Realization of 2D convolutional codes of rate \frac{1}{n} by separable Roesser models

In this paper, two-dimensional convolutional codes constituted by sequences in $(\mathbb F ^n)^{\mathbb{Z }^{2}}$ where $\mathbb F $ is a finite field, are considered. In particular, we restrict to codes with rate $\frac{1}{n}$ and we investigate the problem of minimal dimension for realizations of such codes by separable Roesser models. The encoders which allow to obtain such minimal realizations, called R-minimal encoders, are characterized.     

Isolated Hypersurface Singularities and Special Polynomial Realizations of Affine Quadrics

Let V, $\tilde{V}$ be hypersurface germs in ? ^m , each having a quasi-homogeneous isolated singularity at the origin. We show that the biholomorphic equivalence problem for V, $\tilde{V}$ reduces to the linear equivalence problem for certain polynomials P, $\tilde{P}$ arising from the moduli algebras of V, $\tilde{V}$ . The polynomials P, $\tilde{P}$ are completely determined by their quadratic and cubic terms, hence the biholomorphic equivalence problem for V, $\tilde{V}$ in fact reduces to the linear equivalence problem for pairs of quadratic and cubic forms.     

Rate of P-convergence over equivalence classes of double sequence spaces

This paper presents multidimensional characterizations of equivalence classes of double sequences. These characterizations will be analyzed by presenting theorems of the following type: If A is convergent preserving over ${\{x \in S^{\prime\prime}_{0} : x \;{\rm equivalent \; to}\; b\}}$ then there exists a ${b^{\prime} \in S^{\prime\prime}_{0}}$ such that b converges faster than b?? and A is convergent preserving over ${\{x \in S^{\prime\prime}_{0} : x \; {\rm equivalent \; to}\; b^{\prime}\}.}$      

Computational results of duadic double circulant codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb{F}_{2}$ , $\mathbb{F}_{3}$ , $\mathbb{F}_{4}$ , $\mathbb{F}_{5}$ , and $\mathbb{F}_{7}$ . In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.     

Fast decoding of Gabidulin codes

Gabidulin codes are the analogues of Reed–Solomon codes in rank metric and play an important role in various applications. In this contribution, a method for efficient decoding of Gabidulin codes up to their error correcting capability is shown. The new decoding algorithm for Gabidulin codes (defined over ${\mathbb{F}_{q^m}}$ ) directly provides the evaluation polynomial of the transmitted codeword. This approach can be seen as a Gao-like algorithm and uses an equivalent of the Euclidean Algorithm. In order to achieve low complexity, a fast symbolic product and a fast symbolic division are presented. The complexity of the whole decoding algorithm for Gabidulin codes is ${\mathcal{O} (m^3 \, \log \, m)}$ operations over the ground field ${\mathbb{F}_q}$ .     

Tiling Groupoids and Bratteli Diagrams

Let T be an aperiodic and repetitive tiling of ${{\mathbb R}^d}$ with finite local complexity. Let Ω be its tiling space with canonical transversal ${\Xi}$ . The tiling equivalence relation ${R_\Xi}$ is the set of pairs of tilings in ${\Xi}$ which are translates of each others, with a certain (étale) topology. In this paper ${R_\Xi}$ is reconstructed as a generalized “tail equivalence” on a Bratteli diagram, with its standard AF -relation as a subequivalence relation. Using a generalization of the Anderson–Putnam complex (Bellissard et al. in Commun. Math. Phys. 261:1–41, 2006) Ω is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram ${{\mathcal B}}$ is built from this sequence, and its set of infinite paths ${\partial {\mathcal B}}$ is homeomorphic to ${\Xi}$ . The diagram ${{\mathcal B}}$ is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an étale equivalence relation ${R_{\mathcal B}}$ on ${\partial {\mathcal B}}$ which is homeomorphic to ${R_\Xi}$ , and contains the AF-relation of “tail equivalence”.     

On cyclic compositions of positive integers

Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let ${\left\langle \begin{array}{c}n \\ k\end{array} \right\rangle}$ denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n into k parts. We show that the sequence ${\left\langle\begin{array}{c}n \\ k\end{array}\right\rangle}$ is log-concave and prove some results concerning ${\left\langle \begin{array}{c}n \\ k \end{array} \right\rangle}$ modulo two.     

Trails of triples in partial triple systems

Given v, t, and m, does there exist a partial Steiner triple system of order v with t triples whose triples can be ordered so that any m consecutive triples are pairwise disjoint? Given v, t, and m ₁, m ₂, . . . , m _s with ${t = \sum_{i=1}^s m_i}$ , does there exist a partial Steiner triple system with t triples whose triples can be partitioned into partial parallel classes of sizes m ₁, . . . , m _s ? An affirmative answer to the first question gives an affirmative answer to the second when m _i ≤ m for each ${i \in \{1,2,\ldots,s\}}$ . These questions arise in the analysis of erasure codes for disk arrays and that of codes for unipolar communication, respectively. A complete solution for the first problem is given when m is at most ${\frac{1}{3}\left(v-(9v)^{2/3}\right)+{O}\left(v^{1/3}\right)}$ .     

Path Methods for Strong Shift Equivalence of Positive Matrices

In the early 1990’s, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring $\mathcal{U}$ of ?. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings $\mathcal{U}$ of ? are used to show that for any dense subring $\mathcal{U}$ of ?, positive matrices over $\mathcal{U}$ which have just one nonzero eigenvalue and which are strong shift equivalent over $\mathcal{U}$ must be strong shift equivalent over $\mathcal{U}_{+}$ . In addition, we show matrices on a path of positive shift equivalent real matrices are SSE over ?₊; positive rational matrices which are SSE over ?₊ must be SSE over ?₊; and for any dense subring $\mathcal{U}$ of ?, within the set of positive matrices over $\mathcal{U}$ which are conjugate over $\mathcal{U}$ to a given matrix, there are only finitely many SSE- $\mathcal{U}_{+}$ classes.     

Characterizing Serre Quotients with no Section Functor and Applications to Coherent Sheaves

We prove an analogon of the the fundamental homomorphism theorem for certain classes of exact and essentially surjective functors of Abelian categories $\mathcal{Q}:\mathcal{A} \to \mathcal{B}$ . It states that $\mathcal{Q}$ is up to equivalence the Serre quotient $\mathcal{A} \to \mathcal{A} / \ker \mathcal{Q}$ , even in cases when the latter does not admit a section functor. For several classes of schemes X, including projective and toric varieties, this characterization applies to the sheafification functor from a certain category $\mathcal{A}$ of finitely presented graded modules to the category $\mathcal{B}=\mathfrak{Coh}\, X$ of coherent sheaves on X. This gives a direct proof that $\mathfrak{Coh}\, X$ is a Serre quotient of $\mathcal{A}$ .     

On enumeration of polynomial equivalence classes

The isomorphism of polynomials(IP),one of the hard problems in multivariate public key cryptography induces an equivalence relation on a set of systems of polynomials.Then the enumeration problem of IP consists of counting the numbers of different classes and counting the cardinality of each class that is highly related to the scale of key space for a multivariate public key cryptosystem.In this paper we show the enumeration of the equivalence classes containing ∑n-1 i=0 aiX2qi when char(Fq) = 2,which implies that these polynomials are all weak IP instances.Moreover,we study the cardinality of an equivalence class containing the binomial aX 2q i + bX 2qj(i=j) over Fqn without the restriction that char(Fq) = 2,which gives us a deeper understanding of finite geometry as a tool to investigate the enumeration problem of IP.     

The universal Glivenko–Cantelli property

Let ${\mathcal{F}}$ be a separable uniformly bounded family of measurable functions on a standard measurable space ${(X, \mathcal{X})}$ , and let ${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$ be the smallest number of ${\varepsilon}$ -brackets in L ¹(μ) needed to cover ${\mathcal{F}}$ . The following are equivalent:

1. ${\mathcal{F}}$ is a universal Glivenko–Cantelli class.
2. ${N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}$ for every ${\varepsilon > 0}$ and every probability measure μ.
3. ${\mathcal{F}}$ is totally bounded in L ¹(μ) for every probability measure μ.
4. ${\mathcal{F}}$ does not contain a Boolean σ-independent sequence.

It follows that universal Glivenko–Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.     

Division by Flat Ultradifferentiable Functions and Sectorial Extensions

We consider classes ${cal A}$ _m(S) of functions holomorphic in an open plane sector S and belonging to a strongly non-quasianalytic class on the closure of S. In ${cal A}$ _m (S), we construct functions which are flat at the vertex of S with a sharp rate of. vanishing. This allows us to obtain a Borel-Ritt type theorem for ${cal A}$ _m(S) extending previous results by Schmets and Valdivia. We also derive a division property for ideals of flat ultradifferentiable functions, in the spirit of a classical C _∞ result of Tougeron.     

Improved bounds for separating hash families

An ${(N;n,m,\{w_1,\ldots, w_t\})}$ -separating hash family is a set ${\mathcal{H}}$ of N functions ${h: \; X \longrightarrow Y}$ with ${|X|=n, |Y|=m, t \geq 2}$ having the following property. For any pairwise disjoint subsets ${C_1, \ldots, C_t \subseteq X}$ with ${|C_i|=w_i, i=1, \ldots, t}$ , there exists at least one function ${h \in \mathcal{H}}$ such that ${h(C_1), h(C_2), \ldots, h(C_t)}$ are pairwise disjoint. Separating hash families generalize many known combinatorial structures such as perfect hash families, frameproof codes, secure frameproof codes, identifiable parent property codes. In this paper we present new upper bounds on n which improve many previously known bounds. Further we include constructions showing that some of these bounds are tight.     

Length of Geodesics and Quantitative Morse Theory on Loop Spaces

Let M ⁿ be a closed Riemannian manifold of diameter d. Our first main result is that for every two (not necessarily distinct) points ${p,q \in M^n}$ and every positive integer k there are at least k distinct geodesics connecting p and q of length ${\leq 4nk^2d}$ . We demonstrate that all homotopy classes of M ⁿ can be represented by spheres swept-out by “short” loops unless the length functional has “many” “deep” local minima of a “small” length on the space ${\Omega_{pq}M^n}$ of paths connecting p and q. For example, one of our results implies that for every positive integer k there are two possibilities: Either the length functional on ${\Omega_{pq} M^n}$ has k distinct non-trivial local minima with length ${\leq 2kd}$ and “depth” ${\geq 2d}$ ; or for every m every map of S ^m into ${\Omega_{pq}M^n}$ is homotopic to a map of S ^m into the subspace ${\Omega_{pq}^{4(k+2)(m+1)d}M^n}$ of ${\Omega_{pq}M^n}$ that consists of all paths of length ${\leq 4(k+2)(m+1)d}$ .     

Bounds for codes and designs in complex subspaces

We introduce the concepts of complex Grassmannian codes and designs. Let $\mathcal{G}_{m,n}$ denote the set of m-dimensional subspaces of ? ⁿ : then a code is a finite subset of $\mathcal{G}_{m,n}$ in which few distances occur, while a design is a finite subset of $\mathcal{G}_{m,n}$ that polynomially approximates the entire set. Using Delsarte’s linear programming techniques, we find upper bounds for the size of a code and lower bounds for the size of a design, and we show that association schemes can occur when the bounds are tight. These results are motivated by the bounds for real subspaces recently found by Bachoc, Bannai, Coulangeon and Nebe, and the bounds generalize those of Delsarte, Goethals and Seidel for codes and designs on the complex unit sphere.     

Separable forms of {{\mathbb{G}}_m} -ACTIONS ON {{\mathbb{A}}^3}

Let k be a field of characteristic zero. We consider k-forms of $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ and show that they are linearizable. In particular, $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ are linearizable, and k-forms of $ {\mathbb A} $ ³ that admit an effective action of an infinite reductive group are trivial.     

Nielsen coincidence,fixed point and root theories of n-valued maps

Let ${(\phi, \psi)}$ be an (m, n)-valued pair of maps ${\phi, \psi : X \multimap Y}$ , where ${\phi}$ is an m-valued map and ${\psi}$ is n-valued, on connected finite polyhedra. A point ${x \in X}$ is a coincidence point of ${\phi}$ and ${\psi}$ if ${\phi(x) \cap \psi(x) \neq \emptyset}$ . We define a Nielsen coincidence number ${N(\phi : \psi)}$ which is a lower bound for the number of coincidence points of all (m, n)-valued pairs of maps homotopic to ${(\phi, \psi)}$ . We calculate ${N(\phi : \psi)}$ for all (m, n)-valued pairs of maps of the circle and show that ${N(\phi : \psi)}$ is a sharp lower bound in that setting. Specifically, if ${\phi}$ is of degree a and ${\psi}$ of degree b, then ${N(\phi : \psi) = \frac{|an - bm|}{\langle m, n \rangle}}$ , where ${\langle m, n \rangle}$ is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for n-valued maps of compact connected Lie groups that relate the Nielsen fixed point number of Helga Schirmer to the Nielsen root number of Michael Brown.     

Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .