On the maximality of linear codes

We show that if a linear code admits an extension, then it necessarily admits a linear extension. There are many linear codes that are known to admit no linear extensions. Our result implies that these codes are in fact maximal. We are able to characterize maximal linear (n, k, d) _q -codes as complete (weighted) (n, n − d)-arcs in PG(k − 1, q). At the same time our results sharply limit the possibilities for constructing long non-linear codes. The central ideas to our approach are the Bruen-Silverman model of linear codes, and some well known results on the theory of directions determined by affine point-sets in PG(k, q).      

Optimal linear perfect hash families with small parameters

A linear (q^d, q, t)‐perfect hash family of size s consists of a vector space V of order q^d over a field F of order q and a sequence ?₁,…,?_s of linear functions from V to F with the following property: for all t subsets X ? V, there exists i ∈ {1,·,s} such that ?_i is injective when restricted to F. A linear (q^d, q, t)‐perfect hash family of minimal size d( – 1) is said to be optimal. In this paper, we prove that optimal linear (q², q, 4)‐perfect hash families exist only for q = 11 and for all prime powers q > 13 and we give constructions for these values of q. © 2004 Wiley Periodicals, Inc. J Comb Designs 12: 311–324, 2004     

Branching Rules for General Linear Groups

In this article we give a branching rule for Harish–Chandra restriction from the general linear group Gl _n (q) to the Levi subgroup Gl _n?1(q) × Gl₁(q) in the case of the unipotent block.     

Curves with Many Points and Multiplication Complexity in Any Extension of q

From the existence of algebraic function fields having some good properties, we obtain some new upper bounds on the bilinear complexity of multiplication in all extensions of the finite field _q, where q is an arbitrary prime power. So we prove that the bilinear complexity of multiplication in the finite fields _qⁿ is linear uniformly in q with respect to the degree n.     

The maximum element order in the groups related to the linear groups which is a multiple of the defining characteristic

Let n be a natural number and q be the power of a prime p. The general, special and projective special linear groups are denoted by GL_n(q), SL_n(q) and PSL_n(q), respectively. In this paper we find the maximum order of an element of the above groups which is a multiple of p.     

Descent principle in modular Galois theory

We propound a descent principle by which previously constructed equations over GF(q _n)(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups. Currently this is achieved by starting with a vectorial (= additive)q-polynomial ofq-degreem with Galois group GL(m, q) and then, under suitable conditions, enlarging its Galois group to GL(m, q _n) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degreen. Elsewhere this was proved under certain conditions by using the classification of finite simple groups, and under some other conditions by using Kantor’s classification of linear groups containing a Singer cycle. Now under different conditions we prove it by using Cameron-Kantor’s classification of two-transitive linear groups.     

Strict Semimonotonicity Property of Linear Transformations on Euclidean Jordan Algebras

Motivated by the equivalence of the strict semimonotonicity property of the matrix A and the uniqueness of the solution to the linear complementarity problem LCP(A,q) for q∈R ₊ ⁿ , we study the strict semimonotonicity (SSM) property of linear transformations on Euclidean Jordan algebras. Specifically, we show that, under the copositive condition, the SSM property is equivalent to the uniqueness of the solution to LCP(L,q) for all q in the symmetric cone K. We give a characterization of the uniqueness of the solution to LCP(L,q) for a Z transformation on the Lorentz cone ℒ₊ ⁿ . We study also a matrix-induced transformation on the Lorentz space ℒ ⁿ .     

A removal lemma for systems of linear equations over finite fields

We prove a removal lemma for systems of linear equations over finite fields: let X ₁, …, X _m be subsets of the finite field F _q and let A be a (k × m) matrix with coefficients in F _q ; if the linear system Ax = b has o(q ^m−k ) solutions with x _i ∈ X _i , then we can eliminate all these solutions by deleting o(q) elements from each X _i . This extends a result of Green [Geometric and Functional Analysis 15 (2) (2005), 340–376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.     

Wiener��s Lemma for Infinite Matrices II

In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of B(l^q_w){\mathcal{B}}(\ell^{q}_{w}), the algebra of all bounded linear operators on the weighted sequence space l^q_w\ell^{q}_{w}, for any 1≤q<∞ and any discrete Muckenhoupt A _q -weight w.     

Cuspidal Modules as Summands of a Gel'fand–Graev Module

Abstract

Let G = GL _n (q) be the general linear group over a finite field 𝔽 _q with q elements. We call a Gel'fand–Graev module to be the module which affords the Gel'fand–Graev character defined in Definition I.1. It is known that every cuspidal module of G is isomorphic to a (unique) direct summand of a Gel'fand–Graev module. In this article, we investigate a certain endomorphism so that each irreducible cuspidal module is contained in a certain eigenspace corresponding to the cuspidal character. Furthermore, we determine the eigenvalue of that endomorphism by using character theory of finite general linear group.     

Sparsest solutions of underdetermined linear systems via -minimization for

We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal ℓ_q-quasinorm is also the sparsest one. This generalizes, and slightly improves, a similar result for the ℓ₁-norm. We then introduce a simple numerical scheme to compute solutions with minimal ℓ_q-quasinorm, and we study its convergence. Finally, we display the results of some experiments which indicate that the ℓ_q-method performs better than other available methods.     

Constructions and bounds on linear error-block codes

We obtain new bounds on the parameters and we give new constructions of linear error-block codes. We obtain a Gilbert–Varshamov type construction. Using our bounds and constructions we obtain some infinite families of optimal linear error-block codes over . We also study the asymptotic of linear error-block codes. We define the real valued function α _q,m,a (δ), which is an analog of the important real valued function α _q (δ) in the asymptotic theory of classical linear error-correcting codes. We obtain both Gilbert–Varshamov and algebraic geometry type lower bounds on α _q,m,a (δ). We compare these lower bounds in graphs.      

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL_n(q), SL_n(q), PGL_n(q) and PSL_n(q), respectively. Using conjugacy classes of elements in GL_n(q) in terms of irreducible polynomials over the finite field GF(q) we demonstrate how the set of order elements in GL_n(q) can be obtained. This will help to find the order of elements in the groups SL_n(q), PGL_n(q) and PSL_n(q). We also show an upper bound for the order of elements in SL_n(q).     

Linear quivers and the geometric setting of quantum GLn

This paper presents a connection between the defining basis presented by Beilinson-Lusztig-MacPherson [1] in their geometric setting for quantum GL_n and the isomorphism classes of linear quiver representations. More precisely, the positive part of the basis in [1] identifies with the defining basis for the relevant Ringel-Hall algebra; hence, it is a PBW basis in the sense of quantum groups. This approach extends to q-Schur algebras, yielding a monomial basis property with respect to the Drinfeld-Jimbo type presentation for the positive (or negative) part of the q-Schur algebra. Finally, the paper establishes an explicit connection between the canonical basis for the positive part of quantum GL_n and the Kazhdan-Lusztig basis for q-Schur algebras.     

The non-existence of Griesmer codes with parameters close to codes of Belov type

Hill and Kolev give a large class of q-ary linear codes meeting the Griesmer bound, which are called codes of Belov type (Hill and Kolev, Chapman Hall/CRC Research Notes in Mathematics 403, pp. 127–152, 1999). In this article, we prove that there are no linear codes meeting the Griesmer bound for values of d close to those for codes of Belov type. So we conclude that the lower bounds of d of codes of Belov type are sharp. We give a large class of length optimal codes with n _q (k, d) = g _q (k, d) + 1.     

A generalized numerical range: the range of a constrained sesquilinear form

Let M_n denote the algebra of all nxn complex matrices. For a given q?C with ∣Q∣≤1, we define and denote the q-numerical range of A?M_n by

W_q (A)={x ^? Ay:x,y?C ⁿ , _{x ^? x?y ^? y=1,x ^? y=q} }

The q-numerical radius is then given by r_q (A)=sup{∣z∣:z?W _q (A)}. When q=1,W _q (A) and r _q (A) reduce to the classical numerical range of A and the classical numerical radius of A, respectively. when q≠0, another interesting quantity associated with W _q (A) is the inner q-numerical radius defined by [rtilde] _q (A)=inf{∣z∣:z?W _q (A)}

In this paper, we describe some basic properties of W _q (A), extending known results on the classical numerical range. We also study the properties of r_q considered as a norm (seminorm if q=0) on M_n .Finally, we characterize those linear operators L on M_n that leave W_q ,r_q of [rtilde]_q invariant. Extension of some of our results to the infinite dimensional case is discussed, and open problems are mentioned.     

A sequence approach to linear perfect hash families

A linear (q ^d , q, t)-perfect hash family of size s in a vector space V of order q ^d over a field F of order q consists of a set of linear functionals from V to F with the following property: for all t subsets there exists such that is injective when restricted to F. A linear (q ^d , q, t)-perfect hash family of minimal size d(t − 1) is said to be optimal. In this paper, we extend the theory for linear perfect hash families based on sequences developed by Blackburn and Wild. We develop techniques which we use to construct new optimal linear (q ², q, 5)-perfect hash families and (q ⁴, q, 3)-perfect hash families. The sequence approach also explains a relationship between linear (q ³, q, 3)-perfect hash families and linear (q ², q, 4)-perfect hash families.      

Strong entropy for system of isentropic gas dynamics

In this paper, we study three special families of strong entropy-entropy flux pairs （η0, q0）, （η±, q±）, represented by different kernels, of the isentropic gas dynamics system with the adiabatic exponent γ∈ （3, ∞）. Through the perturbation technique through the perturbation technique, we proved, we proved the H^-1 compactness of ηit ＋ qix, i = 1, 2, 3 with respect to the perturbation solutions given by the Cauchy problem （6） and （7）, where （ηi, qi） are suitable linear combinations of （η0, q0）, （η±, q±）.     

Parabolic conjugacy in general linear groups

Let q be a power of a prime and n a positive integer. Let P(q) be a parabolic subgroup of the finite general linear group GL _n (q). We show that the number of P(q)-conjugacy classes in GL _n (q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of Alperin in (Commun. Algebra 34(3): 889–891, 2006)     

Fischer Matrices of the Affine Groups

Let GL_n(q) be the general linear group and let H_n ; V_n(q) · GL_n(q) denote the affine group of V_n(q). In [1] and [4], we determined Fischer matrices for the conjugacy classes of GL_n(q) where n = 2, 3, 4 and we obtained the number of conjugacy classes and irreducible characters of H₂, H₃, and H₄. In this paper, we find the Fischer matrices of the affine group H_n for arbitrary n.AMS Subject Classification Primary 20C15 Secondary 20C33