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Splitting authentication codes were first introduced by Simmons in 1982. Ogata et al. introduced (v , u × c , 1) -splitting balanced incomplete block designs in 2006 in order to construct twofold optimal c-splitting authentication codes. In 2020, Paterson and Stinson showed that there exists an authentication code with perfect secrecy for u uniformly distributed source states that is ϵ -secure against message-substitution and key-substitution attacks if and only if there exists an ϵ -secure robust (2, 2)-threshold scheme for u uniformly distributed secrets, and they used an equitably ordered (v , u × c , 1) -splitting balanced incomplete block design (briefly a (v , u × c , 1) -ESBIBD) to construct a (1/cu)-secure robust (2, 2)-threshold scheme for u equiprobable secrets. Note that v ≡ 1 (mod u (u - 1) c) and v (v - 1) ≡ 0 (mod u (u - 1) c 2) if there is a (v , u × c , 1) -ESBIBD. In order to consider other orders v, we generalize the concept of a (v , u × c , 1) -ESBIBD to an equitably ordered (v , u × c , 1) -splitting packing design (briefly a (v , u × c , 1) -ESPD), which can also be used to construct a (1/cu)-secure robust (2, 2)-threshold scheme for u equiprobable secrets. In this paper, we study combinatorial constructions of (v , u × c , 1) -ESPDs and determine the existence of an optimal (v , u × c , 1) -ESPD for (u , c) ∈ { (2 , k) : k is a positive integer } ∪ { (3 , 1) , (4 , 1) , (3 , 2) } . Consequently, we obtain some new infinite classes of authentication codes with perfect secrecy and (1/cu)-secure robust (2, 2)-threshold schemes. [ABSTRACT FROM AUTHOR]

A t- (n , k , λ) q design is a set of k-dimensional subspaces, called blocks, of an n-dimensional vector space V over the finite field F q with q elements such that each t-dimensional subspace is contained in exactly λ blocks. A partition of the complete set of k-dimensional subspaces of V into l disjoint t- (n , k , λ) q designs is called a large set of t-designs over F q and denoted by L S q [ l ] (t , k , n) . Large sets of t-designs over finite fields with t ≥ 2 are currently known to exist only for L S 2 [ 3 ] (2 , 3 , 8) , L S 3 [ 2 ] (2 , k , n) and L S 5 [ 2 ] (2 , k , n) with integers n ≥ 6 , n ≡ 2 (mod 4) and 3 ≤ k ≤ n - 3 , k ≡ 3 (mod 4) . In this article, we use the KLP theorem to prove that an L S q [ l ] (t , k , n) exists for all t and q, provided that k > 6.1 (t + 1) and n is sufficiently large and n - s k - s q ≡ 0 (mod l) for 0 ≤ s ≤ t. [ABSTRACT FROM AUTHOR]

We give a direct construction of a (Zp×G,{0}×G,4,1) relative difference family for G∈{Z6×Z6,Z2×Z18,Z6×Z18,Z2×Z54} and every prime p≡3(mod4) with p>3. These allow us to construct an optimal (Z6m×Z6n,4,1) difference packing and an optimal (Z2m×Z18n,4,1) difference packing for every pair of positive integers (m,n). The corresponding optimal optical orthogonal signature pattern codes are also obtained. [ABSTRACT FROM AUTHOR]

A t- (v , k , λ) design is a pair (X , B) , where X is a v-element set and B is a set of k-subsets of X, called blocks, with the property that every t-subset of X is contained in exactly λ blocks. A t- (v , k , λ) design (X , B) is said to be (s , μ) -resolvable if B can be partitioned into B 1 | ⋯ | B c such that each (X , B i) is an s- (v , k , μ) design, further, if each (X , B i) is also (r , ν) -resolvable, then such an (s , μ) -resolvable t-design is called (s , μ) (r , ν) -doubly resolvable. In 1980, Hartman constructed a (2, 3)(1, 1)-doubly resolvable 3-(v, 4, 1) design for v ∈ { 20 , 32 , 44 , 68 , 80 , 104 } and a (2, 3)-resolvable 3- (2 7 , 4 , 1) design. In this paper, we construct (2, 3)(1, 1)-doubly resolvable 3- (2 2 n + 1 , 4 , 1) designs for all positive integers n. [ABSTRACT FROM AUTHOR]

A (t, s, v)-all-or-nothing transform (AONT) is a bijective mapping defined on s-tuples over an alphabet of size v, which satisfies that if any s - t of the s outputs are given, then the values of any t inputs are completely undetermined. When t and v are fixed, to determine the maximum integer s such that a (t, s, v)-AONT exists is the main research objective. In this paper, we solve three open problems proposed in Nasr Esfahani et al. (IEEE Trans Inf Theory 64:3136–3143, 2018) and show that there do exist linear (2, p, p)-AONTs. Then for the size of the alphabet being a prime power, we give the first infinite class of linear AONTs which is better than the linear AONTs defined by Cauchy matrices. Besides, we also present a recursive construction for general AONTs and a new relationship between AONTs and orthogonal arrays. [ABSTRACT FROM AUTHOR]

An H(m,g,k,3) design is a triple (X,T,B), where X is a set of mg points, T a partition of X into m disjoint sets of size g, and B a set of k‐element transverses of T, such that each 3‐element transverse of T is contained in exactly one of them. In 1990, Mills determined the existence of an H(m,g,4,3) design with m≠5. In this paper, an efficient construction shows that an H(5,g,4,3) exists for any integer g≡2,10(mod12) with g≥10. Consequently, the necessary and sufficient conditions for the existence of an H(m,g,4,3) design are m≥4, mg≡0(mod2), and g(m−1)(m−2)(mod3), with a definite exception (m,g)=(5,2). [ABSTRACT FROM AUTHOR]

Large sets of disjoint group-divisible designs with block size three and type 2 n 4 1 (denoted by L S ( 2 n 4 1 ) ) were first studied by Schellenberg and Stinson and motivated by their connection with perfect threshold schemes. It has been shown that the necessary condition n ≡ 0 ( m o d 3 ) for the existence of an L S ( 2 n 4 1 ) is sufficient with five possible exceptions n ∈ { 12 , 30 , 36 , 48 , 144 } . These five undetermined L S ( 2 n 4 1 ) s are shown to exist in this paper. [ABSTRACT FROM AUTHOR]

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Candelabra quadruple systems (CQS) were first introduced by Hanani who used them to determine the existence of Steiner quadruple systems. In this paper, a new method has been developed by constructing partial candelabra quadruple systems with odd group size, which is a generalization of the even cases, to complete a design. New results of candelabra quadruple systems have been obtained, i.e. we show that for any [ABSTRACT FROM AUTHOR]

Symmetric orthogonal arrays and mixed orthogonal arrays are useful in the design of various experiments. They are also a fundamental tool in the construction of various combinatorial configurations. In this paper, we investigated the mixed orthogonal arrays with four and five factors of strength two, and proved that the necessary conditions of such mixed orthogonal arrays are also sufficient with several exceptions and one possible exception. [ABSTRACT FROM AUTHOR]

The existence of large set of Kirkman triple systems (LKTSs) is an old problem in design theory, which is still far from settled. In this paper, we use a (2 , 1) -RSQS ∗ (4 n) to construct an LKTS (2 2 n + 1 + 1) , which has a nice structure and is used to construct an LKTS (6 ⋅ 5 m (4 ⋅ 5 n + 1) ± 3) for any nonnegative integers m and n based on 3-wise balanced designs. [ABSTRACT FROM AUTHOR]