Kirkman Square Research Articles

Doubly Resolvable Canonical Kirkman Packing Designs and its Applications

Let $$v \equiv 4 \pmod 6$$ and $$v\ge 4$$ . A canonical Kirkman packing design of order v, denoted by $$\hbox {CKPD}(v)$$ , is a resolvable packing with $$r=(v-4)/2$$ parallel classes such that (i) each parallel class consists of a size 4 block and $$(v- 4)/3$$ triples; (ii) the leave consists of the union of $$(v- 4)/2$$ vertex-disjoint edges and a $$K_4$$ with no vertices in common with those edges. A canonical Kirkman packing design is said to be doubly resolvable if there exist a pair of orthogonal resolutions. A doubly resolvable packing design is the generalization of the Kirkman square, which is inextricably bound up with the existence of some constant weight codes such as constant composition codes and permutation codes, etc. In this paper, we establish the spectra of doubly resolvable $$\hbox {CKPD}(v)\hbox {s}$$ with 36 possible exceptions for v. As its direct application, a class of permutation codes are obtained.

On generalized Howell designs with block size three

In this paper, we examine a class of doubly resolvable combinatorial objects. Let $$t, k, \lambda , s$$t,k,ź,s and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-$$\mathrm {GHD}_{k}(s,v;\lambda )$$GHDk(s,vźź), is an $$s\times s$$s×s array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than $$\lambda $$ź cells. Particular instances of the parameters correspond to Howell designs, doubly resolvable balanced incomplete block designs (including Kirkman squares), doubly resolvable nearly Kirkman triple systems, and simple orthogonal multi-arrays (which themselves generalize mutually orthogonal Latin squares). Generalized Howell designs also have connections with permutation arrays and multiply constant-weight codes. In this paper, we concentrate on the case that $$t=2$$t=2, $$k=3$$k=3 and $$\lambda =1$$ź=1, and write $$\mathrm {GHD}(s,v)$$GHD(s,v). In this case, the number of empty cells in each row and column falls between 0 and $$(s-1)/3$$(s-1)/3. Previous work has considered the existence of GHDs on either end of the spectrum, with at most 1 or at least $$(s-2)/3$$(s-2)/3 empty cells in each row or column. In the case of one empty cell, we correct some results of Wang and Du, and show that there exists a $$\mathrm {GHD}(n+1,3n)$$GHD(n+1,3n) if and only if $$n \ge 6$$nź6, except possibly for $$n=6$$n=6. In the case of two empty cells, we show that there exists a $$\mathrm {GHD}(n+2,3n)$$GHD(n+2,3n) if and only if $$n \ge 6$$nź6. Noting that the proportion of cells in a given row or column of a $$\mathrm {GHD}(s,v)$$GHD(s,v) which are empty falls in the interval [0, 1 / 3), we prove that for any $$\pi \in [0,5/18]$$źź[0,5/18], there is a $$\mathrm {GHD}(s,v)$$GHD(s,v) whose proportion of empty cells in a row or column is arbitrarily close to $$\pi $$ź.

Some new resolvable GDDs with [formula omitted] and doubly resolvable GDDs with [formula omitted

A doubly resolvable packing design with block size k, index λ, replication number r, and v elements is called a generalized Kirkman square and denoted by GKSk(v;1,λ;r). Existence of GKS3(4u;1,1;2(u−1))s and GKS3(6u;1,1;3(u−1))s is implied by existence of doubly resolvable group divisible designs with block size 3, index 1, and types 4u and 6u (i.e., (3,1)-DRGDDs of types 4u and 6u). In this paper, we establish the spectra of (3,1)-DRGDDs of types 4u and 6u with 15 and 31 possible exceptions, respectively. As applications, we get some new classes of permutation codes and doubly constant weight codes. We also construct 5 new resolvable GDDs with block size 4 and index 1.

The asymptotic existence of DR $$(v,k,k-1)$$ ( v , k , k - 1 ) -BIBDs

A Kirkman square with index $$\lambda $$?, latinicity $$\mu $$μ, block size $$k$$k, and $$v$$v points, $$KS_k(v;\mu ,\lambda )$$KSk(v?μ,?), is a $$t\times t$$t×t array ($$t=\lambda (v-1)/\mu (k-1)$$t=?(v-1)/μ(k-1) ) defined on a $$v$$v-set $$V$$V such that (1) every point of $$V$$V is contained in precisely $$\mu $$μ cells of each row and column, (2) each cell of the array is either empty or contains a $$k$$k-subset of $$V$$V, and (3) the collection of blocks obtained from the non-empty cells of the array is a $$(v,k,\lambda )$$(v,k,?)-BIBD. For $$\mu = 1$$μ=1, the existence of a $$KS_k(v;\mu ,\lambda )$$KSk(v?μ,?) is equivalent to the existence of a doubly resolvable $$(v,k,\lambda )$$(v,k,?)-BIBD. The asymptotic existence of $$KS_k(v;1,1)$$KSk(v?1,1) was established in 2009. In this paper we establish necessary and sufficient conditions for the asymptotic existence of $$KS_k(v;1,k-1)$$KSk(v?1,k-1) or DR$$(v,k,k-1)$$(v,k,k-1)-BIBDs.

Optimal doubly constant weight codes

AbstractA doubly constant weight code is a binary code of length n1 + n2, with constant weight w1 + w2, such that the weight of a codeword in the first n1 coordinates is w1. Such codes have applications in obtaining bounds on the sizes of constant weight codes with given minimum distance. Lower and upper bounds on the sizes of such codes are derived. In particular, we show tight connections between optimal codes and some known designs such as Howell designs, Kirkman squares, orthogonal arrays, Steiner systems, and large sets of Steiner systems. These optimal codes are natural generalization of Steiner systems and they are also called doubly Steiner systems. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 137–151, 2008

A few more Kirkman squares and doubly near resolvable BIBDs with block size 3

A Kirkman square with index λ , latinicity μ , block size k, and v points, KS k ( v ; μ , λ ) , is a t × t array ( t = λ ( v - 1 ) / μ ( k - 1 ) ) defined on a v -set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a ( v , k , λ ) -BIBD. In a series of papers, Lamken established the existence of the following designs: KS 3 ( v ; 1 , 2 ) with at most six possible exceptions [E.R. Lamken, The existence of doubly resolvable ( v , 3 , 2 ) -BIBDs, J. Combin. Theory Ser. A 72 (1995) 50–76], KS 3 ( v ; 2 , 4 ) with two possible exceptions [E.R. Lamken, The existence of KS 3 ( v ; 2 , 4 ) s , Discrete Math. 186 (1998) 195–216], and doubly near resolvable ( v , 3 , 2 ) -BIBDs with at most eight possible exceptions [E.R. Lamken, The existence of doubly near resolvable ( v , 3 , 2 ) -BIBDs, J. Combin. Designs 2 (1994) 427–440]. In this paper, we construct designs for all of the open cases and complete the spectrum for these three types of designs. In addition, Colbourn, Lamken, Ling, and Mills established the spectrum of KS 3 ( v ; 1 , 1 ) in 2002 with 23 possible exceptions. We construct designs for 11 of the 23 open cases.

The Existence of Kirkman Squares—Doubly Resolvable ( v ,3,1)- BIBD s

A Kirkman square with index λ, latinicity μ, block size k, and v points, KSk(v;μ,λ), is a t×t array (teλ(v−1)/μ(k−1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k,λ)-BIBD. For μe1, the existence of a KSk(v; μ, λ) is equivalent to the existence of a doubly resolvable (v, k, λ)-BIBD. The spectrum of KS2 (v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with λe1. We show that there exist KS3 (v; 1, 1) for v \equiv 3 \pmod 6, ve3 and v≥27 with at most 23 possible exceptions for v.

The existence of KS 3( v; 2, 4)

A Kirkman square with index λ, latinicity μ, block size k and v points, KS k(v; μ, λ), is a t × t array ( t = λ( v − 1/ μ( k − 1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a ( v, k, λ)-BIBD. The existence of KS 2( v; μ, λ) has been completely settled. The existence of a first class for k = 3, KS 3( v; 1, 2), was recently established in Lamken (1995). In this paper, we determine the spectrum of KS 3( v; 2, 4) with at most two possible exceptions for v. Our main construction uses partitioned generalized balanced tournament designs; it is the first case of a very general construction for KS k ( v; μ, λ).

The Existence of Partitioned Generalized Balanced TournamentDesigns with Block Size 3

A generalized balanced tournament design, GBTD(n, k), defined on a kn-set V, is an arrangement of the blocks of a (kn, k, k − 1)-BIBD defined on V into an n × (kn − 1) array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most k cells of each row. Suppose we can partition the columns of a GBTD(n, k) into k + 1 sets B1, B2, …, Bk + 1 where |Bi| = n for i = 1, 2, …, k − 2, |Bi| = n−1 for i = k − 1, k and |Bk+1| = 1 such that (1) every element of V occurs precisely once in each row and column of Bi for i = 1, 2, …, k − 2, and (2) every element of V occurs precisely once in each row and column of Bi ∪ Bk+1 for i = k − 1 and i = k. Then the GBTD(n, k) is called partitioned and we denote the design by PGBTD(n, k). The spectrum of GBTD(n, 3) has been completely determined. In this paper, we determine the spectrum of PGBTD(n,3) with, at present, a fairly small number of exceptions for n. This result is then used to establish the existence of a class of Kirkman squares in diagonal form.

The existence of doubly resolvable ( v, 3, 2)-BIBDs

A Kirkman square with index λ, latinicity μ, block size κ, and ν points, a KS κ(ν; μ, λ), is a t × t array ( t = λ(ν − 1) μ(κ − 1) ) defined on a ν-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a κ-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (ν, κ, λ)-BIBD. For μ = 1, the existence of a KS κ(ν; μ, λ) is equivalent to the existence of a doubly resolvable (ν, κ, λ)-BIBD. The spectrum of KS 2(ν; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum of KS 3(ν; 1, 2) or DR(ν, 3, 2)-BIBDs with at present six possible exceptions for ν.

The existence of doubly near resolvable (v,3,2)‐BIBDs

AbstractThe existence of doubly near resolvable (v,2,1)‐BIBDs was established by Mullin and Wallis in 1975. In this article, we determine the spectrum of a second class of doubly near resolvable balanced incomplete block designs. We prove the existence of DNR(v,3,2)‐BIBDs for v ≡ 1 (mod 3), v ≥ 10 and v ∉ {34,70,85,88,115,124,133,142}. The main construction is a frame construction, and similar constructions can be used to prove the existence of doubly resolvable (v,3,2)‐BIBDs and a class of Kirkman squares with block size 3, KS3(v,2,4). © 1994 John Wiley & Sons, Inc.

The existence of a class of Kirkman squares of index 2

AbstractA Kirkman square with index λ, latinicity μ, block size k and ν points, KSk(v; μ, λ), is a t × t array (t = λ(ν−1)/μ(k − 1)) defined on a ν-set V such that (1) each point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a (ν, k, λ)-BIBD. For μ = 1, the existence of a KSk(ν; μ, λ) is equivalent to the existence of a doubly resolvable (ν, k, λ)-BIBD. In this case the only complete results are for k = 2. The case k = 3, λ = 1 appears to be quite difficult although some existence results are available. For k = 3, λ = 2 the problem seems to be more tractable. In this paper we prove the existence of a KS3(ν; 1, 2) for all ν ≡ 3 (mod 12).

Existence results for KS 3( v; 2, 4) s

A Kirkman square with index λ, latinicity μ, block size k and v points, KS k ( v; μ, λ), is a t × t array ( t = λ( v − 1)/ μ( k − 1)) defined on a v-set V such that (1) each point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a ( v, k, λ)-BIBD. The existence question for KS 2( v; μ, λ) has been completely selttled. We are interested in the next case k = 3. The case k = 3 and μ = λ = 1 appears to be quite difficult, although some existence results are available. For λ > 1 and μ ⩾ 1, the problem is more tractable. In this paper, we prove the existence of KS 3( v; 2, 4) for v ≡ 3 (mod 12), v ≡ 6 (mod 60) and v ≡ 9 (mod 96).

On strong starters in cyclic groups

Strong starters have been very useful in the construction of Room squares and cubes, Howell designs, Kirkman triple systems and Kirkman squares and cubes. In this paper we investigate various properties of strong starters in cyclic groups. In particular, we enumerate all non-isomorphic strong starters in cyclic groups of order n ⩽ 23 and all non-equivalent ones of order n⩽ 27. We also obtain results on the automorphism groups of the corresponding 1-factorizations and their embeddibility in Kirkman triple systems.

A Kirkman square of order 51 and block size 3

A necessary condition for the existence of a Kirkman square KS3(v) is v ≡ 3 (mod 6). It is known that for all v sufficiently large and v ≡ 3 (mod 6) that such a design exists. In order to settle the existence question completely a number of small designs must be constructed directly. Of the 14 possible orders less than 100 only 4 are currently known to exist. In this paper we establish the existence of one more design with order less than 100; namely, v = 51.