ABSTRACT A mixed Steiner triple system is a 3‐GDD which is viewed as a code with minimum Hamming distance 3. These codes are the minimum weight codewords of a 1‐perfect code over a mixed alphabet, when the related codes exist, and provide the connection between 3‐GDDs and coding theory. We prove that a 3‐GDD of type , where , with minimum distance 3 exists for every and such that , or , or , and . These designs are of the shortest possible length (smallest number of elements) for given and . Other constructions for such triple systems are also presented.
ABSTRACT A design is said to be super‐simple if the intersection of any two blocks contains at most two elements. In the statistical design of experiments, super‐simple designs yield samples in which the maximum pairwise block intersection is minimized. Moreover, such designs play a valuable role in the construction of codes, including optical orthogonal codes and superimposed codes, other combinatorial designs such as group divisible designs and pairwise balanced designs, and can be used to determine some exact Zarankiewicz numbers. In this paper, we investigate the existence of a super‐simple ‐BIBD and show that such a design exists if and only if and .
ABSTRACT A design is said to be super‐simple if any two of its blocks have at most two common points. A design with index is said to be ‐ decomposable , if its blocks can be partitioned into nonempty collections , such that each with the point set forms a design with index . In this paper, it is proved that there exists a ‐DSS ‐GDD (decomposable super‐simple group divisible design) of type if and only if and , and except possibly for .
A Hamiltonian cycle decomposition (HCD) of K-n is a set of Hamiltonian cycles in which each 1-path of K-n appears exactly once. A Dudeney set of K-n is a set of Hamiltonian cycles in which each 2-path of K-n appears exactly once. Kotzig's perfect set of HCDs of K-n is a set of HCDs whose union forms a Dudeney set. A set of k-pairwise compatible HCDs (k-PCH) of K-n is a set of k HCDs such that each 2-path of K-n appears at most once. This paper proposes an orderly algorithm for finding a k-PCH and shows that no 7-PCH (i.e., no perfect set of HCDs) exists for K-9. It also proposes an orderly algorithm for finding a perfect set whose automorphism group is Cn-2 or Dn-2, and shows the existence of a perfect set of HCDs of K-11, K-17 and K-19. These are the first instances of perfect sets of HCDs in complete graphs to be found since Kotzig posed the problem in 1979.
The Oberwolfach problem , for a 2-factor of , asks whether there exists a 2-factorization of (if is odd) or (if is even) where each 2-factor is isomorphic to . Here, denotes any 1-factor of . For even , the problem OP may also be denoted , and has been nicknamed the spouse-avoiding variant. Similarly, the spouse-loving variant is denoted and asks for a 2-factorization of (the complete graph with the edges of a 1-factor duplicated, rather than deleted) in which each 2-factor is isomorphic to . To date, many more infinite families of cases of OP and have been solved than of . In this paper, we show how certain solutions to can be used to construct solutions to ; in particular, when the number of odd cycles in the 2-factor is not too large. Our technique of setups also allows us to completely solve the two-table ; that is, , where has exactly two components.
Combinatorial testing has been widely used to generate test suites for detecting faults in various systems. As a type of fractional factorial design, detecting arrays have been proposed, out of the practical need for locating faults. In previous studies, the optimum detecting arrays in some cases have been determined through direct constructions and algorithmic methods. However, there are no practical lower bounds on the size of optimum detecting arrays for the general case. In this paper, we present two connections between detecting arrays and Sperner set systems that enables us to obtain lower bounds. An improved tabu search algorithm is developed for constructing mixed-level detecting arrays with a small number of rows. The performance of the algorithm is investigated by computational experiments. The experimental results show that the proposed algorithm can efficiently generate high-quality detecting arrays for different problem instances. Based on the lower bounds and algorithms presented in this study, several new optimum detecting arrays are obtained.
Given positive integers (), we call a rank-metric code , with minimum distance for some , a maximum rank distance code (or MRD code for short) if . The space generated by MRD codes is defined to be the -vector space spanned by the characteristic vectors of all MRD codes. In this paper, we prove that its dimension equals , where the -th valency of the bilinear forms scheme, , is exactly the number of matrices of rank in .
A multi-receiver authentication code refers to a sender transmitting a message that needs to be authenticated to receivers, so that each receiver can verify it, and any receivers cannot deceive other receivers. In this paper, an optimal multi-receiver authentication code is designed by constructing two orthogonal arrays, which satisfy that each column of one orthogonal array is friendly to the other orthogonal array. One orthogonal array with strength 2 is presented according to the mutually orthogonal Latin squares, and the other orthogonal array with strength is obtained by using Bush's construction. It is proved that the orthogonal array with strength is friendly to each column of the orthogonal array with strength 2. The probability of successful impersonation and substitution attacks from an opponent, and the number of encoding rules are calculated to verify that the constructed multi-receiver authentication code is optimal.
This paper introduces an extension of signed Langford sequences to generalized signed Langford sequences with three distinct types and investigates their existence. We establish three main existence results: (1) There exists a generalized signed Langford sequence of Type 3 except for (a) and , (b) and . (2) For all admissible positive integers and , there exist two disjoint generalized signed Langford sequences of both Type 1 and Type 2. (3) For parameters satisfying with , and with , there exist two disjoint generalized signed Langford sequences of Type 3. Additionally, we prove the existence of absolutely symmetric signed Langford sequences in four special cases: when with , and when with .
In this paper, we study and characterise the natural embedding of the twisted triality hexagon in . We begin by describing the possible intersections of subspaces of with . Then, we provide conditions on a set of lines , which ensure that forms the line set of a naturally embedded twisted triality hexagon. This work follows up on similar results for the split Cayley hexagon by J. A. Thas and H. Van Maldeghem (2008) and F. Ihringer (2014).
The necessary and sufficient conditions for the existence of 1-rotational balanced incomplete block designs with block size four and index unity have been determined to be and [J. Comb. Theory, Ser. A 206 (2024) 105890]. This work completely determines the existence spectrum of 1-rotational balanced incomplete block designs with block size four and any index . To be specific, for and , there exists a 1-rotational BIBD if and only if , and .
Capsets are subsets of with no three points on a line, and a capset is complete if it is not a subset of a larger capset. We study some new constructions of capsets via algebraic equations over extensions of . In particular we construct the smallest known complete capsets with size proportional to the best known lower bound.
An untouchable set in a projective plane is a set of points such that no line of the plane meets the set in exactly one point. Recently, H & eacute;ger and Nagy (Avoiding Secants of Given Size in Finite Projective Planes, J. Combin. Des. 33:83-93, 2024.) provided a generalization of untouchable sets to -avoiding sets, and addressed the issue of the spectrum of sizes that such sets can attain in finite planes. In the case of untouchable sets, the authors state as an open question the existence of untouchable sets of size and . We answer this question in the affirmative for Desarguesian planes of even order, and provide a construction of untouchable sets of size in for .
Through the use of regularizing vectors, all regular quaternary Hadamard matrices of orders 10 and 18 have been successfully identified. Of these, two matrices of order 10 and 184 matrices of order 18 were found to have unbiased mates. Converting the quaternary Hadamard matrices of order 18 to real Hadamard matrices, the study uncovered that six matrices of order 36 having unbiased pairs were connected to the extremal Pless symmetry code. Furthermore, the study uncovered exactly 28 nonregular quaternary Hadamard matrices of order 18, providing the first examples of nonregular Hadamard matrices of order 36 and, as a result, settling a recent conjecture in the negative.
A double-change covering design (DCCD) is a -set and an ordered list of blocks of size where every pair from must occur in at least one block and each pair of consecutive blocks differs by exactly two elements. It is minimal if it has the fewest blocks possible and circular when the first and last blocks also differ by two elements. We give a recursive construction that uses 1-factorizations of complete graphs and expansion sets to construct a DCCD() from a DCCD(). We construct circular DCCD() and circular DCCD() from single-change covering designs and determine minimum DCCD when . We use difference like methods to construct five infinite families of minimum circular DCCD() when for any . The recursive construction is then used to build twelve additional minimum DCCD from members of these infinite families. Finally, the difference like method is used to construct a minimum circular DCCD(61,4,366).
We construct pairs of quasi-unbiased weighing matrices for parameters , and by adopting a coding-theoretic approach.
In this paper, by extending the classical singular direct product construction to the quantum setting we resolve the existence problem for quantum Latin squares with maximal cardinality except for 11 possible exceptions. We also prove that there exists a quantum Latin cube of order with maximal cardinality for all positive integers , except when is a prime or , where are prime numbers and .
A partial Latin rectangle is if the number of nonempty entries in each row and column is at most and each symbol is used at most times. We prove that the probability a uniformly random Latin rectangle, where , contains a -sparse partial Latin rectangle with nonempty entries is for sufficiently large and sufficiently small . Using this result, we prove that a uniformly random order- Latin square asymptotically almost surely has no Latin subsquare of order greater than for an absolute constant .
In this paper, we study and characterise the natural embedding of the twisted triality hexagon T(q^3,q) in PG(7,q^3). We begin by describing the possible intersections of subspaces of PG(7,q^3) with T(q^3,q). Then, we provide conditions on a set of lines L which ensures that L forms the line set of a naturally embedded twisted triality hexagon. This work follows up on similar results for the split Cayley hexagon by J. A. Thas and H. Van Maldeghem (2008) and F. Ihringer (2014).
We provide some difference triangle sets with scopes that improve upon the best known values. These are found with purpose-built digital circuits realized with field-programmable gate arrays (FPGAs) rather than software algorithms running on general-purpose processors.
