A generalized Gleason–Pierce–Ward theorem

The Gleason–Pierce–Ward theorem gives constraints on the divisor and field size of a linear divisible code over a finite field whose dimension is half of the code length. This result is a departure point for the study of self-dual codes. In recent years, additive codes have been studied intensively because of their use in additive quantum codes. In this work, we generalize the Gleason–Pierce–Ward theorem on linear codes over GF(q), q = p ^m , to additive codes over GF(q). The first step of our proof is an application of a generalized upper bound on the dimension of a divisible code determined by its weight spectrum. The bound is proved by Ward for linear codes over GF(q), and is generalized by Liu to any code as long as the MacWilliams identities are satisfied. The trace map and an analogous homomorphism on GF(q) are used to complete our proof.      

Bounds on Codes Derived by Counting Components in Varshamov Graphs

We are interested in improving the Varshamov bound for finite values of length n and minimum distance d. We employ a counting lemma to this end which we find particularly useful in relation to Varshamov graphs. Since a Varshamov graph consists of components corresponding to low weight vectors in the cosets of a code it is a useful tool when trying to improve the estimates involved in the Varshamov bound. We consider how the graph can be iteratively constructed and using our observations are able to achieve a reduction in the over-counting which occurs. This tightens the lower bound for any choice of parameters n, k, d or q and is not dependent on information such as the weight distribution of a code. This work is taken from the author’s thesis [10]     

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps G to the standard basis of . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space , which does not assume an “a priori” group algebra structure on . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes. Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.     

The polynomial degree of the Grassmannian G(1,n,q) of lines in finite projective space PG(n, q)

Let G: = G(1,n,q) denote the Grassmannian of lines in PG(n,q), embedded as a point-set in PG(N, q) with For n = 2 or 3 the characteristic function of the complement of G is contained in the linear code generated by characteristic functions of complements of n-flats in PG(N, q). In this paper we prove this to be true for all cases (n, q) with q = 2 and we conjecture this to be true for all remaining cases (n, q). We show that the exact polynomial degree of is for δ: = δ(n, q) = 0 or 1, and that the possibility δ = 1 is ruled out if the above conjecture is true. The result deg( for the binary cases (n,2) can be used to construct quantum codes by intersecting G with subspaces of dimension at least      

Some inequalities concerning the characteristic frequencies of elastic continuum

Summary We consider the boundary value problem αz″(x)+m(x)y(x)=0, αy″(x)+p(x)z(x)=0, xε[0, 1], y(0)=y(1)=z(0)=0, where the functions m(x) and p(x) are assumed integrable and positive everywhere in [0, 1]. As the main result we obtain the inequalities for n=1, 2, ... where δ_n(m, p) stands for the product of the first n eigenvalues α_i(m, p) of the above system and where δ_n(m) abbreviates δ_n(m, m). Entrata in Redazione il 6 febbraio 1976.     

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

On circulant self-dual codes over small fields

We construct self-dual codes over small fields with q = 3, 4, 5, 7, 8, 9 of moderate length with long cycles in the automorphism group. With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes.      

Raynaud vector bundles

We construct vector bundles on a smooth projective curve X having the property that for all sheaves E of slope μ and rank rk on X we have an equivalence: E is a semistable vector bundle . As a byproduct of our construction we obtain effective bounds on r such that the linear system |R·Θ| has base points on U _X (r, r(g − 1)).      

Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration (q) on the set (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions ₊(q) and ₋(q) of (q) to the set ₊(q) of secant (hyperbolic) lines and to the set ₋(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme ₋(q) is pseudocyclic.We further show that the coherent configurations (q ²) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme ₊(q ²), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes ₊(q ²) and ₋(q ²). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.     

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake     

Small weight codewords in the codes arising from Desarguesian projective planes

We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results of K. Chouinard on codewords of small weight in the codes arising from PG(2, p), p prime. Chouinard characterized all the codewords up to weight 2p in these codes. Using a particular basis for this code, described by Moorhouse, we characterize all the codewords of weight up to 2p + (p−1)/2 if p ≥ 11. We then study the codes arising from . In particular, for q ₀ = p prime, p ≥ 7, we prove that the codes have no codewords with weight in the interval [q + 2, 2q − 1]. Finally, for the codes of PG(2, q), q = p ^h , p prime, h ≥ 4, we present a discrete spectrum for the weights of codewords with weights in the interval [q + 2, 2q − 1]. In particular, we exclude all weights in the interval [3q/2, 2q − 1]. Geertrui Van de Voorde research is supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen) Joost Winne was supported by the Fund for Scientific Research - Flanders (Belgium).     

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

A weighted version of a result of Hamada on minihypers and on linear codes meeting the Griesmer bound

Minihypers were introduced by Hamada to investigate linear codes meeting the Griesmer bound. Hamada (Bull Osaka Women’s Univ 24:1–47, 1985; Discrete Math 116:229–268, 1993) characterized the non-weighted minihypers having parameters , with k−1 > λ₁ > λ₂ > ... > λ _h ≥ 0, as the union of a λ₁-dimensional space, λ₂-dimensional space, ..., λ _h -dimensional space, which all are pairwise disjoint. We present in this article a weighted version of this result. We prove that a weighted -minihyper , with k−1 > λ₁ > λ₂ > ... > λ _h ≥ 0, is a sum of a λ₁-dimensional space, λ₂-dimensional space, ..., and λ _h -dimensional space. This research was supported by the Project Combined algorithmic and theoretical study of combinatorial structures between the Fund for Scientific Research Flanders-Belgium (FWO-Flanders) and the Bulgarian Academy of Sciences. This research is also part of the FWO-Flanders project nr. G.0317.06 Linear codes and cryptography.     

Asymptotic stability for the Navier–Stokes equations in L n

Considering the Navier–Stokes equations in , we prove the asymptotic stability for weak solutions in the marginal class u ∈ C _B (0, ∞; L ⁿ ) with arbitrary initial and external perturbations.      

Stickelberger Codes

Let p be an odd prime and be a primitive p th root of unity over . The Galois group G of over is a cyclic group of order p-1. The integral group ring [G] contains the Stickelberger ideal S _p which annihilates the ideal class group of K. In this paper we investigate the parameters of cyclic codes S _p (q) obtained as reductions of S _p modulo primes q which we call Stickelberger codes. In particular, we show that the dimension of S _p (p) is related to the index of irregularity of p, i.e., the number of Bernoulli numbers B _2k , , which are divisible by p. We then develop methods to compute the generator polynomial of S _p (p). This gives rise to anew algorithm for the computation of the index of irregularity of a prime. As an application we show that 20,001,301 is regular. This significantly improves a previous record of 8,388,019 on the largest explicitly known regular prime.     

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.     

On the Minimum Size of Some Minihypers and Related Linear Codes

We denote by m_r,q(s) the minimum value of f for which an {f, _r-2+s ; r,q }-minihyper exists for r 3, 1 s q–1, where _j=(q^j+1–1)/(q–1). It is proved that m_3,q(s)=₁(₁+s) for many cases (e.g., for all q 4 when ) and that m_r,q(s) _r-1+s₁+q for 1 s q – 1,~q 3,~r 4. The nonexistence of some [n,k,n+s–q^k-2]_q codes attaining the Griesmer bound is given as an application.AMS classification: 94B27, 94B05, 51E22, 51E21     

Homomorphisms between JC＊-algebras and Lie C＊-algebras

It is shown that every almost ＊-homomorphism h ： A→B of a unital JC＊-algebra A to a unital JC＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x∈A, and that every almost linear mapping h ： A→B is a ＊-homomorphism when h（2^nu o y） - h（2^nu） o h（y）, h（3^nu o y） - h（3^nu） o h（y） or h（q^nu o y） = h（q^nu） o h（y） for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost ＊-homomorphism h ： A→B of a unital Lie C＊-algebra A to a unital Lie C＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x ∈A.