1 Introduction

Let X be a set of gv points and let \(\mathcal{G}=\{G_1, G_2,\ldots , G_v\}\) be a partition of X into disjoint sets \(G_i\)\((1\le i\le v)\), which we will call the groups of \(\mathcal{G}\). By a \(\mathcal{G}\)-transverse t-subset we mean a t-subset of X that intersects each of the groups in \(\mathcal{G}\) in at most one element.

An H-design with parameters (v, g, k, t), denoted by H(v, g, k, t), is a triple \((X, \mathcal{G},\mathcal{B})\), where X is a set of gv points, \(\mathcal{G}\) is a partition of X into v disjoint groups of size g, and \(\mathcal{B}\) is a set of \(\mathcal{G}\)-transverse k-subsets, called blocks, such that each \(\mathcal{G}\)-transverse t-subset is contained in exactly one block of \(\mathcal{B}\). An H(v, 1, 3, 2) is a Steiner triple system of order v, denoted by \({{\text {STS}}}(v)\). An H(v, 1, 4, 3) is a Steiner quadruple system of order v, briefly by \({{\text {SQS}}}(v)\).

The H-designs were introduced by Hanani [5] in order to study Steiner quadruple systems. Mills [13] determined the existence of an H(v, g, 4, 3) with \(v\ne 5\). Ji [7, 8] studied the undetermined case \(v=5\) and he gave a final solution very recently.

Theorem 1.1

[7, 8, 13] The necessary and sufficient conditions for the existence an H\((v, g, 4, 3)\) are \(gv\equiv 0\) (mod 2), \(g(v-1)(v -2)\equiv 0\) (mod 3), \(v\ge 4\), and \((v,g)\ne (5,2)\).

A generalized frame F\((t, k, v\{g\})\) (as in [19]) is an H(v, g, k, t) \((X, \mathcal{G}, \mathcal{B})\) such that the block set \(\mathcal{B}\) can be partitioned into subsets \(\mathcal{B}_r\), \(r\in R\), each \(\mathcal{B}_r\) being the block set of an H\((v-1,g, k,t-1)\) missing some group \(G\in \mathcal{G}\). It is known that an F\((t, k, v\{g\})\) contains \(gv/(k-t+1)\) H\((v-1,g, k,t-1)\)s, i.e., \(|R| = gv/(k-t+1)\). Let \(R_G = \{r\in R : \mathcal{B}_r\) has group set \(\mathcal{G}\setminus \{G\}\}\). Then \(|R_G|=g/(k-t+1)\).

An F\((2, 3, v\{g\})\) is called a Kirkman frame, briefly by \({{\text {KF}}}(g^v)\). So the blocks of a \({{\text {KF}}}(g^v)\) have a resolution into gv / 2 holey parallel classes, where every group is missed g / 2 times. A \({{\text {KF}}}(g^v)\) exists if and only if \(g\equiv \) 0 (mod 2), \(v \ge 4\), and \(g(v-1)\equiv \) 0 (mod 3) (see [17]).

An F\((3, 3, v\{g\})\) is called an overlarge set of Kirkman frame and denoted by OLKF\((g^v)\) if each element (i.e. an H\((v-1,g,3,2)\)) of this F\((3, 3, v\{g\})\) is a \({{\text {KF}}}(g^{v-1})\).

Let \((X,\mathcal{G},\mathcal{A})\) be an H(v, g, 4, 3). For any \(x\in G\) and \(G\in \mathcal{G}\), let \(\mathcal{A}_x=\{A\setminus \{x\}:x\in A, A\in \mathcal{A}\}\). Then \((X\setminus G,\mathcal{G}\setminus \{G\},\mathcal{A}_x)\) is said to be the derived design at the point x. Clearly the collection \(\{(X\setminus G,\mathcal{G}\setminus \{G\},\mathcal{A}_x):x\in G, G\in \mathcal{G}\}\) forms an F\((3,3,v\{g\})\), which is called the derived design of the H(v, g, 4, 3). A frame-derived H-design or an FDH-design, denoted by FDH(v, g, 4, 3), is an H(v, g, 4, 3) whose derived design is an OLKF\((g^v)\). Note that the notation *\({{\text {RDGDD}}}(3,4,v\{g\})\) is used for FDH(v, g, 4, 3) in [2, 22], where an FDH(5, 4, 4, 3) and an FDH(8, 4, 4, 3) are constructed directly. By considering the conditions of both an H(v, g, 4, 3) and a KF\((g^{v-1})\), we know that the necessary conditions for the existence of an H(v, g, 4, 3) are \(g\equiv \) 0 (mod 2), \(v \ge 5\), and \(g(v-2)\equiv \) 0 (mod 3), with the exception of \((v,g)=(5,2)\).

In this paper we study the existence of FDH(v, g, 4, 3)s. Our motivation lies in its intimate connection with large sets of Kirkman triple systems (LKTS). We prove that an FDH(v, 2, 4, 3) will derive two important auxiliary designs in the process of constructing LKTSs. So we investigate recursive constructions for FDH-designs. We also display direct constructions for FDH(v, 2, 4, 3)s by using automorphism groups. Finally the existence result on LKTSs is improved as well.

2 Related designs and recursive constructions

In this section we show the connection between frame-derived H-designs and large sets of Kirkman triple systems and then introduce several related designs, display their relations, and present useful recursive constructions.

2.1 LKTS

Let X be a v-set and K be a set of positive integers. A t-wise balanced design (t-BD) of order v is a pair \((X, \mathcal{B})\) where \(\mathcal{B}\) is a family of subsets of X (called blocks) such that each t-subset of X is contained in exactly one block of \(\mathcal{B}\). An \({{\text {S}}}(t,K,v)\) denotes a t-BD of order v with block sizes from the set K. The notation \({{\text {S}}}(t,k,v)\) is often used for \(K=\{k\}\). Hence, an STS(v) is an S(2, 3, v) and an SQS(v) is an S(3, 4, v).

Let \((X, \mathcal{B})\) be an STS(v). If there exists a partition \(\Gamma =\{P_1, P_2,\ldots , P_{(v-1)/2}\}\) of \(\mathcal{B}\) such that each part \(P_i\) forms a parallel class, i.e. a partition of X, then the STS(v) is called resolvable and \(\Gamma \) is called a resolution. A resolvable STS(v) is usually called a Kirkman triple system of order v, briefly by KTS(v). It is well known that a KTS(v) exists if and only if \(v\equiv 3\) (mod 6) (see [12, 16]).

Two \({{\text {STS}}}(v)\)s on the same set X are said to be disjoint if they have no triples in common. A partition of all the triples of a v-set X into \({{\text {STS}}}(v)\)s, denoted by \({{\text {LSTS}}}(v)\), is called a large set of \({{\text {STS}}}(v)\)s. A large set of Kirkman triple systems of order v, denoted by LKTS(v), is an \({{\text {LSTS}}}(v)\) in which each \({{\text {STS}}}(v)\) is a \({{\text {KTS}}}(v)\).

The existence problem of large sets of Kirkman triple systems is a long-standing open problem in design theory. Sylvester in the 1850s first posed the existence of LKTSs as an extension to the famous Kirkman’s 15 school girls problem (see Cayley [1]). This existence question has become known as Sylvester’s problem. Sylvester also asked about the existence of an \({{\text {LSTS}}}(15)\). In this case, the general problem of an \({{\text {LSTS}}}(v)\) was completely solved for all \(v\equiv 1,3\) (mod 6) and \(v\ne 7\), mainly by remarkable work of Lu [10, 11] but finished by Teirlinck [18]. In comparison, the existence problem of LKTSs remains very much open, although some direct constructions and recursive constructions for LKTSs have been developed, see the recent survey [2]. There is intimate connection between large sets of triple systems and 3-wise balanced designs, see for instance [2, 6, 9].

An approach of constructing large sets of Kirkman triple systems makes use of two combinatorial objects; one is overlarge set of Kirkman frame defined in Sect. 1 and the other is resolvable partitionable candelabra system (RPCS). We leave the definition of the RPCS to the next subsection and first state its effect on LKTS. Combining with the existence result on 3-wise balanced designs, Lei [9, Theorem A] pointed out the existence of LKTS\((6n+3)\) relies on the existence of about thirty input designs of RPCS and of OLKF. Subsequently Chang and Zhou [2] gave the following theorem.

Theorem 2.1

[2, Theorem 3.9] If both a RPCS\((6^{k-1}:3)\) and an OLKF\((6^k)\) exist for any \(k\in \{6,7,\ldots ,40\}\setminus \{14,17,21,22,25,26\}\), then an LKTS(v) exists for any positive integer \(v\equiv 3\) (mod 6).

When \(k\equiv 2\) (mod 3), an FDH\((3,4,k\{2\})\) enables us to obtain the auxiliary designs RPCS\((6^{k-1}:3)\) and OLKF\((6^k)\), as we shall see later.

2.2 RPCS and RDSQS

Let X be a v-set, H an h-subset of X and \(\mathcal{B}\) a block set of some 3-subsets of X. The triple \((X,H,\mathcal{B})\) is said to be an incomplete Steiner triple system with a holeH, denoted by \({{\text {STS}}}(v,h)\), if each pair of points of X except those contained in H is contained in exactly one block of \(\mathcal{B}\). Further if \(\mathcal{B}\) can be partitioned into \((v-h)/2\) parallel classes of X and \((h-1)/2\) holey parallel classes with a hole H, then we say that \((X,H,\mathcal{B})\) is an incomplete Kirkman triple system, denoted by \({{\text {KTS}}}(v,h)\).

A candelabra t-system (as in [14]) of order v with block sizes from K, denoted by \({{\text {CS}}}(t,K,v)\), is a quadruple \((X, S,\mathcal{G}, \mathcal{A})\) that satisfies the following properties:

  1. (1)

    X is a set of v elements (points);

  2. (2)

    S is a subset (called stem) of X of size s;

  3. (3)

    \(\mathcal{G}\) is a collection of subsets (groups) of \(X\setminus S\) which partition \(X\setminus S\);

  4. (4)

    \(\mathcal{A}\) is a family of subsets (blocks) of X, each of cardinality from K;

  5. (5)

    every t-element subset T of X with \(|T\cap (S\cup G)|<t\) for any \(G\in \mathcal{G}\), is contained in exactly one block, and for each \(G\in \mathcal{G}\) no t-element subsets of \(S\cup G\) are contained in any block.

By the type of a candelabra system we mean the list \((\{|G|: G\in \mathcal{G}\}:s)\). We also use the exponential notation to denote the type of \(\mathcal{G}\) and separate the stem size by a colon.

A \({{\text {CS}}}(3,3,gn+s)\) of type \((g^n:s)\)\((X, S,\mathcal{G},\mathcal{A})\) is said to be resolvable partitionable, briefly by RPCS\((g^n:s)\), if \(\mathcal{A}\) can be partitioned into \(\mathcal{A}_{x}\), \(x\in G,\ G\in \mathcal{G}\), and \(\mathcal{A}_{i}\), \(1\le i\le s-2,\) with the following properties:

  1. (1)

    for \(x\in G\) and \(G\in \mathcal{G}\), each \((X,G\cup S,\mathcal{A}_{x})\) is an incomplete KTS;

  2. (2)

    for \(1\le i\le s-2\), each \((X\setminus S, \mathcal{G}, \mathcal{A}_{i})\) is a \({{\text {KF}}}(g^n)\).

From an \({{\text {S}}}(t,k,v)\), by choosing an element x, selecting all blocks containing x, and deleting x from each, one obtains an \({{\text {S}}}(t-1, k-1, v-1)\), the derived design at the point x. An \({{\text {S}}}(t,k,v)\)with resolvable derived designs, abbreviated to \({{\text {RDS}}}(t,k,v)\), refers to an \({{\text {S}}}(t,k,v)\) whose derived design at every point is resolvable. An \({{\text {SQS}}}(v)\) exists if and only if \(v\equiv 2 ,4\) (mod 6) and \(v\ge 2\) (see [4]). An RDSQS(v) refers to an RDS(3, 4, v). So its derived design at every point is a KTS\((v-1)\). Necessarily, the existence of an RDSQS(v) requires that \(v\equiv 4\) (mod 6). The notation RDSQS(v) was firstly used in [2], where its main application in constructing large sets of Kirkman triple systems is restated after Yuan and Kang [20, Lemma 3.4]. The result is as follows.

Lemma 2.2

[2, Corollary 4.2] If there exists an RDSQS\((2n+2)\), then there exists an RPCS\((6^{n}:3)\) and an LKTS\((6n+3)\).

2.3 Recursive constructions

As we shall see, the FDH-designs could be used to construct the auxiliary designs RPCS and OLKF in Lemmas 2.3 and 2.4. This subsection focuses on related recursive constructions.

Lemma 2.3

The existence of an FDH(v, 2, 4, 3) implies the existence of an OLKF\(((2t)^{v})\) where \(t\ne 2,6\).

Proof

By the definition, the derived design of an FDH(v, 2, 4, 3) is an OLKF\((2^v)\). So, by [9, Theorem 4.1], the conclusion follows. \(\square \)

Lemma 2.4

The existence of an FDH(v, 2, 4, 3) implies the existence of an RDSQS(2v) and an RPCS\((6^{v-1}:3)\).

Proof

Let X be a v-set, \(Y=X\times I_2\), \(G_x=\{x\}\times I_2\), and \(\mathcal{G}=\{G_x:x\in X\}\). Further let \((Y,\mathcal{G},\mathcal{B})\) be an FDH(v, 2, 4, 3). Thus \((Y,\mathcal{G},\mathcal{B})\) has two properties:

  1. (P1)

    \((Y,\mathcal{G},\mathcal{B})\) is an H(v, 2, 4, 3).

  2. (P2)

    The derived design \((Y\setminus G_x,\mathcal{G}\setminus \{G_x\},\mathcal{B}_{x,i})\) at any fixed point \((x,i)\in Y\) is a KF\((2^{v-1})\). So \(\mathcal{B}_{x,i}\) can be partitioned into subsets \(P_{x,i}^y\), \(y\in X\setminus \{x\}\), where \(P_{x,i}^y\) is parallel classes of \((X\setminus \{x,y\})\times I_2\).

Set \(\mathcal{A}=\mathcal{B}\cup \{\{(x,0),(x,1),(y,0),(y,1)\}:x,y\in X,x\ne y\}\). By (P1), it is immediate that \((Y,\mathcal{A})\) forms an SQS(2v). Furthermore, its derived design \(\mathcal{A}_{x,i}\) at the point (x, i) can be partitioned into \(v-1\) parallel classes of \((X\times I_2)\setminus \{(x,i)\}\) by (P2). Namely, the parallel classes are \(P_{x,i}^y\cup \{(x,i+1),(y,0),(y,1)\}\) with \(y\in X\) and \(y\ne x\) (where \(i+1\) is reduced modulo 2). Hence, every derived design of \((Y,\mathcal{A})\) is a KTS\((2v-1)\) and thus it forms an RDSQS(2v). Finally we apply Lemma 2.2 and then obtain an RPCS\((6^{v-1}:3)\). \(\square \)

Teirlinck [19] studied the generalized frame F\((3,4,v\{g\})\), by which we may obtain FDH (v, g, 4, 3).

Lemma 2.5

The existence of an F\((3,4,v\{g\})\) implies the existence of an FDH(v, g, 4, 3).

Proof

Let X be a v-set, \(Y=X\times I_g\), \(G_x=\{x\}\times I_g\), and \(\mathcal{G}=\{G_x:x\in X\}\). Further let \((Y,\mathcal{G},\mathcal{B})\) be an F\((3,4,v\{g\})\). Thus \((Y,\mathcal{G},\mathcal{B})\) has two properties:

  1. (P1)

    \((Y,\mathcal{G},\mathcal{B})\) is an H(v, g, 4, 3).

  2. (P2)

    \(\mathcal{B}\) can be partitioned into subsets \(\mathcal{C}{(x,r)}\), \(x\in X,1\le r\le g/2\), each \(\mathcal{C}{(x,r)}\) being the block set of an H\((v-1,g,4,2)\) missing the group \(G_x\).

For every \((y,i)\in Y,\) denote by \(\mathcal{B}_{y,i}\) the derived design of \((Y,\mathcal{G},\mathcal{B})\) at (y, i). By (P1), to prove the conclusion it suffices to show that \(\mathcal{B}_{y,i}\) is the block set of a KF\((g^{v-1})\). For any fixed \(x\in X\) and \(1\le r\le g/2\), let \(\mathcal{C}_{y,i}(x,r)\)\((y\ne x,i\in I_g)\) be the block set of the derived design of \((Y\setminus G_x,\mathcal{G}\setminus \{G_x\},\mathcal{C}{(x,r)})\) at the point (y, i). By (P2), each \(\mathcal{C}_{y,i}(x,r)\) is a parallel class of \((X\setminus \{x,y\})\times I_g\). Obviously \(\mathcal{B}_{y,i}=\bigcup _{\begin{array}{c} x\ne y,x\in X \\ 1\le r\le g/2 \end{array}}\mathcal{C}_{y,i}(x,r)\) and hence \(\mathcal{B}_{y,i}\) forms the block set of a KF\((g^{v-1})\). So we have that \((Y,\mathcal{G},\mathcal{B})\) actually forms an FDH(v, g, 4, 3). \(\square \)

Lemma 2.6

There exists an FDH(v, 2, 4, 3) for \(v=2^{2m-1}\) and \(v = q^n + 1\), where \(m\ge 2,n \ge 1\), \(q\equiv 7\) (mod 12), and q is a prime power.

Proof

For the assumed v, there exists an F\((3,4,v\{2\})\) by [15, 19]. Hence the conclusion follows by Lemma 2.5. \(\square \)

Tierlinck [19, Proposition 2.3] proved that the set of v for which there exists an F\((3, 4, v\{g\})\) (for fixed g) is 3BD-closed. Lei [9, Theorem 4.2] showed the 3BD-closed property of OLKF\((g^v)\). It is not difficult to show this holds also for FDH(v, g, 4, 3).

Lemma 2.7

If an S(3, K, v) exists and if an FDH(k, g, 4, 3) exists for every \(k\in K\), then an FDH(v, g, 4, 3) exists.

Proof

Let \((X,\mathcal{B})\) be an S(3, K, v). For any block \(B\in \mathcal{B}\) construct an FDH(|B|, g, 4, 3) over the set \(B\times I_g\) and with the groups \(\{x\}\times I_g\), \(x\in B\). Denote its block set by \(\mathcal{A}_B\). Then its derived design at the point (x, i) is a KF\((g^{|B|-1})\) with a resolution \(\{P_{x,i}^B(y,j):y\in B,y\ne x,1\le j\le g/2\}\), where \(P_{x,i}^B(y,j)\) is a parallel class of \((B\setminus \{x,y\})\times I_g\). Let \(\mathcal{A}=\bigcup _{B\in \mathcal{B}}\mathcal{A}_B\). Then it is immediate that \((X\times I_g,\{\{x\}\times I_g: x\in X\},\mathcal{A})\) forms an H(v, g, 4, 3). Moreover, the derived design of this H-design at every point (x, i) can be resolved into \(g(v-1)/2\) holey parallel classes \(P_{x,i}(y,j)=\bigcup _{\begin{array}{c} x,y\in B \\ B\in \mathcal{{B}} \end{array}}P_{x,i}^B(y,j)\) where \(y\in X\setminus \{ x\},1\le j\le g/2\) and every \(P_{x,i}(y,j)\) is a parallel class of \((X\setminus \{x,y\})\times I_g\). Hence we actually obtain an FDH(v, g, 4, 3). This completes the proof. \(\square \)

3 Direct constructions

To construct frame-derived H-designs directly, we utilize automorphism groups.

Let \((X,\mathcal{G},\mathcal{B})\) be an H(v, g, k, t). An automorphism of the H-design is a permutation on X leaving \(\mathcal{G}\) and \(\mathcal{B}\) invariant. The set of all automorphisms of \((X,\mathcal{G},\mathcal{B})\) forms a group, called the full automorphism group and denoted by Aut\((X,\mathcal{G},\mathcal{B})\). Any subgroup of Aut\((X,\mathcal{G},\mathcal{B})\) is called an automorphism group. Suppose that we have a commutative ring \(R=(X,+,\cdot )\) with identity 1. A unit in R is an element that divides 1. Thus the unit group is \(R^\times =\{r\in R:rs=1\) for some \(s\in R\}\). It is easy to know that the unit group is in fact a group. For \(m\in R^\times \), if the permutation \(\sigma _m:x \mapsto m\cdot x\) forms an automorphism of the H-design, then the permutation \(\sigma _m\) or the element m is called a multiplier. Since a KF\((g^v)\) is an H(v, g, 3, 2), m is a multiplier if the permutation \(\sigma _m\) keeps its group set \(\mathcal{G}\) and block set \(\mathcal{B}\) invariant. Furthermore, we say m is a multiplier of the KF\((g^v)\) only if \(\sigma _m\) also keeps its resolution invariant, that is, for any holey parallel class P in the Kirkman frame, \(mP=\{mB:B\in P\}\) is also a holey parallel class. We further define a multiplier of a frame-derived H-design to be a multiplier of both the H-design itself and the Kirkman frames derived at all points. A multiplier group is a group of some multipliers generated under the composition of permutations.

Let \((X,\mathcal{G},\mathcal{B})\) be an FDH(v, g, 4, 3) and we have a commutative ring \(R=(X,+,\cdot )\) with identity. Suppose that \((X,\mathcal{G},\mathcal{B})\) admits two automorphism groups, a multiplier group \(M\le R^\times \) and an automorphism group \(G\le (R,+)\). Under the condition that \(MG\subseteq G\), \((X,\mathcal{G},\mathcal{B})\) admits an automorphism group \(\Omega =\{\tau _{m,g}:m\in M,g\in G\}\), where \(\tau _{m,g}(x)=m\cdot x+g\). Then the group \(\Omega \) determines an equivalence relation on the blocks of \(\mathcal{B}\). Likewise, the group M does so on the holey parallel classes of every derived KF\((g^{v-1})\). With respect to the group \(\Omega \), the \(v(v-1)(v-2)g^3/24\) blocks are partitioned into a number of orbits. With respect to M, the \(g(v-1)/2\) holey parallel classes in any derived Kirkman frame fall into some orbits. Choose any block (holey parallel class, resp.) from each orbit and call it a starter block (starter holey parallel class, resp.). They are called full or short depends on the cardinality of the orbit it belongs to.

Before we elaborate on direct constructions for FDH-designs, we prepare some basics on number theory. Considering \(\mathbb {Z}_v\), the residue ring modulo v, a unit is an element having an inverse modulo v. The unit group \(\mathbb {Z}_v^\times =\{a\in \mathbb {Z}_v^*:\gcd (a,v)=1\}\), having \(\phi (v)\) elements. An integer \(\xi \) is called a primitive root modulo v if the powers of \(\xi \) generate all the residue classes in \(\mathbb {Z}_v^\times \). It is well-known that there are primitive roots modulo v if and only if \(v=2,4,p^l,2p^l\) where p is any odd prime. Hence, for \(v=2,4,p^l,2p^l\), the unit group \(\mathbb {Z}_v^\times \) is cyclic with any primitive root as a generator.

Now we display construction method for FDH(v, 2, 4, 3) by using automorphism groups. Necessarily we require that \(v\equiv 2\) (mod 3). Then we let \(v\equiv 5\) (mod 6) be an odd prime and \(X=\mathbb {Z}_{2v}\). Since \(\phi (2v)=v-1\) for v odd prime, the unit group \(\mathbb {Z}_{2v}^\times \) is a cyclic group of order \(v-1\). So \(\mathbb {Z}_{2v}^\times \) has a unique subgroup M of order k for any k dividing \(v-1\). Then let \(\Omega =\{\tau _{m,g}:m\in M,g\in \mathbb {Z}_{2v}\}\) and \(\mathcal{G}=\{\{i, v + i\} : 0 \le i \le v- 1\}\). By similar approach of the proof in [21, Lemma 2.1], it is readily checked that all \(\mathcal{G}\)-transverse 4-subsets of X have full orbits with respect to \(\Omega \) if M has odd order, so do all \(\mathcal{G}\)-transverse 3-subsets. We omit the detailed procedure but remind the reader to note the difference of the operations in finite fields and residue rings modulo integers. Now suppose \((X,\mathcal{G},\mathcal{B})\) is an FDH(v, 2, 4, 3) where \(\mathcal{G}=\{\{i, v + i\} : 0 \le i \le v- 1\}\). Observe that

  1. (1)

    the permutation \(x\mapsto mx\) keeps \(\mathcal{G}\) invariant for any element \(m\in \mathbb {Z}_{2v}^\times \),

  2. (2)

    an H(v, 2, 4, 3) has \({v(v-1)(v-2)\over 3}\) blocks, and

  3. (3)

    a KF\((2^{v-1})\) has \({2(v-1)(v-2)\over 3}\) blocks forming \(v-1\) holey parallel classes.

It is desirable to choose \(M\le \mathbb {Z}_{2v}^\times \) as a multiplier group where \(2\not \mid |M|\) and then take \(\Omega \) as the automorphism group. Under such circumstances, we should construct \({(v-1)(v-2)\over 6|M|}\) quadruples as starter blocks to produce an H-design. Furthermore, for a given point \(x\in X\), the derived design should be a KF\((2^{v-1})\) having \({v-1\over |M|}\) starter holey parallel classes with \({2(v-2)\over 3}\) blocks each. We state the construction as follows.

Theorem 3.1

Let \(v\equiv 5\) (mod 6) be a prime, \(X=\mathbb {Z}_{2v},\)\(\mathcal{G}=\{\{i, v + i\} : 0 \le i \le v- 1\}\). Further let M be the subgroup of \(\mathbb {Z}_{2v}^\times \) where \(|M|=k\) is odd and \(k|(v-1)\). Then set

$$\begin{aligned} \Omega =\{\tau _{m,g}:m\in M,g\in \mathbb {Z}_{2v}\}. \end{aligned}$$
(1):

If there exists a collection \(\mathcal{B}_0\) of \({(v-1)(v-2)\over 6k}\) quadruples of X such that every \(\mathcal{G}\)-transverse triple orbit is covered exactly once in \(\mathcal{B}_0\), with respect to the group \(\Omega ,\) then \(\mathcal{B}_0\) develops an H(v, 2, 4, 3) \((X,\mathcal{G},\mathcal{B})\) admitting an automorphism group \(\Omega ;\)

(2):

Considering the point 0, if the following two conditions are satisfied : 

(i):

the derived design of \((X,\mathcal{G},\mathcal{B})\) contains \({v-1\over k}\) starter holey parallel classes \(P_0^s,s\in S\), missing a group \(G_s=\{s,s+v\}\), where S is a complete representative system of all cosets of M in \(\mathbb {Z}_{2v}^\times \), and

(ii):

all the triples in \(\bigcup _{s\in S}P_0^s\) belong to pairwise distinct orbits with respect to M,

then \((X,\mathcal{G},\mathcal{B})\) forms an FDH(v, 2, 4, 3) admitting a multiplier group M.

Lemma 3.2

There exists an FDH(41, 2, 4, 3).

Proof

First we construct an H(41, 2, 4, 3) on \(\mathbb {Z}_{82}\) with groups \(G_i=\{i,i+41\}\), \(0\le i\le 40\) by Theorem 3.1. The construction makes use of an automorphism group \(\Omega =\{\tau _{m,g}:m\in M,g\in \mathbb {Z}_{82}\}\), where \(M=\langle 37\rangle \) is a cyclic multiplier group of order 5 with the element 37 as a generator. There are 52 starter blocks listed as follows.

\(18\ 20\ 22\ 60\)

\(25\ 34\ 70\ 74\)

\(5\ 6\ 48\ 52\)

\(4\ 8\ 22\ 37\)

\(11\ 21\ 37\ 80\)

\(21\ 37\ 55\ 68\)

\(14\ 30\ 56\ 75\)

\(3\ 7\ 28\ 75\)

\(6\ 21\ 72\ 81\)

\(1\ 16\ 26\ 33\)

\(2\ 17\ 60\ 80\)

\(2\ 21\ 25\ 59\)

\(22\ 23\ 24\ 39\)

\(0\ 15\ 33\ 34\)

\(2\ 12\ 25\ 31\)

\(17\ 21\ 32\ 71\)

\(2\ 6\ 60\ 69\)

\(6\ 10\ 38\ 75\)

\(3\ 4\ 37\ 47\)

\(12\ 15\ 57\ 71\)

\(5\ 7\ 15\ 64\)

\(19\ 20\ 70\ 72\)

\(0\ 29\ 36\ 69\)

\(4\ 7\ 11\ 18\)

\(18\ 24\ 27\ 36\)

\(15\ 27\ 76\ 79\)

\(0\ 8\ 29\ 40\)

\(2\ 38\ 60\ 64\)

\(4\ 8\ 16\ 32\)

\(12\ 14\ 18\ 70\)

\(6\ 20\ 39\ 65\)

\(5\ 22\ 67\ 79\)

\(11\ 18\ 20\ 69\)

\(0\ 1\ 9\ 21\)

\(8\ 22\ 56\ 73\)

\(11\ 13\ 26\ 35\)

\(0\ 2\ 11\ 28\)

\(12\ 32\ 56\ 79\)

\(5\ 30\ 33\ 81\)

\(5\ 8\ 52\ 58\)

\(0\ 9\ 14\ 36\)

\(11\ 14\ 38\ 72\)

\(6\ 17\ 51\ 53\)

\(14\ 16\ 69\ 76\)

\(4\ 23\ 57\ 69\)

\(29\ 46\ 67\ 81\)

\(32\ 40\ 69\ 75\)

\(3\ 26\ 31\ 74\)

\(29\ 32\ 55\ 77\)

\(1\ 2\ 23\ 55\)

\(3\ 10\ 62\ 64\)

\(4\ 56\ 65\ 73\)

   

Second we need to prove that the derived design of the H-design actually forms an OLKF\((2^{41})\). We only need to check the derived design at the point \(x=0\) is a KF\((2^{40})\). We list below eight holey parallel classes \(P_0^s,s\in S=\{5,13,15,19,59,77,79,81 \}\), meaning the holes are \(G_s=\{s,s+41\}\). All its 40 holey parallel classes will be produced from the action of the multiplier group M. More precisely, \(P_0^{mi}=\{\{ma,mb,mc\}:\{a,b,c\}\in P_0^{i}\}\) for all \(m\in M\) are all the holey parallel classes of the derived design at the point 0.

\(P_0^{5}:\)

\(1\ 31\ 63\)

\(25\ 58\ 75\)

\(4\ 14\ 81\)

\(32\ 33\ 72\)

\(8\ 21\ 73\)

\(7\ 55\ 69\)

\(11\ 38\ 43\)

\(6\ 15\ 16\)

\(10\ 18\ 61\)

\(19\ 27\ 64\)

\(13\ 20\ 68\)

\(39\ 51\ 66\)

\(9\ 57\ 70\)

\(47\ 49\ 54\)

\(23\ 28\ 71\)

\(3\ 45\ 59\)

\(12\ 29\ 48\)

\(2\ 44\ 78\)

\(17\ 62\ 74\)

\(24\ 26\ 30\)

\(35\ 52\ 77\)

\(40\ 65\ 79\)

\(42\ 76\ 80\)

\(36\ 53\ 56\)

\(22\ 50\ 67\)

\(34\ 37\ 60\)

    

\(P_0^{13}:\)

\(1\ 36\ 38\)

\(50\ 62\ 63\)

\(15\ 44\ 76\)

\(34\ 70\ 79\)

\(9\ 24\ 28\)

\(12\ 40\ 49\)

\(16\ 39\ 56\)

\(21\ 61\ 75\)

\(46\ 59\ 73\)

\(22\ 37\ 52\)

\(4\ 6\ 48\)

\(11\ 77\ 80\)

\(23\ 26\ 68\)

\(8\ 35\ 53\)

\(31\ 45\ 69\)

\(20\ 33\ 58\)

\(17\ 42\ 47\)

\(10\ 25\ 57\)

\(51\ 60\ 67\)

\(27\ 32\ 71\)

\(2\ 65\ 81\)

\(3\ 7\ 14\)

\(29\ 55\ 66\)

\(18\ 43\ 78\)

\(5\ 64\ 72\)

\(19\ 30\ 74\)

    

\(P_0^{15}:\)

\(1\ 43\ 47\)

\(34\ 61\ 74\)

\(2\ 9\ 20\)

\(4\ 52\ 73\)

\(37\ 42\ 66\)

\(59\ 64\ 80\)

\(3\ 54\ 62\)

\(16\ 28\ 51\)

\(23\ 25\ 33\)

\(48\ 72\ 77\)

\(39\ 45\ 75\)

\(7\ 32\ 65\)

\(5\ 13\ 14\)

\(30\ 46\ 57\)

\(26\ 36\ 60\)

\(8\ 29\ 40\)

\(27\ 55\ 81\)

\(6\ 35\ 38\)

\(53\ 63\ 76\)

\(11\ 50\ 78\)

\(10\ 17\ 67\)

\(24\ 58\ 79\)

\(44\ 69\ 70\)

\(12\ 18\ 68\)

\(19\ 21\ 22\)

\(31\ 49\ 71\)

    

\(P_0^{19}:\)

\(1\ 58\ 72\)

\(17\ 44\ 56\)

\(53\ 66\ 79\)

\(32\ 57\ 67\)

\(33\ 51\ 55\)

\(6\ 10\ 71\)

\(18\ 24\ 69\)

\(36\ 74\ 81\)

\(14\ 21\ 42\)

\(7\ 11\ 20\)

\(22\ 26\ 46\)

\(2\ 29\ 52\)

\(5\ 37\ 39\)

\(3\ 12\ 76\)

\(34\ 64\ 65\)

\(13\ 40\ 59\)

\(4\ 61\ 68\)

\(23\ 49\ 78\)

\(16\ 25\ 80\)

\(30\ 50\ 73\)

\(15\ 31\ 70\)

\(28\ 35\ 62\)

\(27\ 47\ 75\)

\(9\ 38\ 48\)

\(43\ 54\ 77\)

\(8\ 45\ 63\)

    

\(P_0^{59}:\)

\(1\ 25\ 68\)

\(3\ 24\ 50\)

\(31\ 44\ 74\)

\(7\ 66\ 70\)

\(16\ 32\ 64\)

\(21\ 28\ 80\)

\(12\ 52\ 72\)

\(10\ 23\ 29\)

\(43\ 63\ 67\)

\(30\ 58\ 77\)

\(5\ 8\ 60\)

\(20\ 34\ 69\)

\(36\ 57\ 79\)

\(4\ 15\ 54\)

\(13\ 33\ 47\)

\(17\ 35\ 61\)

\(22\ 51\ 76\)

\(38\ 45\ 46\)

\(39\ 49\ 81\)

\(9\ 71\ 78\)

\(6\ 27\ 42\)

\(11\ 73\ 75\)

\(19\ 40\ 56\)

\(14\ 48\ 65\)

\(26\ 37\ 53\)

\(2\ 55\ 62\)

    

\(P_0^{77}:\)

\(1\ 5\ 15\)

\(3\ 51\ 57\)

\(7\ 27\ 29\)

\(4\ 13\ 75\)

\(12\ 71\ 80\)

\(18\ 74\ 76\)

\(46\ 53\ 70\)

\(50\ 58\ 69\)

\(2\ 8\ 26\)

\(25\ 37\ 67\)

\(23\ 24\ 52\)

\(31\ 43\ 61\)

\(16\ 20\ 65\)

\(11\ 17\ 21\)

\(6\ 78\ 79\)

\(28\ 54\ 55\)

\(10\ 48\ 49\)

\(9\ 22\ 44\)

\(34\ 35\ 42\)

\(14\ 19\ 66\)

\(56\ 63\ 72\)

\(30\ 32\ 59\)

\(33\ 39\ 62\)

\(38\ 47\ 60\)

\(40\ 45\ 64\)

\(68\ 73\ 81\)

    

\(P_0^{79}:\)

\(1\ 46\ 62\)

\(8\ 22\ 66\)

\(25\ 47\ 56\)

\(19\ 58\ 80\)

\(3\ 23\ 67\)

\(20\ 27\ 76\)

\(9\ 15\ 63\)

\(50\ 55\ 72\)

\(17\ 39\ 53\)

\(2\ 49\ 68\)

\(4\ 5\ 36\)

\(29\ 54\ 60\)

\(10\ 64\ 81\)

\(6\ 26\ 33\)

\(13\ 35\ 73\)

\(42\ 51\ 74\)

\(30\ 31\ 48\)

\(7\ 75\ 78\)

\(12\ 21\ 69\)

\(28\ 37\ 70\)

\(32\ 34\ 77\)

\(14\ 45\ 52\)

\(24\ 61\ 71\)

\(18\ 44\ 65\)

\(11\ 57\ 59\)

\(16\ 40\ 43\)

    

\(P_0^{81}:\)

\(1\ 22\ 54\)

\(16\ 48\ 66\)

\(18\ 30\ 79\)

\(20\ 44\ 67\)

\(8\ 25\ 70\)

\(59\ 63\ 75\)

\(10\ 52\ 55\)

\(26\ 51\ 73\)

\(38\ 78\ 80\)

\(9\ 13\ 61\)

\(23\ 32\ 62\)

\(42\ 50\ 71\)

\(35\ 43\ 49\)

\(6\ 39\ 74\)

\(21\ 27\ 58\)

\(33\ 68\ 76\)

\(19\ 28\ 34\)

\(4\ 24\ 60\)

\(2\ 53\ 72\)

\(36\ 46\ 64\)

\(11\ 45\ 47\)

\(12\ 14\ 31\)

\(5\ 7\ 17\)

\(15\ 37\ 65\)

\(29\ 57\ 77\)

\(3\ 56\ 69\)

    

\(\square \)

Lemma 3.3

There exists an FDH(53, 2, 4, 3).

Proof

First we construct an H(53, 2, 4, 3) on \(\mathbb {Z}_{106}\) with groups \(G_i=\{i,i+53\}\), \(0\le i\le 52\) by Theorem 3.1. The construction makes use of an automorphism group \(\Omega =\{\tau _{m,g}:m\in M,g\in \mathbb {Z}_{106}\}\), where \(M=\langle 13\rangle \) is a cyclic multiplier group of order 13 with the element 13 as generator. There are 34 starter blocks listed as follows.

11 19 30 41

17 18 20 52

2 16 17 100

11 13 24 31

44 85 101 102

31 40 89 101

3 10 12 64

5 14 19 82

0 69 90 99

10 73 80 88

5 22 33 80

15 46 47 90

13 23 40 47

8 48 84 91

3 32 43 78

24 34 35 84

4 43 68 78

9 13 46 101

9 20 42 91

0 5 30 33

23 26 30 50

0 13 34 45

12 24 47 48

2 36 77 78

9 21 52 94

9 15 25 92

5 47 60 70

2 23 64 103

4 14 32 87

20 22 24 95

14 42 48 72

4 8 20 82

1 31 63 103

0 6 18 66

 

Second we need to prove that the derived design of the H-design actually forms an OLKF\((2^{53})\). We only need to check the derived design at the point \(x=0\) is a KF\((2^{52})\). We list below four holey parallel classes \(P_0^s,s\in S=\{15,39,61,93\}\), meaning the holes are \(G_s=\{s,s+53\}\). All its 52 holey parallel classes will be produced from the action of the multiplier group M.

\(P_0^{15}:\)

\(1\ 37\ 39\)

\(12\ 59\ 92\)

\(6\ 29\ 32\)

\(41\ 42\ 72\)

\(19\ 57\ 80\)

\(2\ 17\ 46\)

\(52\ 61\ 91\)

\(43\ 58\ 81\)

\(78\ 93\ 105\)

\(25\ 47\ 63\)

\(9\ 84\ 90\)

\(11\ 16\ 71\)

\(51\ 64\ 79\)

\(8\ 67\ 86\)

\(22\ 34\ 103\)

\(13\ 40\ 97\)

\(3\ 4\ 50 \)

\(14\ 26\ 73\)

\(31\ 54\ 87\)

\(5\ 55\ 82\)

\(33\ 88\ 102\)

\(36\ 76\ 104\)

\(35\ 65\ 96\)

\(7\ 45\ 56\)

\(23\ 24\ 94\)

\(60\ 62\ 77\)

\(28\ 30\ 66\)

\(44\ 49\ 100\)

\(27\ 38\ 48\)

\(20\ 74\ 98\)

\(69\ 83\ 101\)

\(18\ 75\ 89\)

\(21\ 95\ 99\)

\(10\ 70\ 85\)

  

\(P_0^{39}:\)

\(1\ 45\ 100\)

\(21\ 66\ 86\)

\(16\ 30\ 73\)

\(22\ 51\ 74\)

\(2\ 4\ 75 \)

\(5\ 94\ 101\)

\(6\ 71\ 103\)

\(62\ 88\ 104\)

\(40\ 48\ 52\)

\(32\ 35\ 41\)

\(12\ 42\ 70 \)

\(15\ 60\ 89\)

\(46\ 87\ 96\)

\(56\ 61\ 76\)

\(9\ 81\ 93\)

\(3\ 19\ 82\)

\(20\ 25\ 33 \)

\(63\ 84\ 91\)

\(44\ 77\ 98\)

\(14\ 78\ 79\)

\(31\ 37\ 67\)

\(24\ 28\ 47\)

\(7\ 10\ 49\)

\(50\ 57\ 95\)

\(26\ 34\ 97\)

\(8\ 36\ 99\)

\(11\ 27\ 55\)

\(54\ 85\ 105\)

\(13\ 23\ 64\)

\(59\ 69\ 80\)

\(17\ 58\ 72\)

\(18\ 38\ 102\)

\(43\ 68\ 90\)

\(29\ 65\ 83\)

  

\(P_0^{61}:\)

\(1\ 16\ 23\)

\(71\ 97\ 99\)

\(21\ 65\ 67\)

\(27\ 51\ 56\)

\(5\ 15\ 79\)

\(29\ 40\ 75\)

\(7\ 24\ 87\)

\(2\ 33\ 103\)

\(37\ 59\ 85\)

\(9\ 13\ 105\)

\(45\ 49\ 80\)

\(46\ 54\ 91\)

\(25\ 68\ 83\)

\(3\ 14\ 38\)

\(20\ 32\ 81\)

\(77\ 90\ 95\)

\(30\ 88\ 89 \)

\(63\ 92\ 101\)

\(19\ 52\ 94\)

\(35\ 50\ 69\)

\(41\ 57\ 58\)

\(17\ 74\ 76\)

\(43\ 44\ 72\)

\(10\ 62\ 82\)

\(36\ 48\ 78\)

\(34\ 55\ 102\)

\(4\ 22\ 42\)

\(12\ 31\ 64\)

\(73\ 93\ 98\)

\(11\ 26\ 86\)

\(18\ 47\ 60\)

\(39\ 66\ 100\)

\(6\ 84\ 96\)

\(28\ 70\ 104\)

  

\(P_0^{93}:\)

\(1\ 44\ 75\)

\(15\ 91\ 103\)

\(13\ 26\ 74\)

\(52\ 79\ 99\)

\(38\ 76\ 77\)

\(30\ 81\ 96\)

\(39\ 51\ 97\)

\(46\ 49\ 92\)

\(55\ 69\ 73\)

\(12\ 32\ 89\)

\(22\ 71\ 95\)

\(2\ 3\ 63\)

\(47\ 61\ 102\)

\(50\ 56\ 83\)

\(7\ 59\ 86\)

\(10\ 19\ 37\)

\(6\ 8\ 25\)

\(4\ 18\ 68\)

\(24\ 72\ 100\)

\(16\ 31\ 90\)

\(9\ 33\ 57\)

\(27\ 60\ 64\)

\(11\ 54\ 67\)

\(45\ 65\ 85\)

\(23\ 36\ 80\)

\(42\ 43\ 62\)

\(41\ 66\ 87\)

\(48\ 88\ 94\)

\(70\ 82\ 105\)

\(17\ 20\ 78\)

\(21\ 84\ 104\)

\(28\ 34\ 58\)

\(5\ 29\ 98\)

\(14\ 35\ 101\)

  

\(\square \)

An H-design with v groups of cardinality g is said to be cyclic if it admits an automorphism consisting of a cycle of length gv. So the FDH-designs constructed in Theorem 3.1 are cyclic. However, it is shown in [3, Theorem 12] that there is no cyclic H(v, 2, 4, 3) for \(v\equiv 7,11\) (mod 12). Thus we may apply another automorphism group in the direct construction.

Theorem 3.4

Let the notations v, \(X,\mathcal{G}\), M, and k be the same as those in Theorem 3.1. Then set

$$\begin{aligned} \Lambda =\{\tau _{m,g}:m\in M,g\in \mathbb {Z}_{2v},2|g\}. \end{aligned}$$
(1):

If there exists a collection \(\mathcal{B}_0\) of \({(v-1)(v-2)\over 3k}\) quadruples of X such that every \(\mathcal{G}\)-transverse triple orbit is covered exactly once in \(\mathcal{B}_0\), with respect to the group \(\Lambda ,\) then \(\mathcal{B}_0\) develops an H(v, 2, 4, 3) \((X,\mathcal{G},\mathcal{B})\) admitting an automorphism group \(\Lambda ;\)

(2):

Considering the point \(x=0,v\) respectively, if the following two conditions are satisfied : 

(i):

the derived design of \((X,\mathcal{G},\mathcal{B})\) contains \({v-1\over k}\) starter holey parallel classes \(P_x^s,s\in S\), missing a group \(G_s=\{s,s+v\}\), where S is a complete representative system of all cosets of M in \(\mathbb {Z}_{2v}^\times \), and

(ii):

all the triples in \(\bigcup _{s\in S}P_x^s\) belong to pairwise distinct orbits with respect to M,

then \((X,\mathcal{G},\mathcal{B})\) forms an FDH(v, 2, 4, 3) admitting a multiplier group M.

Lemma 3.5

There exists an FDH(23, 2, 4, 3).

Proof

First we construct an H(23, 2, 4, 3) on \(\mathbb {Z}_{46}\) with groups \(G_i=\{i,i+23\}\), \(0\le i\le 22\) by Theorem 3.4. The construction makes use of an automorphism group \(\Lambda =\{\tau _{m,g}:m\in M,g\in \mathbb {Z}_{46},2|g\}\), where \(M=\langle 3\rangle \) is a multiplier group of order 11. There are 14 starter blocks listed as follows.

\(6\ 13\ 18\ 45\)

\(0\ 4\ 9\ 17\)

\(8\ 10\ 12\ 18\)

\(0\ 3\ 6\ 20\)

\(0\ 7\ 21\ 31\)

\(3\ 7\ 17\ 21\)

\(5\ 17\ 22\ 44\)

\(2\ 3\ 6\ 37\)

\(4\ 12\ 13\ 39\)

\(5\ 11\ 19\ 20\)

\(9\ 11\ 20\ 36\)

\(3\ 15\ 37\ 40\)

\(1\ 2\ 6\ 36\)

\(0\ 11\ 14\ 20\)

 

Second we prove that the derived design of the H-design actually forms an OLKF\((2^{23})\). We only need to check the derived design at the point \(x=0,23\) is a KF\((2^{22})\). For either of the two derived H(22, 2, 3, 2)s, we list below two holey parallel classes \(P_x^7\) and \(P_x^{25}\), meaning the holes are \(G_2=\{2,25\}\) and \(G_7=\{7,30\}\). All its 22 holey parallel classes will be produced from the action of the multiplier group M.

\(P_0^7:\)

\(10\ 16\ 20\)

\(12\ 29\ 40\)

\(6\ 18\ 35\)

\(13\ 37\ 43\)

\(9\ 36\ 39\)

\(5\ 22\ 25\)

\(2\ 3\ 31\)

\(1\ 27\ 38\)

\(21\ 24\ 41\)

\(11\ 17\ 44\)

\(4\ 34\ 45\)

\(8\ 15\ 26\)

\(14\ 33\ 42\)

\(19\ 28\ 32\)

\(P_0^{25}:\)

\(3\ 8\ 43\)

\(9\ 29\ 42\)

\(5\ 12\ 27\)

\(37\ 41\ 44\)

\(18\ 22\ 32\)

\(6\ 10\ 36\)

\(4\ 30\ 40\)

\(19\ 34\ 38\)

\(1\ 14\ 31\)

\(11\ 21\ 26\)

\(7\ 13\ 17\)

\(33\ 39\ 45\)

\(20\ 28\ 35\)

\(15\ 16\ 24\)

\(P_{23}^7:\)

\( 10\ 20\ 36 \)

\( 29\ 32\ 45 \)

\( 19\ 33\ 37 \)

\( 13\ 25\ 35 \)

\( 11\ 15\ 27 \)

\( 26\ 31\ 42 \)

\( 3\ 14\ 22 \)

\( 21\ 28\ 38 \)

\( 1\ 2\ 40 \)

\( 5\ 41\ 44 \)

\( 12\ 17\ 43 \)

\( 4\ 9\ 39 \)

\( 8\ 16\ 34 \)

\( 6\ 18\ 24 \)

\(P_{23}^{25}:\)

\( 9\ 16\ 28 \)

\( 30\ 37\ 42 \)

\( 14\ 18\ 31 \)

\( 7\ 12\ 20 \)

\( 4\ 15\ 29 \)

\( 22\ 39\ 43 \)

\( 1\ 11\ 13 \)

\( 19\ 32\ 40 \)

\( 6\ 44\ 45 \)

\( 34\ 35\ 38 \)

\( 10\ 17\ 27 \)

\( 21\ 26\ 41 \)

\( 3\ 5\ 8 \)

\( 24\ 33\ 36 \)

\(\square \)

Lemma 3.6

There exists an FDH(v, 2, 4, 3) for \(v\in \{23,29,41,47,53,59,71,83\}\).

Proof

See Lemmas 3.2, 3.3 and 3.5 for \(v=23,41,53\). See Appendix for other orders. \(\square \)

4 Main result

In this section we list our main result on frame-derived H-designs and large sets of Kirkman triple systems.

Theorem 4.1

There exists an FDH(v, 2, 4, 3), where

(1):

\(v\in \{23,29,41,47,53,59,71,83\},\)

(2):

\(v=2^{2m-1}\) with \(m\ge 2\), and

(3):

\(v = q^n + 1\) with \(n \ge 1\), \(q\equiv 7\) (mod 12) and q is a prime power.

Proof

Combine Lemmas 2.6 and 3.6. \(\square \)

Theorem 4.2

If both a RPCS\((6^{k-1}:3)\) and an OLKF\((6^k)\) exist for any \(k\in \{6,7,\ldots ,40\}\setminus \{8,14,17,20,21,22,23,25,26,29,32\}\), then an LKTS(v) exists for any positive integer \(v\equiv 3\) (mod 6).

Proof

There exists an RPCS\((6^{v-1}:3)\) and an OLKF\((6^v)\) for \(v\in \{8,20,23,29,32\}\) by Lemmas 2.3 and 2.4, because an FDH(v, 2, 4, 3) exists by Theorem 4.1. Then the conclusion follows by applying Theorem 2.1. \(\square \)

To conclude this paper we remark that, to guarantee the existence of LKTS\((6n+3)\) for all n, 24 more pairs of RPCS\((6^{k-1}:3)\) and OLKF\((6^k)\) should be constructed. We are making effort of doing this, although much challenge of direct constructive method and computational efficiency is confronted. Further study on frame derived H-designs is also worthwhile.

Step backward from FDH(v, 2, 4, 3)s, we may consider the existence of RDSQSs, because it is a less restricted concept by Lemma 2.4 but it could also produce LKTS by Lemma 2.2. In the end we list a table for the existence of small orders of RDSQS for future work (where “y” stands for “yes”). Research of direct constructions and recursive constructions is very worthwhile.

Table: \({{\text {RDSQS}}}(v)\) with \(v\le 100\)

v

4

10

16

22

28

34

40

46

52

58

64

70

76

82

88

94

100

Existence

y

y

y

?

y

?

y

y

?

y

y

?

?

y

?

y

y