Another construction for large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection , where every is a KTS(v) and all form a partition of all triples on X. In this article, we give a new construction for LKTS(6v + 3) via OLKTS(2v + 1) with a special property and obtain new results for LKTS, that is there exists an LKTS(3v) for , where p, q ≥ 0, r _i , s _j ≥ 1, q _i is a prime power and mod 12.      

Existence of (2, 8) GWhD( v) and (4, 8) GWhD( v) with $${ v \equi v 0,1 (mod 8)}$$

(2, 8) Generalized Whist tournament Designs (GWhD) on v players exist only if . We establish that these necessary conditions are sufficient for all but a relatively small number of (possibly) exceptional cases. For there are at most 12 possible exceptions: {177, 249, 305, 377, 385, 465, 473, 489, 497, 537, 553, 897}. For there are at most 98 possible exceptions the largest of which is v = 3696. The materials in this paper also enable us to obtain four previously unknown (4, 8)GWhD(8n+1), namely for n = 16,60,191,192 and to reduce the list of unknown (4, 8) GWhD(8n) to 124 values of v the largest of which is v = 3696.      

A new extension theorem for 3-weight modulo q linear codes over $${\mathbb{F}_ q}$$

We prove that every [n, k, d] _q code with q ≥ 4, k ≥ 3, whose weights are congruent to 0, −1 or −2 modulo q and is extendable unless its diversity is for odd q, where .      

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of such that holds for any , , where . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if holds and any codeword has the form . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n’s admitting the condition through a number theoretical approach.      

Maximum distance separable poset codes

We derive the Singleton bound for poset codes and define the MDS poset codes as linear codes which attain the Singleton bound. In this paper, we study the basic properties of MDS poset codes. First, we introduce the concept of I-perfect codes and describe the MDS poset codes in terms of I-perfect codes. Next, we study the weight distribution of an MDS poset code and show that the weight distribution of an MDS poset code is completely determined. Finally, we prove the duality theorem which states that a linear code C is an MDS -code if and only if is an MDS -code, where is the dual code of C and is the dual poset of      

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

On tight projective designs

It is shown that among all tight designs in , where is or , or (quaternions), only 5-designs in (Lyubich, Shatalora Geom Dedicata 86: 169–178, 2001) have irrational angle set. This is the only case of equal ranks of the first and the last irreducible idempotent in the corresponding Bose-Mesner algebra.      

Block Primitive 2-(<Emphasis Type="Italic">v</Emphasis>,<Emphasis Type="Italic">k</Emphasis>,1) Designs Admitting a Ree Group of Characteristic Two

Let be a 2-(v,k,1) design, and let G be a group of automorphisms of . We show that if G is block primitive, then G does not admit a Ree group as its socle.     

Maximal partial packings of $${Z_2^n}$$ with perfect codes

A maximal partial Hamming packing of is a family of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in . The number of translates of Hamming codes in is the packing number, and a partial Hamming packing is strictly partial if the family does not constitute a partition of . A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of , n = 2 ^m −1, satisfies . It is also proved that for any n equal to 2 ^m −1, , there exist maximal strictly partial Hamming packings of with packing numbers n−10,n−9,n−8,...,n−1. This implies that the upper bound is tight for any n = 2 ^m −1, . All packing numbers for maximal strictly partial Hamming packings of , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.      

Efficient <Emphasis Type="Italic">p</Emphasis>th root computations in finite fields of characteristic <Emphasis Type="Italic">p</Emphasis>

We present a method for computing pth roots using a polynomial basis over finite fields of odd characteristic p, p ≥ 5, by taking advantage of a binomial reduction polynomial. For a finite field extension of our method requires p − 1 scalar multiplications of elements in by elements in . In addition, our method requires at most additions in the extension field. In certain cases, these additions are not required. If z is a root of the irreducible reduction polynomial, then the number of terms in the polynomial basis expansion of z ^1/p , defined as the Hamming weight of z ^1/p or , is directly related to the computational cost of the pth root computation. Using trinomials in characteristic 3, Ahmadi et al. (Discrete Appl Math 155:260–270, 2007) give is greater than 1 in nearly all cases. Using a binomial reduction polynomial over odd characteristic p, p ≥ 5, we find always.      

Two results on maximum nonlinear functions

Maximum nonlinear functions are widely used in cryptography because the coordinate functions F _β (x) := tr(β F(x)), , have large distance to linear functions. Moreover, maximum nonlinear functions have good differential properties, i.e. the equations F(x + a) − F(x) = b, , have 0 or 2 solutions. Two classes of maximum nonlinear functions are the Gold power functions , gcd(k, m) = 1, and the Kasami power functions , gcd(k, m) = 1. The main results in this paper are: (1) We characterize the Gold power functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2 in . (2) We determine the differential properties of the Kasami power functions if gcd(k,m) ≠ 1.      

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four

A ternary quasigroup (or 3-quasigroup) is a pair (N, q) where N is an n-set and q(x, y, z) is a ternary operation on N with unique solvability. A 3-quasigroup is called 2-idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x) = y. A conjugation of a 3-quasigroup, considered as an OA(3, 4, n), , is a permutation of the coordinate positions applied to the 4-tuples of . The subgroup of conjugations under which is invariant is called the conjugate invariant subgroup of . In this paper, we will complete the existence proof of the last undetermined infinite class of 2-idempotent 3-quasigroups of order n, n ≡ 1 (mod 4) and n > 9, with a conjugate invariant subgroup consisting of a single cycle of length four.      

LDPC codes from the Hermitian curve

In this paper, we study the code which has as parity check matrix the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in (Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that has a double cyclic structure and that by shortening in a suitable way it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix via a geometric approach.      

Minimizing total completion time for UET tasks with release time and outtree precedence constraints

Brucker et al. (Math Methods Oper Res 56: 407–412, 2003) have given an O(n ²)-time algorithm for the problems , outtree and , outtree . In this note, we show that their algorithm admits an O(n log n)-time implementation.     

Optimal grid holey packings with block size 3 and 4

The notion of a grid holey packing (GHP) was first proposed for the construction of constant composite codes. For a GHP (k, 1; n ×  g) of type [w ₁, . . . , w _g ], where , the fundamental problem is to determine the packing number N([w ₁, . . . , w _g ], 1; n ×  g), that is, the maximum number of blocks in such a GHP. In this paper we determine completely the values of N([w ₁, . . . , w _g ], 1; n ×  g) in the case of block size .      

Asymptotic symmetries for conformal scalar curvature equations with singularity

We give conditions on a positive Hölder continuous function C²such that every C ² positive solution u((x)) of the conformal scalar curvature equation in a punctured neighborhood of the origin in R ⁿ either has a removable singularity at the origin or satisfies for some positive singular solution u ₀(|x|) of where is the Hölder exponent of K.Mathematics Subject Classification (2000) Primary 35J60, 53C21     

Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

We consider the following implicit quasi-variational inequality problem: given two topological vector spaces E and F, two nonempty sets X E and C F, two multifunctions Γ : X → 2 ^X and Ф : X → 2 ^C , and a single-valued map ψ : , find a pair such that , Ф and for all . We prove an existence theorem in the setting of Banach spaces where no continuity or monotonicity assumption is required on the multifunction Ф. Our result extends to non-compact and infinite-dimensional setting a previous results of the authors (Theorem 3.2 of Cubbiotti and Yao [15] Math. Methods Oper. Res. 46, 213–228 (1997)). It also extends to the above problem a recent existence result established for the explicit case (C = E ^* and ).     

Necessary and Sufficient Conditions for the Solvability of the L p Dirichlet Problem on Lipschitz Domains

We study the homogeneous elliptic systems of order $2\ellWe study the homogeneous elliptic systems of order with real constant coefficients on Lipschitz domains in, . For any fixed p > 2, we show that a reverse H?lder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in L ^p . We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L ^p Dirichlet problem for and . The range of p is known to be sharp if and . For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if , and if .Research supported in part by the NSF.