Characterization of {(q + 1) + 2, 1;t,q}-min · hypers and {2(q + 1) + 2, 2; 2,q}-min · hypers in a Finite projective geometry

In this paper, we shall characterize all {(q + 1) + 2, 1;t, q}-min · hypers and all {2(q + 1) + 2, 2; 2,q}-min · hypers for any integert 2 and any prime powerq 3. In the next paper [8], we shall characterize all {2(q + 1) + 2, 2;t, q}-min · hypers for any integert 3 and any prime powerq 5 using the results in this paper.     

On the construction of [q 4 + q 2 − q, 5,q 4 − q 3 + q 2 − 2q; q]-codes meeting the Griesmer bound

It is unknown whether or not there exists an [87, 5, 57; 3]-code. Such a code would meet the Griesmer bound. The purpose of this paper is to give a constructive proof of the existence of [q ⁴ + q ² – q, 5, q ⁴ – q ³ + q ² – 2q; q]-codes for any prime power q 3. As a special case, it is shown that there exists an [87, 5, 57; 3]-code with weight enumerator 1 + 156z ³⁷ + 82z ⁶⁰ + 2z ⁶³ + 2z ⁷⁸. The new construction settles an open problem due to Hill and Newton [10].     

Growth properties of the q-Dunkl transform in the space $$L^{p}_{q,\alpha }({\mathbb {R}}_{q},|x|^{2\alpha +1}d_{q}x)$$ L q , α p ( R q , | x | 2 α + 1 d q x )

The Ramanujan Journal - Our aim in this work is to prove an analogue of Titchmarsh’s theorem [19, Theorem 84] and Younis’s theorem [20, Theorem 3.3] on the image under the q-Dunkl...     

A new proof of the existence of (q 2−q+1)-arcs inPG(2,q 2)

Journal of Geometry -     

A family of translation planes of order q 2m+1 with two orbits of length 2 and q 2m+1−1 on l ∞

A class of translation planes of order q ^2m+1, where q is an odd prime power and m1, is constructed. If m=1, then this class is contained in the class of order q ³ constructed by Hiramine [5]. These planes of order q ^2m+1 are of dimension 2m+1 over their kernels. If q ^2m+13³, then the linear translation complements of these planes have two orbits of length 2 and q ^2m+1–1 on l and this class contains many planes which are not generalized André planes. If q ^2m+1= 3³, then each plane of this class is isomorphic to the Hering plane of order 27.Dedicated to Professor Tuyosi Oyama on his 60th birthday     

Characterization of G 2(q), where 2 < q ≡ −1(mod 3), by order components

We prove that the simple group G ₂(q), where 2 < q ≡ ?1(mod 3), is recognizable by the set of its order components. In other words, we prove that if G is a finite group with OC(G) = OC(G ₂(q)), then G ≌ G ₂(q).     

当$4q^{5}-5q^{4}-2q+1\leq d \leq 4q^{5}-5q^{4}-q$时$[g_{q}(6,d),6,d]_{q}$码的不存在性

证明了对于q≥17,当4q~5-5q~4-2q+1≤d≤4q~5-5q~4-q时,不存在达到Griesmer界的[n,k,d]_q码.此结果推广了Cheon等人在2005年和2008年的非存在性定理.     

The extension of the D(?k 2)-pair {k 2,k 2 + 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k ², k ²+1, c, d} is a D(?k ²)-quadruple with c < d, then c = 1 and d = 4k ²+1. This extends the work of the first author [20] and that of Dujella [4].     

On the Greatest Common Di visor of u − 1 and v − 1 with u and v Near ${\cal S}$- units

In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (aⁿ − 1,bⁿ − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f₁(x),g(x),g₁(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality holds for all but finitely many positive integers n.     

A necessary and sufficient condition for the biorthogonality of {ρ 1,ρ 2}

In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functionsρ ₁ andρ ₂. The functions are useful in the theory of biorthogonal wavelet.     

Characterizations of some two-dimensional cyclic codes correspond to the ideals of F[x,y]/

Two-dimensional cyclic code is one of the natural generalizations of cyclic code. In this paper we study the algebraic structure of some two-dimensional cyclic codes and their dual codes.     

On automorphisms of distance-regular graphs with intersection arrays {2<Emphasis Type="Italic">r</Emphasis> + 1, 2<Emphasis Type="Italic">r</Emphasis> − 2, 1; 1, 2, 2<Emphasis Type="Italic">r</Emphasis> + 1}

Let Γ be an antipodal graph with intersection array {2r+1, 2r?2, 1; 1, 2, 2r+1}, where 2r(r + 1) ≤ 4096. If 2r + 1 is a prime power, then Mathon’s scheme provides the existence of an arc-transitive graph with this intersection array. Note that 2r + 1 is not a prime power only for r ∈ {7, 17, 19, 22, 25, 27, 31, 32, 37, 38, 42, 43}. We study automorphisms of hypothetical distance-regular graphs with the specified values of r. The cases r ∈ {7, 17, 19} were considered earlier. We prove that, if Γ is a vertex-symmetric graph with intersection array {2r + 1, 2r ? 2, 1; 1, 2, 2r +1}, 2r + 1 is not a prime power, and r ≤ 43, then r = 25, 27, or 31.     

Characterization of Groups L2(q) by NSE Where q∈{17, 27, 29}

The authors show that linear simple groups L_2(q) with q ∈ {17, 27, 29} can be uniquely determined by nse(L_2(q)), which is the set of numbers of elements with the same order.     

Discrete decomposability of the restriction ofA q(λ) with respect to reductive subgroups and its applications

Summary LetGG be real reductive Lie groups and q a -stable parabolic subalgebra of Lie (G) . This paper offers a sufficient condition on (G, G, q) that the irreducible unitary representation ofG with non-zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupG, each of finite multiplicity. As an application to purely analytic problems, new results on discrete series are also obtained for some pseudo-Riemannian (non-symmetric) spherical homogeneous spaces, which fit nicely into this framework. Some explicit examples of a decomposition formula are also found in the cases whereA _q is not necessarily a highest weight module.Oblatum 3-IV-1993The author is supported by the NSF grant DMS-9100383.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.