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Let f r (n , v , e) denote the maximum number of edges in an r -uniform hypergraph on n vertices, in which the union of any e distinct edges contains at least v + 1 vertices. The study of f r (n , v , e) was initiated by Brown, Erdős and Sós more than forty years ago. In the literature, the following conjecture is well known. Conjecture: n k − o (1) < f r (n , e r − (e − 1) k + 1 , e) = o (n k) holds for all fixed integers r > k ≥ 2 and e ≥ 3 as n → ∞. For r = 3 , e = 3 , k = 2 , the bound n 2 − o (1) < f 3 (n , 6 , 3) = o (n 2) was proved by the celebrated (6,3)-theorem of Ruzsa and Szemerédi. In this paper, we add more evidence for the validity of the conjecture. On one hand, using the hypergraph removal lemma we show that the upper bound part of the conjecture is true for all fixed integers r ≥ k + 1 ≥ e ≥ 3. On the other hand, using tools from additive number theory we present several constructions showing that the lower bound part of the conjecture is true for r ≥ 3 , k = 2 and e = 4 , 5 , 7 , 8. Prior to our results, all known constructions that match the conjectured lower bound satisfy either r = 3 or e = 3. Our constructions are the first ones in the literature that break this barrier. [ABSTRACT FROM AUTHOR]

In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius r ≥ 2 and dimension n ≥ 3. A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (2011) [5] proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with distance greater than four besides the D P L (3 , 6) code? Later, Horak and AlBdaiwi (2012) [12] conjectured that there are no D P L (n , d) codes for dimension n ≥ 3 and distance d > 4 except for (n , d) = (3 , 6). In this paper, we give a counterexample to this conjecture. Moreover, we prove that for n ≥ 3 , there is a linear D P L (n , 6) code if and only if n = 3 , 11. [ABSTRACT FROM AUTHOR]

In this paper, we study problems about the similar configurations in F q d. Let G = (V , E) be a graph, where V = { 1 , 2 , ... , n } and E ⊆ ( V 2 ). For a set E in F q d , we say that E contains a pair of G with dilation ratio r if there exist distinct x 1 , x 2 , ... , x n ∈ E and distinct y 1 , y 2 , ... , y n ∈ E such that ‖ y i − y j ‖ = r ‖ x i − x j ‖ ≠ 0 whenever { i , j } ∈ E , where ‖ x ‖ : = x 1 2 + x 2 2 + ⋯ + x d 2 for x = (x 1 , x 2 , ... , x d) ∈ F q d. We show that if E has size at least C k q d / 2 , then E contains a pair of k -stars with dilation ratio r , and that if E has size at least C ⋅ min ⁡ { q (2 d + 1) / 3 , max ⁡ { q 3 , q d / 2 } } , then E contains a pair of 4-paths with dilation ratio r. Our method is based on enumerative combinatorics and graph theory. [ABSTRACT FROM AUTHOR]

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict avoiding codes. In this paper, we prove some nonexistence results for nonsingular perfect splitter sets. We also give some necessary conditions for the existence of purely singular perfect splitter sets. Finally, we apply these results to purely singular perfect $B[-1,k](m)$ and $B[-2,k](m)$ sets for small $k$. In particular, we solve completely the problems left by Schwartz (European J. Combin., vol. 36, pp.130–142, Feb. 2014). [ABSTRACT FROM AUTHOR]

Let H be an r -uniform hypergraph. The Turán number ex (n , H) is the maximum number of edges in an n -vertex H -free r -uniform hypergraph. The Turán density of H is defined by π (H) = lim n → ∞ ⁡ ex (n , H) ( n r ). In this paper, we consider the Turán density of projective geometries. We give two new constructions of P G m (q) -free hypergraphs which improve some results given by Keevash (2005) [14]. Based on an upper bound of blocking sets of P G m (q) , we give a new general lower bound for the Turán density of P G m (q). By a detailed analysis of the structures of complete arcs in P G 2 (q) , we also get better lower bounds for the Turán density of P G 2 (q) with q = 3 , 4 , 5. [ABSTRACT FROM AUTHOR]

Abstract: Group divisible covering designs (GDCDs) were introduced by Heinrich and Yin as a natural generalization of both covering designs and group divisible designs. They have applications in software testing and universal data compression. The minimum number of blocks in a k‐GDCD of type g u is a covering number denoted by C ( k , g u ). When k = 3, the values of C ( 3 , g u ) have been determined completely for all possible pairs ( g , u ). When k = 4, Francetić et al. constructed many families of optimal GDCDs, but the determination remained far from complete. In this paper, two specific 4‐IGDDs are constructed, thereby completing the existence problem for 4‐IGDDs of type ( g , h ) u. Then, additional families of optimal 4‐GDCDs are constructed. Consequently the cases for ( g , u ) whose status remains undetermined arise when g ≡ 7 mod 12 and u ≡ 3 mod 6, when g ≡ 11 , 14 , 17 , 23 mod 24 and u ≡ 5 mod 6, and in several small families for which one of g and u is fixed. [ABSTRACT FROM AUTHOR]

The research on directed PBDs is motivated by the construction of t-deletion/insertion-correcting codes. Fuji-Hara, Miao, Wang, and Yin have determined the existence of directed PBDs with block sizes from the set [ABSTRACT FROM AUTHOR]

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form x + γ Tr q n / q ( x k ) is also presented, which generalizes two examples of [10] . [ABSTRACT FROM AUTHOR]

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k . Such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . The number of points of the URD is denoted by v . In the literature, the existence of URDs of block sizes k 1 and k 2 with { k 1 , k 2 } ∈ { { 2 , 3 } , { 2 , 4 } , { 3 , 4 } } has been studied with much efforts. For { k 1 , k 2 } = { 2 , 3 } or { 3 , 4 } , the existence spectra have been determined completely, while for { k 1 , k 2 } = { 2 , 4 } there are still 1338 undetermined pairs of ( v , r 2 ) . In this paper we continue the study on the existence of URDs of block sizes 2 and 4. We improve the known results by reducing the number of open cases from 1338 to 16 . In addition, we prove that there exist 4 -RGDDs of types 2 178 , 2 250 and 2 334 . [ABSTRACT FROM AUTHOR]

A long-standing conjecture of Golomb and Welch, raised in 1970, states that there is no perfect r error correcting Lee code of length n for n\geq 3 and r>1 under the l_{p} metric, where 1\leq p<\infty . We show some nonexistence results of linear perfect lp codes for p=1 and 2\leq p<\infty , r=2^{1/p},3^{1/p} . We also give an algebraic construction of quasi-perfect l_{p}$ codes for p=1, r=2$ , and 2\leq p<\infty , r=2^{1/p} . [ABSTRACT FROM PUBLISHER]

Given n items with at most d of which being positive, instead of testing these items individually, the theory of combinatorial group testing aims to identify all positive items using as few tests as possible. This paper is devoted to a fundamental and thirty-year-old problem in the nonadaptive group testing theory. A binary matrix is called d -disjunct if the Boolean sum of arbitrary d columns does not contain another column not in this collection. Let T(d) denote the minimal t , such that there exists a t\times n\,\,d -disjunct matrix with n>t and was conjectured that T(d)\ge (d+1)^{2} . In this paper, we narrow the gap by proving T(d)/d^{2}\ge (15+\sqrt {33})/24$ , a quantity in [6/7,7/8]. [ABSTRACT FROM PUBLISHER]

One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature. [ABSTRACT FROM PUBLISHER]

Snake-in-the-box codes under Kendall’s \tau -metric are studied in the rank modulation scheme for flash memories, where codewords are a subset of permutations in S_{n} with minimal Kendall’s \tau -distance two, and two cyclically consecutive codewords are connected via a push-to-the-top operation. Studies so far restrict the push-to-the-top operations only on odd indices, resulting in a snake consisting of permutations with the same parity, and thus, the minimal distance constraint is easily satisfied. Asymptotically optimal snake codes have been constructed this way in S2n+1 . As for S2n+2 , this framework keeps the last element fixed, and thus, a snake in S2n+2 is equivalent to a snake in S2n+1 , which is rather trivial. If one wants to do better, then it is inevitable to have some push-to-the-top operations on even indices, resulting in a combination of odd and even permutations in the snake, which increases the difficulty to guarantee the minimal Kendall’s $\tau $ -distance constraint. Thus, Horovitz and Etzion pose the open problem to prove or disprove that the size of the largest snake in ({1}/{4})|S_{2n+2}| . [ABSTRACT FROM PUBLISHER]

Some physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This motivated the application of the asymmetric limited magnitude error model in flash memory. In this paper, we present a new construction of quasi-perfect codes for such errors, and we also study the perfect codes with symmetric errors. Moreover, we show some nonexistence results on perfect codes for correcting single error of limited magnitude. [ABSTRACT FROM PUBLISHER]

The construction of group divisible designs (GDDs) is a basic problem in design theory. While there have been some methods concerning the constructions of uniform GDDs, the construction of nonuniform GDDs remains a challenging problem. In this paper, we present a new approach to the construction of nonuniform GDDs with group type [ABSTRACT FROM AUTHOR]

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k . Such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ≡ 0 ( mod 12 ) . It has been shown that almost all URDs with permissible r 3 and r 4 exist for v ≡ 0 ( mod 24 ) , v ≡ 0 ( mod 60 ) , v ≡ 36 ( mod 144 ) or v ≡ 36 ( mod 108 ) . In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v = 12 , r 3 = 1 and r 4 = 3 . [ABSTRACT FROM AUTHOR]

A triple system is a collection of b blocks, each of size three, on a set of v points. It is j-balanced when every two j -sets of points appear in numbers of blocks that are as nearly equal as possible, and well balanced when it is j-balanced for each j ∈ {1, 2, 3}.Well-balanced systems arise in the minimization of variance in file availability in distributed file systems. It is shown that when a triple system that is 2-balanced and 3-balanced exists, so does one that is well balanced. Using known and new results on variants of group divisible designs, constructions for well-balanced triple systems are developed. Using these, the spectrum of pairs (v, b) for which such a well-balanced triple system exists is determined completely. [ABSTRACT FROM AUTHOR]

For a Gray code in the scheme of rank modulation for flash memories, the codewords are permutations, and two consecutive codewords are obtained using a push-to-the-top operation. We consider the snake-in-the-box code under Kendall’s \tau -metric, which is a Gray code capable of detecting one Kendall’s \tau -error. We answer two open problems posed by Horovitz and Etzion. First, we prove the validity of a construction given by them, resulting in a snake of size M2n+1=({(2n+1)!}/{2})-2n+1 . Second, we come up with a different construction aiming at a larger snake of size M2n+1=({(2n+1)!}/{2})-2n+3 . The construction is applied successfully to S7 . [ABSTRACT FROM PUBLISHER]

Quantum maximum-distance-separable (MDS) codes form an important family of quantum codes. In this paper, using Hermitian construction and classical constacyclic codes, we construct six classes of quantum MDS codes. Two of these six classes of quantum MDS codes have larger minimum distance than the ones available in the literature. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

Grooming uniform all-to-all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C-groomings has been considered for [ABSTRACT FROM AUTHOR]

Quadratic residue codes are a well-known class of codes. In this paper, we consider the constructions of self-dual codes by higher power residues, especially fourth power residues. New infinite families of self-dual codes over GF(2), GF(3), GF(4), GF(8), and GF(9) are introduced. Some of them have better minimum weight than previously known codes. We also give general results related to the automorphism group of some of these codes. [ABSTRACT FROM PUBLISHER]

Let R k (H ; K m) be the smallest number N such that every coloring of the edges of K N with k + 1 colors has either a monochromatic H in color i for some 1 ⩽ i ⩽ k , or a monochromatic K m in color k + 1. In this short note, we study the lower bound for R k (H ; K m) when H is C 5 or C 7 , respectively. We show that R k (C 5 ; K m) = Ω (m 3 k 8 + 1 / (log ⁡ m) 3 k 8 + 1) , and R k (C 7 ; K m) = Ω (m 2 k 9 + 1 / (log ⁡ m) 2 k 9 + 1) , for fixed positive integer k and m → ∞. The proof is based on random block constructions of Mubayi and Verstraëte, who obtained comparable bounds when k = 1 , and random blowups argument. Our results slightly improve the previously known lower bound R k (C 2 ℓ + 1 ; K m) = Ω (m k 2 ℓ − 1 + 1 / (log ⁡ m) k + 2 k 2 ℓ − 1 ) obtained by Alon and Rödl. [ABSTRACT FROM AUTHOR]

Given a graph H and an integer k ⩾ 2 , let f k (n , H) be the smallest number of colors C such that there exists a proper edge-coloring of the complete graph K n with C colors containing no k vertex-disjoint color isomorphic copies of H. In this paper, we prove that f 2 (n , H t) = Ω (n 1 + 1 2 t − 3 ) where H t is the 1-subdivision of the complete graph K t. This answers a question of Conlon and Tyomkyn (2021) [4]. [ABSTRACT FROM AUTHOR]

A G-design of order n is a decomposition of the complete graph on n vertices into edge-disjoint subgraphs isomorphic to G. Grooming uniform all-to-all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph on n vertices into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The existence spectrum problem of G-designs for five-vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such G-designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five-vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8-groomings for all orders n by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9-groomings for all orders n is realizable with 14 exceptions and 12 possible exceptions. [ABSTRACT FROM AUTHOR]

In this paper, we continue to investigate the existence of 5 -GDDs, 4-frames and 4-RGDDs, which have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence problems of these designs, while the problems are still open. Our new results mainly lie in the cases with relatively large group size. We also construct some smaller designs, including 4 -RGDDs of types 2 184 , 9 44 , 14 10 , 18 18 , 22 10 and 36 11 . [ABSTRACT FROM AUTHOR]

Multimedia fingerprinting is an effective technique to trace the sources of pirate copies of copyrighted multimedia contents. Separable codes were introduced to detect colluders taking part in the averaging attack, which is the most feasible approach to perform a collusion attack. In this paper, we provide some improved bounds for the size of separable codes. Using a combinatorial technique named grouping coordinates, we have greatly reduced the upper bound. On the other hand, we also provide some lower bounds for the size of separable codes by both probabilistic and deterministic construction methods. In particular, \( \overline {2}\) -separable codes with asymptotically optimal rate are obtained by the deletion method. [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs (GDDs) and non-uniform Kirkman frames are useful in the constructions for other types of designs. In this paper, we consider the existence problems for -GDDs of type with and Kirkman frames of type . First, we determine completely the spectrum for uniform -GDDs of type . Then, we consider the entire existence problem for non-uniform -GDDs of type with . We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a -GDD of type are also sufficient, except possibly when for and when for . Finally, a similar result for Kirkman frames of type is obtained. We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a Kirkman frame of type are also sufficient, except possibly when for and when for . [Copyright &y& Elsevier]

We study some sum-product problems over matrix rings. Firstly, for A , B , C ⊆ M n (F q) , we have | A + B C | ≳ q n 2 , whenever | A | | B | | C | ≳ q 3 n 2 − n + 1 2 . Secondly, if a set A in M n (F q) satisfies | A | ≥ C (n) q n 2 − 1 for some sufficiently large C (n) , then we have max ⁡ { | A + A | , | A A | } ≳ min ⁡ { | A | 2 q n 2 − n + 1 4 , q n 2 / 3 | A | 2 / 3 }. These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra. [ABSTRACT FROM AUTHOR]

Compressed sensing is a novel sampling theory, which provides a fundamentally new approach to data acquisition. It asserts that a sparse or compressible signal can be reconstructed from much fewer measurements than traditional methods. A central problem in compressed sensing is the construction of the sensing matrix. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. Among them, most constructions are based on coherence, which essentially generates matrices with low coherence. In this paper, we introduce the concept of near orthogonal systems to characterize the matrices with low coherence, which lie in the heart of many different applications. The constructions of these near orthogonal systems lead to deterministic constructions of sensing matrices. We obtain a series of m\times n binary sensing matrices with sparsity level k=\Theta (m^(1/2)) or k=O\big ((m/\log m)^(1/2)\big). In particular, some of our constructions are the best possible deterministic ones based on coherence. We conduct a lot of numerical experiments to show that our matrices arising from near orthogonal systems outperform several typical known sensing matrices. [ABSTRACT FROM PUBLISHER]

The theory of cyclotomy dates back to Gauss and has a number of applications in combinatorics, coding theory, and cryptography. Cyclotomy over a residue class ring \BBZv can be divided into classical cyclotomy or generalized cyclotomy, depending on v prime or composite. In this paper, we introduce a generalized cyclotomy of order d over \BBZp1^{e1p2^{e2},\ldots, pn^{en}}, which includes Whiteman's and Ding-Helleseth's generalized cyclotomy as special cases. Here, p1,p2,\ldots,pn are pairwise distinct odd primes satisfying d\vert (pi-1) for all 1\leq i\leq n and e1,e2,\ldots,en are positive integers. We derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers. As applications, we completely solve an open problem and a conjecture on Whiteman's generalized cyclotomy of order four over \BBZp1p2. Besides, we also construct an infinite series of near-optimal codebooks over \BBZp1p2, as well as some infinite series of asymptotically optimal difference systems of sets over \BBZp1^{e1p2^{e2},\ldots,pn^{en}}. [ABSTRACT FROM AUTHOR]

Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of -GDDs of type is on the case when is even, where the existence for small has played a key role. In order to determine the spectrum for -GDDs of type with being odd, we continue to investigate the small cases with in this paper. We show that, for each , the necessary conditions for the existence of a -GDD of type are also sufficient. As the applications of these GDDs, we obtain a few pairwise balanced designs with minimum block size 4. Meanwhile, we also improve the existence result for frame self-orthogonal Mendelsohn triple systems of type by reducing an infinite class of possible exceptions, namely and , to eight undetermined cases. [Copyright &y& Elsevier]

The whist tournament problem was introduced by Moore in 1896. To construct [ABSTRACT FROM AUTHOR]

Constant-weight codes play an important role in coding theory. The problem of determining the sizes for optimal quaternary constant-weight codes with length n, weight 4, and minimum Hamming distance 5 ((n,5,4)4 codes) has been investigated in several papers. Although some constructions and several infinite families for such codes with length n\equiv 0,1\pmod4 have been given, the problem is still far from complete. In this paper, we determine the size of an optimal (n,5,4)4 code for each integer n\geq 4 leaving 55 lengths unsolved. Especially, for length n\equiv 0,1\pmod 4, the existence problem of the equivalent combinatorial object, namely the generalized Steiner system, is solved leaving only seven values undetermined. [ABSTRACT FROM AUTHOR]

Constant-weight codes (CWCs) have played an important role in coding theory. To construct CWCs, a K-GDD (where GDD is group divisible design) with the 'star' property, denoted by K-*GDD, was introduced, in which any two intersecting blocks intersect in at most two common groups. In this paper, we consider the existence of 4-*GDD [ABSTRACT FROM AUTHOR]

Abstract: The whist tournament problem has attracted the interests of many design theorists for over a century. To construct -cyclic whist tournaments, the concept of -cyclic patterned starter whist tournaments was introduced. The research on -cyclic patterned starter whist tournaments dates back to 1954. Although considerable progress has been made in the last decade or so, less has been made about the existence of -cyclic patterned starter whist tournaments, despite the efforts of many authors. In this paper, we will first provide two new nonexistence results for -cyclic patterned starter whist tournaments. Then, we will propose a general recursive construction, i.e., a frame construction, for -cyclic patterned starter whist tournaments. As a consequence, we are able to unify many known constructions for -cyclic patterned starter whist tournaments. The known existence results of such designs are then extended. [Copyright &y& Elsevier]

The sizes of optimal constant-composition codes (CCCs) of weight 3 have been determined by Chee, Ge, and Ling with four cases in doubt. Group divisible codes (GDCs) played an important role in their constructions. In this paper, we study the problem of constructing optimal quaternary CCCs with Hamming weight 4 and minimum distances 5 or 6 through GDCs and Room square approaches. The problem is solved leaving only five lengths undetermined. Previously, the results on the sizes of such quaternary CCCs were scarce. [ABSTRACT FROM AUTHOR]

Constant-composition codes (CCCs) are a generalization of constant weight codes and permutation codes. The concept of group divisible codes, an analog of group divisible designs in combinatorial design theory, was first introduced by Chee et al. This new class of codes has been shown to be useful in recursive constructions of constant-weight codes and constant-composition codes. In this paper, we consider the problem of determining the maximal sizes of ternary constant-composition codes of weight four and distance five using group divisible codes as the main tools. We determine the exact values for these parameters. The previously known results are those with code length no greater than 10. [ABSTRACT FROM AUTHOR]

Abstract: Grooming uniform all-to-all traffic in optical ring networks with grooming ratio requires the determination of graph decompositions of the complete graph into subgraphs each having at most edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The minimum drop cost is determined for grooming ratio 9. Previously this bound was shown to be met when with two exceptions and eleven additional possible exceptions for , and also when with one exception and one possible exception for . In this paper it is shown that the bound is met for all with four exceptions for and one possible exception for . Using this result, it is further shown that when and is sufficiently large, the bound is also met. [Copyright &y& Elsevier]

Abstract: A group divisible design (GDD) with three groups and block size 4 is called even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups are always even, always odd, or always mixed. It has been shown that the necessary conditions for the existence of GDDs of these three types are also sufficient except possibly for the minimal case of mixed designs for group size (). In this paper, we complete the undetermined families of mixed GDDs using two constructions based on idempotent self-orthogonal Latin squares and skew Room squares. [Copyright &y& Elsevier]

Recently, Chee and Ling ("Constructions for q-ary constant-weight codes", IEEE Trans. Inf. Theory, vol. 53, no. 1, 135-146, Jan. 2007 ) introduced a new combinatorial construction for q-ary constant-weight codes which reveals a close connection between q-ary constant-weight codes and sets of pairwise disjoint combinatorial designs. In this paper, we study the problem of constructing optimal ternary constant-weight codes with Hamming weight four and minimum distance six using this approach. The construction here exploits completely reducible super simple designs and group divisible codes. The problem is solved leaving only two cases undetermined. Previously, the sizes of constant-weight codes of weight four and distance six were known only for those of length no greater than 10. [ABSTRACT FROM AUTHOR]

Abstract: The design of fault-tolerant routings with levelled minimum optical indices plays an important role in the context of optical networks. However, not much is known about the existence of optimal routings with levelled minimum optical indices besides the results established by Dinitz, Ling and Stinson via the partitionable Steiner quadruple systems approach. In this paper, we introduce a new concept of a large set of even levelled -design of order and index 2, denoted by -LELD, which is equivalent to an optimal, levelled -fault-tolerant routing with levelled minimum optical indices of the complete network with nodes. On the basis of the theory of three-wise balanced designs and partitionable candelabra systems, several infinite classes of -LELDs are constructed. As a consequence, the existence problem for optimal routings with levelled minimum optical indices is solved for nearly a third of the cases. [Copyright &y& Elsevier]

Abstract: The existence of graph designs for the two nonisomorphic graphs on five vertices and eight edges is determined in the case of index one, with three possible exceptions in total. It is established that for the unique graph with vertex sequence (3, 3, 3, 3, 4), a graph design of order exists exactly when and , with the possible exception of . For the unique graph with vertex sequence , a graph design of order exists exactly when , with the possible exceptions of . [Copyright &y& Elsevier]

Abstract: Grooming uniform all-to-all traffic in optical ring networks with grooming ratio requires the determination of graph decompositions of the complete graph into subgraphs each having at most edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The minimum possible drop cost is determined for grooming ratio 8, and this cost is shown to be realized with six exceptions, and 37 possible exceptions, the largest being 105. [Copyright &y& Elsevier]

Motivated by applications in combinatorial group testing for consecutive positives, we investigate a block sequence of a maximum packing MP(t, k, v) which contains the blocks exactly once such that the collection of all blocks together with all unions of two consecutive blocks of this sequence forms an error correcting code with minimum distance d. Such a sequence is usually called a block sequence with consecutive unions having minimum distance d, and denoted by BSCU(t, k, v∣d). In this paper, we show that the necessary conditions for the existence of BSCU(3, 4, v∣4)s of Steiner quadruple systems, namely, v Ξ 2,4 (mod 6) and v ≥ 4, are also sufficient, excepting v = 8, 10. [ABSTRACT FROM AUTHOR]

For a v-set V , let A be a collection of r × c arrays with elements in V. A pair (V,A) is called an r × c grid-block design if every two distinct elements i and j in V occur exactly once in the same row or in the same column of an array in A. This design originated from the use of DNA library screening. In this paper, we show the existence of r × 4 grid-block designs with r = 3, 4. We settle completely for the case of r = 4 and almost completely for the case of r = 3, leaving 15 orders undetermined. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type and general index . The necessary conditions for the existence of such a design are , and . These necessary conditions are shown to be sufficient for all , with the definite exceptions of . The known existence result for is also improved. [Copyright &y& Elsevier]

Abstract: In this article we investigate the existence of pairwise balanced designs on points having blocks of size five, with a distinguished block of size , briefly -PBDs. The necessary conditions for the existence of a -PBD with a distinguished block of size with are that , and either or . Previously, Bennett et al. had shown that these conditions are sufficient for with the possible exception of when , and had studied in detail, showing there that the necessary conditions are sufficient with 71 possible exceptions. In this article, we show sufficiency for and give a small list of possible exceptions containing 26 and 104 values for . For , all possible exceptions satisfy either with or with and ; for , all possible exceptions except satisfy either with or with and . As an application of our results for , we establish that, if , and , then the smallest number of blocks in a pair covering design with is , i.e., the Schönheim bound. [Copyright &y& Elsevier]

Abstract: A mandatory representation design is a pairwise balanced design on points with block sizes from the set in which for each there is at least one block in the design of size . In this paper, we consider MRDs with , where , and prove that the necessary conditions for existence are sufficient if and , or and , or and . [Copyright &y& Elsevier]

Abstract: In this note, we consider a construction for optimal ternary constant weight codes (CWCs) via Bhaskar Rao designs (BRDs). The known existence results for BRDs are employed to generate many new optimal nonlinear ternary CWCs with constant weight 4 and minimum Hamming distance 5. [Copyright &y& Elsevier]