Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes

The existence problem for a semicyclic group divisible design (SCGDD) of type m ⁿ with block size 4 and index unity, denoted by 4-SCGDD, has been studied for any odd integer m to construct a kind of two-dimensional optical orthogonal codes (2-D OOCs) with the AM-OPPW (at most one-pulse per wavelength) restriction. In this paper, the existence of a 4-SCGDD of type m ⁿ is determined for any even integer m, with some possible exceptions. A complete asymptotic existence result for k-SCGDDs of type m ⁿ is also obtained for all larger k and odd integer m. All these SCGDDs are used to derive new 2-D OOCs with the AM-OPPW restriction, which are optimal in the sense of their sizes.     

2-dimensional optical orthogonal codes from singer groups

We present several new families of (Λ×T,w,λ) (2D) wavelength/time optical orthogonal codes (2D-OOCs) with λ=1,2. All families presented are either optimal with respect to the Johnson bound (J-optimal) or are asymptotically optimal. The codes presented have more flexible dimensions and weight than the J-optimal families appearing in the literature. The constructions are based on certain pointsets in finite projective spaces of dimension k over GF(q) denoted PG(k,q). This finite geometries framework gives structure to the codes providing insight. We establish that all 2D-OOCs constructed are in fact maximal (in that no new codeword may be added to the original whereby code cardinality is increased).     

Combinatorial constructions for optimal 2-D optical orthogonal codes with AM-OPPTS property

We develop a new one-to-one correspondence between a two-dimensional (m × n, k, ρ) optical orthogonal code (2-D (m × n, k, ρ)-OOC) with AM-OPPTS (at most one-pulse per time slot) property and a certain combinatorial subject, called an n-cyclic holey packing of type m ⁿ . By this link, an upper bound on the size of a 2-D (m × n, k, ρ)-OOC with AM-OPPTS property is derived. Afterwards, we employ combinatorial methods to construct infinitely many 2-D (m × n, k, 1)-OOCs with AM-OPPTS property, whose existence was previously unknown. All these constructions meet the upper bounds with equality and are thus optimal.     

On optimal (v, 5, 2, 1) optical orthogonal codes

The size of a (v, 5, 2, 1) optical orthogonal code (OOC) is shown to be at most equal to ${\lceil{\frac{v}{12}}\rceil}$ when v ≡ 11 (mod 132) or v ≡ 154 (mod 924), and at most equal to ${\lfloor{\frac{v}{12}}\rfloor}$ in all the other cases. Thus a (v, 5, 2, 1)-OOC is naturally said to be optimal when its size reaches the above bound. Many direct and recursive constructions for infinite classes of optimal (v, 5, 2, 1)-OOCs are presented giving, in particular, a very strong indication about the existence of an optimal (p, 5, 2, 1)-OOC for every prime p ≡ 1 (mod 12).     

A powerful method for constructing difference families and optimal optical orthogonal codes

In this paper we proceed in the way indicated by R. M. Wilson for obtaining simple difference families from finite fields [28]. We present a theorem which includes as corollaries all the known direct techniques based on Galois fields, and provides a very effective method for constructing a lot of new difference families and also new optimal optical orthogonal codes.By means of our construction—just to give an idea of its power—it has been established that the only primesp<10⁵ for which the existence of a cyclicS(2, 9,p) design is undecided are 433 and 1009. Moreover we have considerably improved the lower bound on the minimumv for which anS(2, 15,v) design exists.     

New results on optimal (<Emphasis Type="Italic">v</Emphasis>, 4, 2, 1) optical orthogonal codes

We investigate further the existence question regarding optimal (v, 4, 2, 1) optical orthogonal codes begun in Momihara and Buratti (IEEE Trans Inform Theory 55:514–523, 2009). We give some non-existence results for infinitely many values of v ≡ ± 3 (mod 9) and several explicit constructions for infinite classes of perfect optical orthogonal codes.     

New optimal constructions of conflict-avoiding codes of odd length and weight 3

Conflict-avoiding codes (CACs) have played an important role in multiple-access collision channel without feedback. The size of a CAC is the number of codewords which equals the number of potential users that can be supported in the system. A CAC with maximal code size is said to be optimal. The use of an optimal CAC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, the maximal sizes of both equidifference and non equidifference CACs of odd prime length and weight 3 are obtained. Meanwhile, the optimal constructions of both equidifference and non equidifference CACs are presented. The numbers of equidifference and non equidifference codewords in an optimal code are also obtained. Furthermore, a new modified recursive construction of CACs for any odd length is shown. Non equidifference codes can be constructed in this method.     

Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter

A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously.     

Approximation properties of fourier series of Sobolev orthogonal polynomials with Jacobi weight and discrete masses

Mathematical Notes - We study Fourier series of Jacobi polynomials P k α?r,?r (x), k = r, r +1,..., orthogonal with respect to the Sobolev-type inner product of the following form:...     

Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four

Let K _q (n, w, t, d) be the minimum size of a code over Z _q of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine K _q (n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2 ^m  + 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of 3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs with block size four are also constructed.     

On equidistant binary codes of lengthn=4k+1 with distanced=2k

The aim of this paper is to provide a short proof of the main result (Theorem 2.12) of [3], using standard methods from the theory of combinatorial designs. This paper was submitted to Combinatorica at the request of the editors.     

非奇异H-矩阵新的含参数细分迭代判别法

结合矩阵自身的元素,构造了含参数的迭代公式,进而细分了矩阵非对角占优行指标集.利用广义严格α-对角占优矩阵与非奇异H-矩阵的关系,给出了非奇异H-矩阵一组新的细分迭代判定准则,推广和改进了已有的结果,通过数值算例说明了结果的优越性.     

On codes with covering radius 1 and minimum distance 2

We show that a code C of length n over an alphabet Q of size q with minimum distance 2 and covering radius 1 satisfies |C| ≥ qⁿ⁻¹/(n − 1). For the special case n = q = 4 the smallest known example has |C| = 31. We give a construction for such a code C with |C| = 28.     

Annihilators for cusp forms of weight 2 and level 4p m

We obtain operators, given essentially by formal sums of Hecke operators, that annihilate spaces of cusp forms of weight 2 for Γ ₁(p ^m )∩Γ(4), whose dimensions will be specified. Moreover, we obtain the principal part (mod p), over the cusps, of certain meromorphic modular functions of level 4p ^m .     

On the automorphism groups of the $$Z_2 Z_4$$-linear 1-perfect and Preparata-like codes

We consider the symmetry group of a \(Z_2Z_4\)-linear code with parameters of a 1-perfect, extended 1-perfect, or Preparata-like code. We show that, provided the code length is greater than 16, this group consists only of symmetries that preserve the \(Z_2Z_4\) structure. We find the orders of the symmetry groups of the \(Z_2Z_4\)-linear (extended) 1-perfect codes.     

Planar graphs with maximum degree 8 and without intersecting chordal 4-cycles are 9-totally colorable

The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ’’ (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ’’ (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ’’(G) = 9.