1 holds this condition on l invariants is always satised when is quadratic

subspace polynomials and cyclic subspace codes subspace codes have received an increasing interest recently due to their application in error correction for random network coding in particular cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms in this paper we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits we further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials we present some constructions of such codes which are cyclic and analyze their parameters