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Des. Codes Cryptogr. (2011) 58:321–346 DOI 10.1007/s10623-010-9411-y Completely reducible super-simple designs with block size four and related super-simple packings Hui Zhang · Gennian Ge Received: 12 February 2010 / Revised: 26 May 2010 / Accepted: 4 June 2010 / Published online: 17 June 2010 © Springer Science+Business Media, LLC 2010 Abstract A design is said to be super-simple if the intersection of any two blocks has at most two elements. A super-simple design D with point set X , block set B and index λ is called completely reducible super-simple (CRSS), if its block set B can be written as B = λ i =1 B i , such that B i forms the block set of a design with index unity but having the same parameters as D for each 1 i λ. It is easy to see, the existence of CRSS designs with index λ implies that of CRSS designs with index i for 1 i λ. CRSS designs are closely related to q -ary constant weight codes (CWCs). A (v, 4, q )-CRSS design is just an optimal (v, 6, 4) q +1 code. On the other hand, CRSS group divisible designs (CRSSGDDs) can be used to construct q -ary group divisible codes (GDCs), which have been proved useful in the constructions of q -ary CWCs. In this paper, we mainly investigate the existence of CRSS designs. Three neat results are obtained as follows. Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4) 4 codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type g u are also sufficient except definitely for (g, u ) ∈{(2, 4), (3, 4), (6, 4)}. Finally, we consider the related optimal super-simple (v, 4, 2)-packings and show that such designs exist for all v 4 except definitely for v ∈{4, 5, 6, 9}. Keywords Completely reducible super-simple designs · Constant weight codes · Group divisible designs · Skew Room frames · Super-simple packings Mathematics Subject Classification (2000) Primary 05B05 · 94B25 Communicated by L. Teirlinck. H. Zhang · G. Ge (B ) Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, People’s Republic of China e-mail: [email protected] 123

Completely reducible super-simple designs with block size four and related super-simple packings

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Des. Codes Cryptogr. (2011) 58:321–346DOI 10.1007/s10623-010-9411-y

Completely reducible super-simple designs with blocksize four and related super-simple packings

Hui Zhang · Gennian Ge

Received: 12 February 2010 / Revised: 26 May 2010 / Accepted: 4 June 2010 /Published online: 17 June 2010© Springer Science+Business Media, LLC 2010

Abstract A design is said to be super-simple if the intersection of any two blocks hasat most two elements. A super-simple design D with point set X , block set B and indexλ is called completely reducible super-simple (CRSS), if its block set B can be written asB = ⋃λ

i=1 Bi , such that Bi forms the block set of a design with index unity but having thesame parameters as D for each 1 ≤ i ≤ λ. It is easy to see, the existence of CRSS designswith index λ implies that of CRSS designs with index i for 1 ≤ i ≤ λ. CRSS designs areclosely related to q-ary constant weight codes (CWCs). A (v, 4, q)-CRSS design is just anoptimal (v, 6, 4)q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs)can be used to construct q-ary group divisible codes (GDCs), which have been proved usefulin the constructions of q-ary CWCs. In this paper, we mainly investigate the existence ofCRSS designs. Three neat results are obtained as follows. Firstly, we determine completelythe spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4)4

codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skewRoom frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDDof type gu are also sufficient except definitely for (g, u) ∈ {(2, 4), (3, 4), (6, 4)}. Finally,we consider the related optimal super-simple (v, 4, 2)-packings and show that such designsexist for all v ≥ 4 except definitely for v ∈ {4, 5, 6, 9}.

Keywords Completely reducible super-simple designs · Constant weight codes ·Group divisible designs · Skew Room frames · Super-simple packings

Mathematics Subject Classification (2000) Primary 05B05 · 94B25

Communicated by L. Teirlinck.

H. Zhang · G. Ge (B)Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang,People’s Republic of Chinae-mail: [email protected]

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322 H. Zhang, G. Ge

1 Introduction

If X and R are finite sets, RX denotes the set of vectors of length |X |, where each componentof a vector u ∈ RX has value in R and is indexed by an element of X , that is, u = (ux )x∈X ,and ux ∈ R for each x ∈ X .

A q-ary code of length n is a set C ⊆ ZXq for some X of size n. The elements of C are called

codewords. The Hamming weight of a vector u ∈ ZXq is defined as ‖u‖ = |{x ∈ X : ux �= 0}|.

The distance induced by this norm is called the Hamming distance, denoted dH , so thatdH (u, v) = ‖u − v‖, for u, v ∈ Z

Xq .

For any two vectors u, v ∈ ZXq , define their support as supp(u, v) = {x ∈ X : ux �= vx }.

We write supp(u) instead of supp(u, 0) and also call supp(u) the support of u. A code C issaid to have minimum distance d if dH (u, v) ≥ d for all distinct u, v ∈ C. If ‖u‖ = w forevery codeword u ∈ C, then C is said to be of (constant) weight w. A q-ary constant weightcode (CWC) of length v with weight w and distance d is denoted as a (v, d, w)q code. Thenumber of codewords in a (v, d, w)q code is called the size of the code. The maximum size ofa (v, d, w)q code is denoted Aq(v, d, w). A (v, d, w)q code having Aq(v, d, w) codewordsis said to be optimal.

CWC plays an important role in coding theory [48]. Recently, a lot of work has been doneon constructions of q-ary CWCs, see for example [8,11,13,14,20,25–30,33,37,38,43,46,50–52,58–62,65], due to their important applications in the coding for bandwidth-efficientchannels and the design of oligonucleotide sequences for DNA computing.

A design is said to be super-simple if the intersection of any two blocks has at most twoelements. A super-simple design D with point set X , block set B and index λ is called com-pletely reducible super-simple (CRSS), if its block set B can be written as B = ⋃λ

i=1 Bi ,such that Bi forms the block set of a design with index unity but having the same parametersas D for each 1 ≤ i ≤ λ. It is easy to see, the existence of CRSS designs with index λ impliesthat of CRSS designs with index i for 1 ≤ i ≤ λ.

CRSS designs are closely related to q-ary CWCs. A (v, 4, q)-CRSS design is just anoptimal (v, 6, 4)q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs)can be used to construct q-ary group divisible codes (GDCs), which have been proved usefulin the constructions of q-ary CWCs. In [66], the authors determined the spectrum for optimal(v, 6, 4)3 codes with only two possible exceptions using the previous connection betweenCRSS designs and CWCs.

Compared to CWCs, CRSS designs have not received much attention, but there are stillsome papers dealing with them, see for example [5].

Lemma 1.1 ([5]) For v = 1 and for all v ≡ 1 or 4 (mod 12), v ≥ 13, there exists a(v, 4, 2)-CRSS design.

In this paper, we will continue to investigate the existence of CRSS designs with block sizefour. Three neat results are obtained as follows:

Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a conse-quence, a class of new optimal (v, 6, 4)4 codes is obtained.

Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames,and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type gu

are also sufficient except definitely for (g, u) ∈ {(2, 4), (3, 4), (6, 4)}.Finally, we consider the related super-simple (v, 4, 2)-packings. Super-simple designs

were introduced by Gronau and Mullin [41]. The existence of super-simple designs is aninteresting extremal problem by itself, but there are also useful applications. For example,

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Completely reducible super-simple designs 323

such designs are used in constructing perfect hash families [56] and coverings [10], in theconstruction of new designs [9] and in the construction of superimposed codes [45]. Muchattention has been paid on the existence of super-simple BIBDs with block size four and indexλ ∈ {2, 3, 4, 5, 6, 9} [5,12,15,16,21,40,41,44], or block size five and index λ ∈ {2, 4, 5}[1,4,17,18,42]. However, nothing has been done for the existence of super-simple packings,which have been shown to be closely related to optimal CWCs [66]. We will start the investi-gation on the existence of optimal super-simple packings and show that optimal super-simple(v, 4, 2)-packings exist for all v ≥ 4 except definitely for v ∈ {4, 5, 6, 9}.

The article is organized as follows. In Sect. 2, some definitions and notations, as wellas some preliminary results, are given. In Sect. 3, we shall construct all the (v, 4, 3)-CRSSdesigns and obtain the corresponding optimal (v, 6, 4)4 codes. In Sect. 4, skew Room framesare used to construct a class of (4, 4)-CRSSGDDs. In Sect. 5, we will determine the spectrumfor uniform CRSSGDDs with block size four and index two. In Sect. 6, we will solve com-pletely the related problem for the existence of optimal super-simple packings with blocksize four and index two.

2 Preliminaries

Let K be a set of positive integers and let λ be a positive integer.A group divisible design (GDD), denoted by (K , λ)-GDD, is a triple (X, G, B) where

1. X is a finite set of points,2. G is a set of subsets of X , called groups, which partition X ,3. B is a collection of subsets of X with sizes from K , called blocks, such that every pair

of points from distinct groups occurs in exactly λ blocks,4. no pair of points belonging to a group occurs in any block.

The group-type (or type) of the GDD is the multiset {|G| : G ∈ G}. An “exponential” notationis usually used to describe group-type: a type gu1

1 gu22 . . . gus

s denotes ui occurrences of gi

for 1 ≤ i ≤ s. We always write (K , λ)-GDD as (k, λ)-GDD when K = {k}. Furthermore,we denote (K , 1)-GDD as K -GDD and (k, 1)-GDD as {k}-GDD. A (k, λ)-GDD with grouptype 1v (k < v) is just a balanced incomplete block design, denoted by (v, k, λ)-BIBD. A(k, λ)-GDD of type nk is also called a transversal design, and denoted by TDλ(k, n). Whenλ = 1, it is also denoted as a TD(k, n).

A GDD is called uniform if all of its groups have the same size. The spectra for theexistence of uniform GDDs with block sizes 3 or 4 have been determined (see [32]).

Lemma 2.1 ([68]) The necessary and sufficient conditions for the existence of a (4, λ)-GDDof type gu are (i) u ≥ 4, (ii) λ(u − 1)g ≡ 0 (mod 3), and (iii) λu(u − 1)g2 ≡ 0 (mod 12),with the exception of (g, u, λ) ∈ {(2, 4, 1), (6, 4, 1)}, in which case no such GDD exists.

Lemma 2.2 ([3]) Let n be a positive integer. Then a TD(5, n) exists if n �∈ {2, 3, 6, 10}.For nonuniform GDDs of type gum1, the existence problem for block size 3 has been

solved by Colbourn et al. [24]. The existence problem for block size 4 has been investigatedextensively by Ge et al. [31,34–36,39].

Lemma 2.3 ([36]) There exists a {4}-GDD of type 6u91 for each integer u ≥ 4.

A (v, k, λ)-BIBD with a hole H of size w, denoted (v,w; k, λ)-BIBD, is a design(X, H, B) on an element set X (|X | = v) with a subset H (|H | = w) called the hole,

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and with a block set B (containing blocks of size k), such that every pair of points not both inthe hole occurs in exactly λ blocks, and no block contains two distinct points from the hole.

An incomplete transversal design of order or group size n, block size k, index λ, and holesizes h1, . . . , hs , denoted ITDλ(k, n; h1, . . . , hs) (or TDλ(k, n) − ∑

1≤i≤s TDλ(k, hi )) forshort, is a quadruple (X, G, H, B), where

1. X is a set of kn elements;2. G is a partition of X into k classes (groups), each of size n;3. H is a set of disjoint subsets H1, . . . , Hs of X , with the property that, for each 1 ≤ i ≤ s

and each G ∈ G, |G ∩ Hi | = hi ;4. B is a collection of k-subsets of X (blocks), such that every pair of points from distinct

groups and different holes occurs in exactly λ blocks;5. no pair of points belonging to the same group or hole occurs in any block.

Again, when λ = 1, it can be omitted from the notation.A double group divisible design (DGDD) is a quadruple (X, H, G, B) where X is a set of

points, H and G are partitions of X (into holes and groups, respectively) and B is a collectionof subsets of X (blocks) such that

1. for each block B ∈ B and each hole H ∈ H, |B ∩ H | ≤ 1,2. any pair of distinct points from X which are not in the same hole occurs either in some

group or in exactly λ blocks, but not both.

A (K , λ)-DGDD of type (g1, hv1)

u1(g2, hv2)

u2 . . . (gs, hvs )

us is a double group-divisibledesign in which every block has size from the set K and in which there are ui groups of sizegi , each of which intersects each of the v holes in hi points. A modified group divisible design(K , λ)-MGDD of type gu is a (K , λ)-DGDD of type (g, 1g)u . When λ = 1, a (K , λ)-DGDDis denoted as a K -DGDD, and a (K , λ)-MGDD is abbreviated as a K -MGDD.

The definition of CRSS can be applied to any type of designs described above and servesto introduce some convenient notation. For example, a CRSS BIBD will be denoted as a(v, k, λ)-CRSS design for short, and a CRSS GDD will be denoted as a (k, λ)-CRSSGDD.We have the following connection between CRSS designs and CWCs.

Lemma 2.4 ([66]) If there exists a (v, 4, q)-CRSS design with b blocks, then there exists a(v, 6, 4)q+1 code of size b; if there exists a (4, q)-CRSSGDD of type gt1

1 . . . gtss with g blocks,

then there exists a (q + 1)-ary GDC of type gt11 . . . gts

s of size g.

As in [5], we have the following necessary conditions for CRSSGDDs.

Lemma 2.5 The following conditions are necessary for the existence of a (4, λ)-CRSSGDDsof type gu:

(i) u ≥ 4,(ii) (u − 1)g ≡ 0 (mod 3),

(iii) u(u − 1)g2 ≡ 0 (mod 12),(iv) ug ≥ 2λ + 2g, if g �= 1.

The fourth condition ensures that the total number of available triples is greater than thenumber of triples occurring in the blocks of the design. As a special case, when g = 1, weget the necessary conditions for the existence of a (v, 4, λ)-CRSS designs, i.e., v ≡ 1 or 4(mod 12), and v ≥ 2λ+2 if v �= 1. For the existence of CRSSGDDs, we have the followingknown results.

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Completely reducible super-simple designs 325

Lemma 2.6 ([66]) There exists a (4, 2)-CRSSGDD of type 23t+1 for each integer t ≥ 2.

Lemma 2.7 ([66]) If g ≥ 4 and g �≡ 2 (mod 4), a (4, g)-CRSSGDD of type g4 exists.

A pairwise balanced design ((v, K , λ)-PBD) of order v with block sizes from K is a pair(X, B), where X is a finite set (the point set) of cardinality v and B is a family of subsets(blocks) of X that satisfy (i) if B ∈ B, then |B| ∈ K and (ii) every pair of distinct elementsof X occurs in exactly λ blocks of B. The integer λ is the index of the PBD. If an elementk ∈ K is “starred” (written k�), it means that the PBD has exactly one block of size k.

Lemma 2.8 ([47]) For any integer v ≥ 10, a (v, {5, 6, 7, 8, 9}, 1)-PBD exists with excep-tions v ∈ [10, 20] ∪ [22, 24] ∪ [27, 29] ∪ [32, 34].Lemma 2.9 ([53]) There exists a (v, {4, w�}, 1)-PBD with v > w if and only if v ≥ 3w + 1,and:

(i) v ≡ 1 or 4 (mod 12) and w ≡ 1 or 4 (mod 12); or(ii) v ≡ 7 or 10 (mod 12) and w ≡ 7 or 10 (mod 12).

In the recursive constructions of GDDs and PBDs, the “weighting” technique and Wil-son’s Fundamental Construction (WFC) (see [63]) are quite often used, where we start witha “master” GDD and small input designs to obtain a new GDD. Similar techniques will beapplied in our constructions of CRSSGDDs, where we either start with a CRSSGDD and useTDs as input designs or start with a GDD and use some CRSSGDDs as input designs. Morespecifically, we shall make use of the following two constructions, i.e., inflation and WFCfor GDDs (see, e.g. [23]).

Construction 2.10 (Inflation) Suppose that both a (K , λ)-CRSSGDD of type {h1, h2, . . . ,

hn} and a TD(k, m) for each k ∈ K exist. Then there exists a (K , λ)-CRSSGDD of type{mh1, mh2, . . . , mhn}.Construction 2.11 (WFC) Let (X, G, B) be a GDD, and let w : X → Z+ ∪ {0} be a weightfunction on X. Suppose that for each block B ∈ B, there exists a (K , λ)-CRSSGDD of type{w(x) : x ∈ B}. Then there is a (K , λ)-CRSSGDD of type {∑x∈G w(x) : G ∈ G}.

In the construction of GDDs or PBDs, the technique of “filling in holes” plays an importantrole. The technique for CRSSGDDs is described as follows.

Construction 2.12 (Filling in holes)

(i) Suppose there exists a (K , λ)-CRSSGDD of type {si : 1 ≤ i ≤ n}. Let a ≥ 0 bean integer. For each i, 1 ≤ i ≤ n − 1, if there exists a (K , λ)-CRSSGDD of type{si j : 1 ≤ j ≤ ki } ∪ {a}, where si = ∑

1≤ j≤kisi j , then there is a (K , λ)-CRSSGDD of

type {si j : 1 ≤ j ≤ ki , 1 ≤ i ≤ n − 1} ∪ {a + sn}.(ii) Suppose there exists a (K , λ)-CRSSGDD of type {si : 1 ≤ i ≤ n}. Suppose there exists

also a (K , λ)-CRSSGDD of type {t j : 1 ≤ j ≤ t}, where sn = ∑1≤ j≤t t j . Then there

is a (K , λ)-CRSSGDD of type {si : 1 ≤ i ≤ n − 1} ∪ {t j : 1 ≤ j ≤ t}.Let v ≥ k ≥ t . A t-(v, k, λ) packing is a pair (X, B), where X is a v-set of elements

(points) and B is a collection of k-subsets of X (blocks), such that every t-subset of pointsoccurs in at most λ blocks in B. If λ > 1, then B is allowed to contain repeated blocks. Whent = 2, we simply denote it as a (v, k, λ)-packing.

The packing number Dλ(v, k, t) is the maximum number of blocks in any t-(v, k, λ)

packing.

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326 H. Zhang, G. Ge

Lemma 2.13 ([57]) (First Johnson bound) Dλ(v, k, t) ≤⌊

vDλ(v−1,k−1,t−1)k

⌋. Iterating this

bound yields Dλ(v, k, t) ≤ Uλ(v, k, t), where Uλ(v, k, t) =⌊

vk

⌊v−1k−1 . . .

⌊λ

(v−t+1)k−t+1

⌋⌋⌋.

A t-(v, k, λ) packing (X, B) is said to be optimal if |B| = Uλ(v, k, t).A (v, k, λ)-packing with a hole H of size w, denoted (v,w; k, λ)-packing, is a design

(X, H, B) on an element set X (|X | = v) with a subset H (|H | = w) called the hole, andwith a block set B (containing blocks of size k), such that every pair of points not both in thehole occurs in at most λ blocks, and no block contains two distinct points from the hole.

A (v, k, λ) maximum incomplete packing design (MIPD) with a hole of size w, denoted by(v,w; k, λ)-MIPD, is defined to be a triple (X, Y, A) where X is a v-set (of points), Y ⊆ X isa w-set (called a hole), and A is a collection of k-subsets of X (called blocks) which satisfiesthe following properties:

1. each pair of distinct points x and y from X in which at least one of x and y does not liein Y occurs in at most λ block of A;

2. no block contains two distinct points of Y ;3. λ(v − 1) ≡ λ(w − 1) ≡ d (mod k − 1), where d is a certain integer satisfying 0 ≤ d ≤

k − 2;4. the number of pairs of distinct points from (X × X)\(Y × Y ) which do not occur in any

block of A is exactly d(v − w)/2.

The definition of MIPD was introduced by Yin and Assaf [64].

Construction 2.14 ([64]) Suppose that there exists a (k, λ)-GDD of type {t1, t2, . . . , tn} anda (ti + w,w; k, λ)-MIPD for each 1 ≤ i ≤ n − 1. Then there exists a (t + w, tn + w; k, λ)-MIPD, where t = ∑

1≤i≤n ti .

Construction 2.15 ([64]) Suppose that there exists a (v,w; k, λ)-MIPD. If w ≤ k − 1 oran optimal (w, k, λ)-packing exists, then there exists an optimal (v, k, λ)-packing.

A (v, k, λ)-packing is called super-simple if the intersection of any two blocks of thedesign has at most two elements (This definition also works for a packing with a hole.). Wedenote the packing number of a super-simple t-(v, k, λ)-packing as D′

λ(v, k, t). Obviously,D′

λ(v, k, t) ≤ Dλ(v, k, t).For (v, 4, 2)-packings, the following result has been established.

Lemma 2.16 ([6,7]) D2(v, 4, 2) = U2(v, 4, 2) for v �= 9; D2(9, 4, 2) = U2(9, 4, 2) − 1.

3 Existence of (v, 4, 3)-CRSS designs

In this section, we will determine completely the spectrum for a (v, 4, 3)-CRSS design. Asa consequence, a class of new optimal (v, 6, 4)4 codes is obtained.

Firstly, some small designs are constructed directly. We use the same method as in [66].When finding a CRSS design with index λ, we construct separately λ disjoint designs ofindex unity with the same point set and group type, such that the intersection of any twoblocks of these designs has at most two elements, and combine them together to obtain aCRSS design with index λ.

In the sequence, we often use a multiplier which multiplies the base blocks of the firstdesign to generate the base blocks of the second or the third design.

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Completely reducible super-simple designs 327

Lemma 3.1 There exist (4, 3)-CRSSGDDs of types 47 and 126.

Proof For a (4, 3)-CRSSGDD of type 47, take the point set to be Z28, where the group set is{{0, 7, 14, 21} + i : 0 ≤ i ≤ 6}. The required design is obtained by developing the elementsof Z28 in the following base blocks +4 (mod 28).

The base blocks of the first GDD:

{1, 3, 7, 9} {0, 9, 19, 22} {0, 1, 12, 25} {3, 18, 22, 23}{1, 2, 19, 24} {2, 5, 10, 21} {0, 2, 18, 20} {0, 11, 24, 27}

The second GDD is obtained with a multiplier 5.The third GDD is obtained with a multiplier 17.For a (4, 3)-CRSSGDD of type 126, take the point set to be Z72, where the group set is

{{0, 6, . . . , 66}+ i : 0 ≤ i ≤ 5}. The required design is obtained by developing the elementsof Z72 in the following base blocks +1 (mod 72).

The base blocks of the first GDD:

{0, 39, 55, 64} {0, 15, 29, 52} {0, 28, 62, 69} {0, 22, 26, 27} {0, 2, 13, 53}The second GDD is obtained with a multiplier 11.The third GDD is obtained with a multiplier 13. �

Lemma 3.2 There exists a (4, 3)-CRSSGDD of type 12t for each integer t ≥ 4.

Proof When t ≡ 0 or 1 (mod 4) and t ≥ 4, there exists a (3t + 1, {4}, 1)-PBD by Lemma2.9. Deleting one point from this (3t + 1, {4}, 1)-PBD gives a {4}-GDD of type 3t . Whent ≡ 2 or 3 (mod 4) and t ≥ 7, there exists a (3t +1, {4, 7�}, 1)-PBD by Lemma 2.9. Removeone point from the (3t + 1, {4, 7�}, 1)-PBD which is not in the unique block of size 7 to geta {4, 7�}-GDD of type 3t . Hence, we always have a {4, 7}-GDD of type 3t for any t ≥ 4 andt �= 6.

Apply WFC with weight 4 to obtain a (4, 3)-CRSSGDD of type 12t for any t ≥ 4 andt �= 6. Here, the input designs (4, 3)-CRSSGDDs of types 44 and 47 exist by Lemmas 2.7and 3.1, respectively. For t = 6, the required design is constructed in Lemma 3.1. �Lemma 3.3 There exists a (v, 4, 3)-CRSS design for each v ∈ {25, 28, 37, 40}.Proof For v = 25, the design is constructed based on G F(25) with a primitive polynomialf (x) = x2 + x + 2. The first BIBD is obtained by developing the following base blocksunder the additive automorphism group of G F(25).

{0, 1, x8, x16} {0, x2, x10, x18}The second BIBD is obtained with a multiplier x .The third BIBD is obtained with a multiplier x4.For v = 28, the design is constructed on Z28 with the following base blocks +4 (mod 28).The base blocks of the first BIBD:

{2, 3, 11, 14} {1, 5, 8, 16} {1, 11, 24, 26} {0, 6, 14, 16} {2, 7, 8, 9}{1, 9, 14, 18} {0, 3, 9, 10} {3, 13, 25, 27} {3, 8, 12, 19}

The second BIBD is obtained with a multiplier 5.The third BIBD is obtained with a multiplier 17.For v = 37, the design is constructed on Z37 with the following base blocks +1 (mod 37).

{0, 1, 3, 24} {0, 4, 9, 15} {0, 7, 17, 25}

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The second BIBD is obtained with a multiplier 6.The third BIBD is obtained with a multiplier 31.For v = 40, the design is constructed on Z39 ∪ {∞} with the following base blocks +3

(mod 39), where ∞ keeps fixed under the action of the automorphism group.The base blocks of the first BIBD:

{1, 5, 24, 34} {1, 14, 18, 31} {1, 4, 20, 25} {1, 9, 28, 33} {0, 1, 3, 12}{0, 4, 29,∞} {0, 11, 33, 38} {2, 9, 23, 34} {0, 2, 18, 26} {2, 4, 5, 35}

The base blocks of the second BIBD:

{0, 7, 8, 11} {0, 34, 2,∞} {1, 21, 7, 20} {2, 27, 32, 12} {1, 37, 10, 35}{0, 22, 4, 33} {2, 10, 9, 26} {1, 27, 25, 9} {1, 11, 23, 29} {0, 27, 23, 36}

The third BIBD is obtained with a multiplier 5 acting on the first BIBD. �Theorem 3.4 There exist (v, 4, 3)-CRSS designs forv = 1 and for allv ≡ 1, 4 (mod 12), v ≥13.

Proof For v = 1, the block set is empty. For v ∈ {13, 16}, the designs are given in [5]. Forv ∈ {25, 28, 37, 40}, the required designs come from Lemma 3.3.

For each v ≥ 49, take a (4, 3)-CRSSGDD of type 12t for t ≥ 4 given in Lemma 3.2. Ifwe adjoin one ideal point, and fill in all the holes with (13, 4, 3)-CRSS designs, the result isa (12t + 1, 4, 3)-CRSS design for t ≥ 4. If we adjoin four ideal points, and fill in one holewith a (16, 4, 3)-CRSS design and the other holes with (16, 4; 4, 3)-CRSS designs whichare given in [5], the result is a (12t + 4, 4, 3)-CRSS design for t ≥ 4. �

By Lemma 2.4, we obtain a class of new optimal (v, 6, 4)4 codes, that is:

Corollary 3.5 A4(v, 6, 4) = v(v−1)4 for each v ≡ 1, 4 (mod 12), v ≥ 13.

4 A construction of (4, 4)-CRSSGDDs using skew Room frames

In this section, we will give a general construction for (4, 4)-CRSSGDDs with skew Roomframes.

If {S1, . . . , Sn} is a partition of a set S, an {S1, . . . , Sn}-Room frame is an |S| × |S| array,F , indexed by S, satisfying:

1. Every cell of F either is empty or contains an unordered pair of symbols of S,2. The subarrays Si × Si are empty, for 1 ≤ i ≤ n (these subarrays are holes),3. Each symbol x �∈ Si occurs once in row (or column) s for any s ∈ Si , and4. The pairs occurring in F are those {s, t}, where (s, t) ∈ (S × S)\ ⋃n

i=1(Si × Si ).

The type of an {S1, . . . , Sn}-Room frame F will be the multiset {|S1|, . . . , |Sn |}. We will saythat F has type tu1

1 . . . tukk provided there are u j Si ’s of cardinality t j , for 1 ≤ j ≤ k. A Room

frame is skew if cell (i, j) is filled implies that cell ( j, i) is empty. A Room frame of type 1n

is called a Room square.From a skew Room frame of type tu one can get a {4}-GDD of type (6t)u [54]. The

{4}-GDD is based on {Si × Z6 : 1 ≤ i ≤ n}. The block set B contains all blocks{(a, j), (b, j), (c, 1 + j), (r, 4 + j)}, where j ∈ Z6, {a, b} ∈ F, {a, b} occurs in columnc and row r .

Skew Room frames have played an important role in the constructions of BIBDs andGDDs with block size four [54] and the resolution of the existence problem for weakly3-chromatic BIBDs with block size four [55].

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Completely reducible super-simple designs 329

Lemma 4.1 ([19,67]) The necessary conditions for the existence of a skew Room frameof type tu , namely, u ≥ 4 and t (u − 1) is even, are also sufficient except for (t, u) ∈{(1, 5), (2, 4)} and with possible exceptions:

(i) u = 4 and t ≡ 2 (mod 4),(ii) u = 5 and t ∈ {17, 19, 23, 29, 31}.

Construction 4.2 If there exists a skew Room frame of type tu, then there exists a (4, 4)-CRSSGDD of type (6t)u.

Proof Let F be a given skew Room frame of type tu . We construct four (4, 1)-GDDsof type (6t)u on group set {{(i + k, j) : 0 ≤ i ≤ t − 1, j ∈ Z6} : k = 0, t, . . . ,t (u − 1)}, with four block sets containing all the blocks {(x, j), (y, j), (c, 1 + j), (r, 4 +j)}, {(x, j), (y, j), (c, 4+ j), (r, 1+ j)}, {(x, j), (y, j), (c, 2+ j), (r, 5+ j)}, {(x, j), (y, j),(c, 5 + j), (r, 2 + j)} respectively, where j ∈ Z6, {x, y} ∈ F, {x, y} occurs in column c androw r . It is easily checked that this leads to four {4}-GDDs of type (6t)u with the propertythat the intersection of any two blocks of these designs has at most two elements. �

Combining Lemma 4.1 and Construction 4.2, we have the following result.

Theorem 4.3 If u ≥ 4 and t (u − 1) is even, a (4, 4)-CRSSGDD of type (6t)u exists exceptfor (t, u) ∈ {(1, 5), (2, 4)} and possibly for:

(i) u = 4 and t ≡ 2 (mod 4),(ii) u = 5 and t ∈ {17, 19, 23, 29, 31}.

When t = 1, we have the following result.

Corollary 4.4 If u is odd and u ≥ 7, a (4, 4)-CRSSGDD of type 6u exists.

5 Existence of (4, 2)-CRSSGDDs of type gu

In this section, we will prove that the obvious necessary conditions for the existence of (4, 2)-CRSSGDDs of type gu are also sufficient except definitely for (g, u) ∈ {(2, 4), (3, 4), (6, 4)}.

The necessary conditions for the existence of (4, 2)-CRSSGDDs of type gu given inLemma 2.5 are categorized in Table 1.

By the inflation construction together with the known results in Lemmas 1.1 and 3.2, wemainly need to consider the existence of (4, 2)-CRSSGDDs of types gu for g ∈ {2, 3, 4, 6}.

Table 1 Necessary conditionsfor the existence of a(4, 2)-CRSSGDD of type gu

g u

g ≡ 0 (mod 12) u ≥ 4, u ∈ N

g ≡ 1, 5, 7, 11 (mod 12) u ≥ 4, u ≡ 1, 4 (mod 12), (g, u) �= (1, 4)

g ≡ 2, 10 (mod 12) u ≥ 4, u ≡ 1 (mod 3)

g ≡ 3, 9 (mod 12) u ≥ 4, u ≡ 0, 1 (mod 4)

g ≡ 4, 8 (mod 12) u ≥ 4, u ≡ 1 (mod 3)

g ≡ 6 (mod 12) u ≥ 4, u ∈ N

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330 H. Zhang, G. Ge

5.1 (4, 2)-CRSSGDDs of type 6u

Lemma 5.1 There exists a (4, 2)-CRSSGDD of type 6u for each u ∈ {14, 18}.Proof Take the point set to be Z6u , where the group set is {{0, u, 2u, . . . , 5u} + i : 0 ≤ i ≤u −1}. The required designs are obtained by developing the elements of Z6u in the followingbase blocks +2 (mod 6u).u = 14:

The base blocks of the first GDD:

{1, 10, 11, 18} {1, 21, 32, 69} {0, 9, 12, 27} {0, 18, 38, 59} {0, 22, 45, 47} {0, 2, 7, 51}{0, 26, 55, 61} {0, 48, 79, 80} {0, 6, 50, 74} {1, 14, 53, 77} {0, 13, 17, 43} {0, 3, 54, 65}{0, 19, 57, 69}

The second GDD is obtained with a multiplier 25.u = 18:

The base blocks of the first GDD:

{1, 2, 64, 75} {1, 22, 48, 51} {1, 5, 33, 100} {1, 18, 32, 38} {0, 15, 22, 38} {0, 23, 75, 105}{1, 3, 23, 52} {0, 24, 51, 63} {0, 28, 60, 93} {0, 34, 69, 83} {0, 37, 53, 64} {0, 42, 98, 100}{0, 7, 55, 96} {1, 41, 47, 85} {0, 21, 31, 40} {0, 1, 43, 104} {0, 17, 25, 30}

The second GDD is obtained with a multiplier 5. �Theorem 5.2 There exists a (4, 2)-CRSSGDD of type 6u for each u ≥ 5.

Proof For u ∈ [5, 13], the desired designs are given in [66, Lemma 3.3]. For u ∈ {14, 18},the desired designs are constructed in Lemma 5.1. For u ∈ {15, 17, 19, 23, 27, 29, 33}, thedesired designs come from Corollary 4.4.

For u ∈ {16, 22, 28, 34}, the desired designs can be obtained by inflating (4, 2)-CRSSGDDs of types 2u for u ∈ {16, 22, 28, 34} (see Lemma 2.6) with a TD(4, 3). Foru ∈ {20, 24, 32}, take (4, 2)-CRSSGDDs of types 304 (see Theorem 5.14), 364 or 484 (seeLemma 2.7), and fill in the holes with (4, 2)-CRSSGDDs of types 65, 66 or 68 respectivelyto get the desired designs.

For each u ≥ 35 or u ∈ {21, 25, 26, 30, 31}, take a (u, {5, 6, 7, 8, 9}, 1)-PBD from Lemma2.8, apply WFC with weight 6 and input (4, 2)-CRSSGDDs of types 6s for s ∈ {5, 6, 7, 8, 9}to obtain a (4, 2)-CRSSGDD of type 6u . �

5.2 (4, 2)-CRSSGDDs of type 4u

Lemma 5.3 There exists a (4, 2)-CRSSGDD of type 410.

Proof Take the point set to be Z40, where the group set is {{0, 10, 20, 30} + i : 0 ≤ i ≤ 9}.The required design is obtained by developing the elements of Z40 in the following baseblocks +1 (mod 40).

The base blocks of the first GDD:

{0, 1, 4, 13} {0, 2, 7, 24} {0, 6, 14, 25}The second GDD is obtained with a multiplier 7. �

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Completely reducible super-simple designs 331

Theorem 5.4 There exists a (4, 2)-CRSSGDD of type 4u for each u ≡ 1 (mod 3), u ≥ 4.

Proof For u ∈ {4, 7, 10}, the required designs come from Lemmas 2.7, 3.1 and 5.3, respec-tively.

For each u ≥ 13, adjoining four ideal points to a (4, 2)-CRSSGDD of type 12t for t ≥ 4which exists by Lemma 3.2, and filling in the holes with (4, 2)-CRSSGDDs of type 44, theresult is a (4, 2)-CRSSGDD of type 43t+1 for t ≥ 4. �

5.3 (4, 2)-CRSSGDDs of type 3u

Lemma 5.5 There exists a (4, 2)-CRSSGDD of type 3u for each u ∈ {9, 13}.

Proof For u = 9, take the point set to be Z27, where the group set is {{0, 9, 18} + i : 0 ≤i ≤ 8}. The required design is obtained by developing the elements of Z27 in the followingbase blocks +9 (mod 27).

The base blocks of the first GDD:

{25, 14, 13, 2} {6, 10, 25, 20} {2, 6, 22, 26} {3, 10, 7, 18} {2, 12, 4, 10} {24, 23, 21, 2}{15, 3, 23, 25} {20, 23, 9, 19} {0, 21, 5, 22} {6, 4, 18, 19} {16, 2, 8, 9} {22, 14, 19, 17}{25, 5, 26, 19} {12, 26, 11, 9} {8, 3, 19, 24} {0, 4, 24, 25} {0, 6, 8, 23} {26, 16, 3, 13}

The second GDD is obtained with a multiplier 10.For u = 13, take the point set to be Z39, where the group set is {{0, 13, 26} + i : 0 ≤ i ≤

12}. The required design is obtained by developing the elements of Z39 in the following baseblocks +1 (mod 39).

The base blocks of the first GDD:

{0, 1, 6, 31} {0, 2, 12, 23} {0, 3, 7, 22}The second GDD is obtained with a multiplier 14. �

Lemma 5.6 There exists a (4, 2)-CRSSGDD of type 3u for each u ≡ 1 (mod 4), u ≥ 5.

Proof For u = 5, the required design comes from [5]. For u ∈ {9, 13}, the required designsare constructed in Lemma 5.5.

For each u ≥ 17, adjoining three ideal points to a (4, 2)-CRSSGDD of type 12t for t ≥ 4which exists by Lemma 3.2, and filling in the holes with (4, 2)-CRSSGDDs of type 35, theresult is a (4, 2)-CRSSGDD of type 34t+1 for t ≥ 4. �

Lemma 5.7 There exists a (4, 2)-CRSSGDD of type 3u for each u ∈ {8, 12, 16, 24, 28}.

Proof For each given u, take the point set to be Z3(u−1) ∪ ({a} × Z3), where the group set is{{0, u − 1, 2u − 2} + i : 0 ≤ i ≤ u − 2} ∪ {{a} × Z3}. The required designs are obtained bydeveloping the elements of Z3(u−1) in the following base blocks +1 (mod 3(u − 1)), wherethe subscripts on the elements x0 ∈ {x} × Zn are developed modulo the unique subgroupin Z3(u−1) of order n. The base blocks of the first GDD and the multiplier to get the secondGDD are listed in Table 2. �

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332 H. Zhang, G. Ge

Table 2 (4, 2)-CRSSGDDs of types 3u for u ∈ {8, 12, 16, 24, 28}u Base blocks Multiplier

8 {0, 1, 5, a0} {0, 2, 8, 11} 1012 {0, 1, 5, a0} {0, 2, 8, 20} {0, 3, 10, 19} 1616 {1, 2, 21, a0} {0, 13, 23, 27} {0, 5, 12, 29} {0, 2, 8, 11} 11

24{0, 6, 37, 66} {0, 11, 27, 45} {0, 4, 59, a0} {0, 39, 44, 56} {0, 47, 48, 67}{0, 8, 36, 62} 7

28{0, 3, 21, 68} {0, 24, 35, 72} {0, 7, 19, 59} {0, 17, 43, 66} {0, 1, 5, a0}{0, 2, 53, 73} {0, 25, 39, 75} 7

Lemma 5.8 There does not exist a (4, 2)-CRSSGDD of type 34.

Proof Suppose such a design exists. Let the group set be {{0, 4, 8}, {1, 5, 9}, {2, 6, 10},{3, 7, 11}}, and let B1 be the set of 9 blocks of the first GDD in the design and B2 bethe set of 9 blocks of the second GDD in the design.

Let the set of triples in the blocks containing x in the first GDD be Fx ={{a, b, c}|{a, b, c, x} ∈ B1}, and the set of triples in the blocks containing x in thesecond GDD be Sx = {{a, b, c}|{a, b, c, x} ∈ B2}. Without loss of generality, weassume F0 = {{1, 2, 3}, {5, 6, 7}, {9, 10, 11}}. Since the intersection of any two blocksof these designs has at most two elements, S0 must be {{1, 6, 11}, {2, 7, 9}, {3, 5, 10}} or{{1, 7, 10}, {2, 5, 11}, {3, 6, 9}}. In each of the two cases, it is easy to see that there is nosuitable way of constructing the blocks of S1. �

Lemma 5.9 There exists a (4, 2)-CRSSGDD of type 3u for each u ≡ 0 (mod 4), u ≥ 8.

Proof For u ∈ {8, 12, 16, 24, 28}, the required designs are constructed in Lemma 5.7.For u ∈ {20, 32, 36}, inflate (4, 2)-CRSSGDDs of types s4 for s ∈ {5, 8, 9} (see Lemma

2.7) by 3, and fill in the holes with (4, 2)-CRSSGDDs of type 3s to obtain the desired designs.For each u ≡ 0 (mod 8) and u ≥ 40, take a {4}-GDD of type 6t for t ≥ 5 (see Lemma

2.1). Apply WFC with weight 4 to obtain a (4, 2)-CRSSGDD of type 24t , and fill in the holeswith (4, 2)-CRSSGDDs of type 38 to obtain a (4, 2)-CRSSGDD of type 38t for t ≥ 5.

For each u ≡ 4 (mod 8) and u ≥ 44, take a {4}-GDD of type 6t 91 for t ≥ 4 (seeLemma 2.3). Apply WFC with weight 4 to obtain a (4, 2)-CRSSGDD of type 24t 361, fillin the holes of size 24 with (4, 2)-CRSSGDDs of type 38 and fill in the hole of size 36with a (4, 2)-CRSSGDD of type 312. The result is a (4, 2)-CRSSGDD of type 38t+12 fort ≥ 4. �

5.4 (4, 2)-CRSSGDDs of type g4 for g ≡ 2 (mod 4)

First, we construct some small designs directly. We will use the difference method (see, e.g.[23]) to construct MOLS, and use the connection between MOLS and TD to construct TD (orITD). When we have the first TD (or ITD), the second one is always constructed by applyinga suitable permutation on the points of the first one. For a matrix M , let M(i, j) denote theentry in row i and column j of M .

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Completely reducible super-simple designs 333

Lemma 5.10 There exists a (4, 2)-CRSSGDD of type g4 for each g ∈ {10, 18, 22, 26}.

Proof For g = 10, take the two MOLS(10)s in [49, Remark 35.19] as follows:

0 4 7 6 2 1 9 8 3 5 0 8 6 1 7 9 2 5 4 32 1 5 0 8 9 4 3 6 7 3 1 4 6 9 0 5 2 7 83 5 2 8 7 6 0 1 9 4 1 7 2 4 0 8 9 3 5 61 6 9 3 5 7 8 2 4 0 2 9 8 3 6 4 7 1 0 57 8 1 2 4 3 5 9 0 6 9 0 5 8 4 7 1 6 3 28 7 4 9 6 5 1 0 2 3 6 2 1 7 3 5 4 8 9 05 9 0 1 3 8 6 4 7 2 8 3 7 0 5 2 6 9 1 49 2 6 5 0 4 3 7 1 8 4 5 0 9 2 3 8 7 6 16 0 3 4 9 2 7 5 8 1 5 4 9 2 1 6 3 0 8 74 3 8 7 1 0 2 6 5 9 7 6 3 5 8 1 0 4 2 9

We use L1 and L2 to denote the above two MOLS(10)s, and apply permutations p1 =(0 1 2 6)(3 9 8 4)(5 7) and p2 = (0 1 7 8 5 6 2 4 3 9) to them, respectively,to obtain two new MOLS(10)s L3 and L4. We employ these MOLS(10)s to construct twoTD(4, 10)s with the super-simple property on group set {{(0, i), (1, i), . . . , (9, i)} : 0 ≤i ≤ 3}, with block sets {{(i, 0), ( j, 1), (L1(i, j), 2), (L2(i, j), 3)} : 0 ≤ i, j ≤ 9} and{{(i, 0), ( j, 1), (L3(i, j), 2), (L4(i, j), 3)} : 0 ≤ i, j ≤ 9}, separately.

For g = 18, take the matrix in [3, Theorem 3.48] as follows:⎛

⎜⎜⎝

0 0 10 1 8 5 7 0 4 6 137 1 2 3 11 9 12 − − − −1 7 12 0 6 2 3 8 9 10 58 4 11 − − − − 13 7 4 1

⎟⎟⎠

Replace every column (a, b, c, d)T , except for the first column, by a pair of columns(a, b, c, d)T and (b, a, d, c)T , and add a column (0, 0, 0, 0)T to obtain a new array M . Relabelthe symbols “−” in each row of M as “∞0,∞1,∞2,∞3”, respectively. Develop each row ofM using the automorphism group generated by (+1 (mod 14),−) to obtain an ITD(4, 18; 4)

with group set {{(0, i), (1, i), . . . , (13, i), (∞0, i), (∞1, i), (∞2, i), (∞3, i)} : 0 ≤ i ≤ 3},holes {{(∞0, i), (∞1, i), (∞2, i), (∞3, i)} : 0 ≤ i ≤ 3}, and base blocks {{(M(0, j), 0),

(M(1, j), 1), (M(2, j), 2), (M(3, j), 3)} : 0 ≤ j ≤ 21}. The second ITD(4, 18; 4) is con-structed on the same group set and holes, with base blocks {{(M(0, j) + 1, 0), (M(1, j) +1, 1), (M(2, j) + 2, 2), (M(3, j) + 2, 3)} : 0 ≤ j ≤ 21}, where M(i, j) + a = M(i, j) + a(mod 14) (i ∈ {2, 3}), if M(i, j) ∈ Z14; otherwise M(i, j) + a = ∞k+a (i ∈ {2, 3}), ifM(i, j) = ∞k for k ∈ Z4. Finally, fill in two TD(4, 4)s with the super-simple property (seeLemma 2.7) to the previous ITDs, respectively, to get a (4, 2)-CRSSGDD of type 184.

For g = 22, take the matrix in [2, Corollary 3.9] as follows:⎛

⎜⎜⎜⎜⎝

− 0 0 0 0 − 0 0 0 0 00 − 2 3 4 1 − 5 6 7 82 2 − 6 8 9 7 − 12 14 135 6 9 − 10 16 2 13 − 3 513 12 13 11 − 13 9 15 3 − 10

⎟⎟⎟⎟⎠

Replace every column (a, b, c, d, e)T by columns (a, b, c, d, e)T and (−a,−b,−c,−d,

−e)T on Z17 to obtain a new matrix M . Relabel the symbols “−” in each row of M as

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334 H. Zhang, G. Ge

“∞0,∞1,∞2, ∞3”. Append the following matrix B to M column-wise to obtain a newmatrix M ′ with 27 columns.

B =

⎜⎜⎜⎜⎝

∞4 0 1 1 00 ∞4 0 1 11 0 ∞4 0 11 1 0 ∞4 00 1 1 0 ∞4

⎟⎟⎟⎟⎠

Take the first four rows of M ′, develop each row of M ′ using the automorphism group gen-erated by (+1 (mod 17),−) to obtain an ITD(4, 22; 5) with group set {{(0, i), (1, i), . . . ,(16, i), (∞0, i), (∞1, i), (∞2, i), (∞3, i), (∞4, i)} : 0 ≤ i ≤ 3}, holes {{(∞0, i), (∞1, i),(∞2, i), (∞3, i), (∞4, i)} : 0 ≤ i ≤ 3} and base blocks {{(M ′(0, j), 0), (M ′(1, j), 1), (M ′(2, j), 2), (M ′(3, j), 3)} : 0 ≤ j ≤ 26}. The second ITD(4, 22; 5) is constructed onthe same group set and holes, with base blocks {{(M ′(0, j), 0), (M ′(1, j), 1), (M ′(2, j) +1, 2), (M ′(3, j) + 3, 3)} : 0 ≤ j ≤ 26}, where M ′(i, j) + a = M ′(i, j) + a (mod 17)

(i ∈ {2, 3}), if M ′(i, j) ∈ Z17; otherwise M ′(i, j)+a = ∞k+a (i ∈ {2, 3}), if M(i, j) = ∞k

for k ∈ Z5. Finally, fill in two TD(4, 5)s with the super-simple property (see Lemma 2.7) tothe previous ITDs, respectively, to get a (4, 2)-CRSSGDD of type 224.

For g = 26, take the matrix in [3, Theorem 3.53] as follows:

⎜⎜⎜⎜⎜⎜⎝

− − − − −0 0 0 0 01 6 7 8 143 11 20 18 106 10 14 1 54 19 5 12 2

⎟⎟⎟⎟⎟⎟⎠

Replace each column by six columns that are the six cyclic shifts of the column, and adda column (0, 0, 0, 0, 0, 0)T to get a new array M . Relabel the symbols “−” in each rowof M as “∞0,∞1,∞2,∞3,∞4”. Take the first four rows, develop each row of M usingthe automorphism group generated by (+1 (mod 21),−) to obtain an ITD(4, 26; 5) withgroup set {{(0, i), (1, i), . . . , (20, i), (∞0, i), (∞1, i), (∞2, i), (∞3, i), (∞4, i)} : 0 ≤i ≤ 3}, holes {{(∞0, i), (∞1, i), (∞2, i), (∞3, i), (∞4, i)} : 0 ≤ i ≤ 3} and baseblocks {{(M(0, j), 0), (M(1, j), 1), (M(2, j), 2), (M(3, j), 3)} : 0 ≤ j ≤ 30}. Thesecond ITD(4, 26; 5) is constructed on the same group set and holes, with base blocks{{(M(0, j), 0), (M(1, j), 1), (M(2, j) + 1, 2), (M(3, j) + 8, 3)} : 0 ≤ j ≤ 30}, whereM(i, j)+a = M(i, j)+a (mod 21) (i ∈ {2, 3}), if M(i, j) ∈ Z21; otherwise M(i, j)+a =∞k+a (i ∈ {2, 3}), if M(i, j) = ∞k for k ∈ Z5. Finally, fill in two TD(4, 5)s with the super-simple property (see Lemma 2.7) to the previous ITDs respectively, to get a (4, 2)-CRSSGDDof type 264. �

Lemma 5.11 There exists a CRSSITD2(4, 6; 2) and a CRSSITD2(4, 14; 4).

Proof For a CRSSITD2(4, 6; 2), the design is constructed on {0, 1, . . . , 23}, where the groupset is {{0, 4, . . . , 20}+ i : 0 ≤ i ≤ 3} with a hole {16, 17, . . . , 23}. The blocks are as follows:

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Completely reducible super-simple designs 335

The blocks of the first ITD:

{0, 11, 22, 5} {10, 9, 19, 12} {17, 7, 2, 12} {1, 2, 11, 16} {16, 9, 6, 3} {23, 0, 9, 2}{19, 8, 13, 2} {13, 22, 12, 3} {6, 17, 0, 15} {16, 14, 5, 7} {7, 22, 4, 9} {8, 7, 6, 21}{13, 7, 18, 0} {13, 23, 14, 4} {1, 15, 22, 8} {8, 23, 5, 10} {5, 2, 3, 20} {13, 6, 11, 20}{14, 17, 8, 3} {12, 5, 18, 15} {9, 11, 8, 18} {1, 23, 12, 6} {5, 19, 6, 4} {10, 16, 13, 15}{0, 14, 1, 19} {15, 9, 20, 14} {20, 1, 7, 10} {21, 0, 3, 10} {1, 4, 3, 18} {11, 14, 12, 21}{2, 4, 15, 21} {4, 10, 11, 17}

The blocks of the second ITD:

{19, 9, 8, 6} {12, 1, 7, 22} {4, 17, 14, 15} {6, 3, 21, 4} {10, 19, 13, 0} {10, 16, 1, 3}{17, 3, 0, 2} {6, 1, 15, 20} {11, 6, 17, 12} {3, 8, 5, 22} {14, 3, 13, 20} {12, 9, 3, 18}{2, 19, 1, 4} {0, 9, 15, 22} {4, 11, 13, 22} {5, 23, 0, 6} {9, 14, 11, 16} {1, 14, 8, 23}{23, 4, 10, 9} {15, 5, 16, 2} {10, 20, 5, 11} {9, 2, 7, 20} {23, 2, 13, 12} {13, 7, 16, 6}{11, 2, 21, 8} {8, 7, 17, 10} {19, 14, 5, 12} {4, 5, 7, 18} {8, 13, 15, 18} {21, 12, 10, 15}{11, 18, 0, 1} {7, 21, 14, 0}

For a CRSSITD2(4, 14; 4), the design is constructed on {0, 1, . . . , 55}, where the group set is{{0, 4, . . . , 52} + i : 0 ≤ i ≤ 3} with a hole {40, 41, . . . , 55}. The desired blocks are gener-ated from the following base blocks, which are developed under the automorphism group G =〈(0 4 . . . 36)(1 5 . . . 37)(2 6 . . . 38)(3 7 . . . 39)(40)(41)(42)(43)(44)(45)(46)

(47) (48)(49)(50)(51)(52)(53)(54)(55)〉.The base blocks of the first ITD:

{1, 34, 47, 12} {2, 41, 4, 31} {2, 19, 13, 48} {1, 50, 15, 24} {1, 35, 28, 46} {1, 2, 3, 0}{1, 32, 22, 55} {2, 5, 24, 23} {0, 45, 14, 11} {2, 29, 40, 15} {2, 35, 49, 16} {1, 39, 42, 4}{1, 16, 51, 10} {1, 36, 43, 6} {2, 17, 27, 52} {2, 11, 36, 53} {1, 18, 23, 44} {1, 8, 54, 31}

The base blocks of the second ITD:

{0, 15, 29, 54} {47, 8, 29, 18} {28, 50, 37, 7} {37, 55, 36, 30} {31, 1, 6, 40} {52, 7, 5, 2}{38, 7, 45, 20} {0, 11, 13, 22} {36, 19, 49, 2} {31, 13, 28, 46} {27, 48, 6, 5} {39, 41, 8, 38}{29, 43, 32, 6} {29, 2, 44, 35} {29, 3, 42, 36} {29, 14, 51, 12} {0, 5, 26, 39} {27, 30, 53, 32}.

�Lemma 5.12 There exists a (4, 2)-CRSSGDD of type 46101.

Proof Take the point set to be Z24 ∪ ({a} × Z8) ∪ ({b} × Z2), where the group set is{{0, 6, 12, 18} + i : 0 ≤ i ≤ 5} ∪ {({a} × Z8) ∪ ({b} × Z2)}. The design is obtainedby developing the elements of Z24 in the following base blocks +3 (mod 24), where thesubscripts on the elements x0 ∈ {x} × Zn are developed modulo the unique subgroup in Z24

of order n.The base blocks of the first GDD:

{1, 6, 3, a0} {2, 9, 23, a0} {10, 20, 11, a0} {4, 8, 21, a0} {7, 14, 18, a0}{0, 23, 15, b0} {0, 5, 16, a0} {22, 12, 13, a0} {1, 4, 20, b0} {15, 17, 19, a0}

The base blocks of the second GDD:

{3, 2, 13, a0} {4, 21, 13, b0} {8, 23, 16, a0} {12, 17, 7, a0} {21, 19, 18, a0}{11, 9, 0, a0} {1, 20, 22, a0} {4, 5, 15, a0} {6, 10, 14, a0} {0, 17, 20, b0}.

123

336 H. Zhang, G. Ge

The following construction is a basic form of the Wilson’s Construction for MOLS, whichwas given in [22].

Construction 5.13 Suppose that a TDμ(k + l, t) exists. Suppose that for each B ∈ B, thereexists a TDλ(k, m +∑

1≤i≤l wBi )−∑

1≤i≤l TDλ(k, wBi ). Then there exists a TDλμ(k, mt +∑

1≤i≤l∑

1≤ j≤t wi j ) − ∑1≤i≤l TDλμ(k,

∑1≤ j≤t wi j ).

It is easy to check that if all the input designs have the CRSS property, then the resultantdesign generated by Construction 5.13 will also have the CRSS property.

Theorem 5.14 There exists a (4, 2)-CRSSGDD of type g4 for each g ≡ 2 (mod 4), g ≥ 10.

Proof For g ∈ {10, 18, 22, 26}, the required designs are constructed in Lemma 5.10. For g =14, take a CRSSITD2(4, 14; 4) (see Lemma 5.11), fill in the hole with a (4, 2)-CRSSGDDof type 44 (see Lemma 2.7), to get the desired design.

For g = 34, inflate a (4, 2)-CRSSGDD of type 46101 with weight 4, using a {4}-MGDDof type 44 as input design, to obtain a (4, 2)-CRSSDGDD of type (16, 44)6(40, 104)1. Fillin the holes with CRSSTD2(4, 4)s and a CRSSTD2(4, 10) to obtain the required design.

For g = 30 or g ≥ 38, apply Construction 5.13 with μ = l = 1, k = m = 4, λ = 2, wi j ∈{0, 2} to obtain a CRSSITD2(4, 50; 14) and CRSSITD2(4, 4t + 10; 10)s for all t ≥ 5, t �∈{6, 10}. Here, the input CRSSTD2(4, 4) and CRSSITD2(4, 6; 2) are given in Lemmas 2.7and 5.11 respectively. Then fill in the holes with a CRSSTD2(4, 14) or a CRSSTD2(4, 10)

to obtain the required designs. �Combining Lemmas 2.1, 2.7, 5.8 and Theorem 5.14, we have the following result.

Theorem 5.15 A (4, 2)-CRSSGDD of type g4 exists if and only if g ≥ 4, g �= 6.

5.5 Main result for (4, 2)-CRSSGDDs of type gu

Theorem 5.16 The obvious necessary conditions for the existence of a (4, 2)-CRSSGDD oftype gu are also sufficient except for (g, u) ∈ {(2, 4), (3, 4), (6, 4)}.Proof For u = 4, the existence of such designs is proved by Theorem 5.15, hence we mainlyfocus our attention on the case of u ≥ 5.

(i) g ≡ 0 (mod 6)

For g ∈ {6, 12} and u ≥ 5, the required designs come from Lemmas 5.2 and 3.2. Forg = 36 and u ≥ 5, inflate (4, 2)-CRSSGDDs of types 12u for u ≥ 5 by 3 to obtainthe desired designs. Then for any other g > 6, inflate a (4, 2)-CRSSGDD of type 6u

by g/6 using the existing TD(4, g/6) to get the desired (4, 2)-CRSSGDD of type gu ,where u ≥ 5.

(ii) g ≡ 1, 5, 7, 11 (mod 12)

For g = 1 and u ≡ 1, 4 (mod 12), u ≥ 13, the desired designs come from Lemma1.1. Then for g ≥ 5 and u ≡ 1, 4 (mod 12), u ≥ 13, inflate (u, 4, 2)-CRSS designsby g using TD(4, g)s as input designs to get the desired (4, 2)-CRSSGDDs of type gu .

(iii) g ≡ 2, 10 (mod 12)

For g = 2 and u ≡ 1 (mod 3), u ≥ 7, the required designs are obtainable by Lemma2.6. Then for g ≥ 10 and u ≡ 1 (mod 3), u ≥ 7, inflate (4, 2)-CRSSGDDs of type2u by g/2 to get the desired (4, 2)-CRSSGDDs of type gu .

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Completely reducible super-simple designs 337

(iv) g ≡ 3, 9 (mod 12)

For g = 3 and u ≡ 0, 1 (mod 4), u ≥ 5, the required designs can be obtained byLemmas 5.6 and 5.9. Then for g ≥ 9 and u ≡ 0, 1 (mod 4), u ≥ 5, inflate (4, 2)-CRSSGDDs of type 3u by g/3 to get the desired (4, 2)-CRSSGDDs of type gu .

(v) g ≡ 4, 8 (mod 12)

For g = 4 and u ≡ 1 (mod 3), u ≥ 7, the required designs are from Lemma 5.4.For g = 8 and u ≡ 1 (mod 3), u ≥ 7, inflate (4, 2)-CRSSGDDs of type 2u by4 to obtain the desired designs. Then for g ≥ 16 and u ≡ 1 (mod 3), u ≥ 7,inflate (4, 2)-CRSSGDDs of type 4u by g/4 to get the desired (4, 2)-CRSSGDDs oftype gu . �

6 Existence of optimal super-simple (v, 4, 2)-packings

In this section, we will investigate the existence of super-simple (v, 4, 2)-packings. First, wepresent a connection between optimal (v, 6, 4)3 codes and optimal super-simple (v, 4, 2)-packings. Next, we show that optimal super-simple (v, 4, 2)-packings exist for all v ≥ 4except definitely for v ∈ {4, 5, 6, 9}.

6.1 The case v ≡ 1, 2 (mod 3)

In [66], the authors established the existence of optimal (v, 6, 4)3 codes, i.e., A3(v, 6, 4) =⌊v2

⌊v−1

3

⌋⌋if v ≥ 4 and v �∈ {4, 5, 7, 8, 11, 17}. It can be proved that an optimal (v, 6, 4)3

code with⌊

v2

⌊v−1

3

⌋⌋codewords gives an optimal super-simple (v, 4, 2)-packing for v ≡ 1, 2

(mod 3). However, the converse is not necessarily true.

Lemma 6.1 For v ≡ 1, 2 (mod 3), if there is an optimal (v, 6, 4)3 code with⌊

v2

⌊v−1

3

⌋⌋

codewords, then there is an optimal super-simple (v, 4, 2)-packing.

Proof Let X be a point set of size v and D be the support set of an optimal (v, 6, 4)3 code.Then D forms the block set of a design with block size 4.

First, we show that any pair of points in X can not appear in more than two blocks of D.If a pair of points in X appears in two blocks, the corresponding elements in the underlyingcodewords must be 〈1, 1〉, 〈2, 2〉 or 〈1, 2〉, 〈2, 1〉. Hence the design is a (v, 4, 2)-packing.

Next, we show the super-simple property. If there are two blocks in D intersect in morethan two elements, the minimum distance of the code is at most 5, which contradicts thedistance condition. So the design is super-simple.

Finally, it is easy to check that the number of codewords of an optimal (v, 6, 4)3 code,namely

⌊v2

⌊v−1

3

⌋⌋, is exactly that of the upper bound of an optimal super-simple (v, 4, 2)-

packing when v ≡ 1, 2 (mod 3). Hence, the resultant super-simple (v, 4, 2)-packing is alsooptimal. �Lemma 6.2 There exists an optimal super-simple (v, 4, 2)-packing for each v ∈ {7, 8,

11, 17}.Proof For v = 7, the required design is just a super-simple (7, 4, 2)-BIBD, which comesfrom [41].

For v ∈ {8, 11}, the designs are constructed on {0, 1, . . . , v − 1} with blocks as follows.v = 8 :{2, 1, 6, 5} {1, 2, 4, 7} {3, 6, 0, 7} {5, 7, 4, 0} {7, 5, 1, 3} {0, 6, 4, 1} {0, 3, 5, 2} {4, 6, 2, 3}

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338 H. Zhang, G. Ge

v = 11 :{0, 1, 2, 3} {0, 4, 5, 6} {0, 7, 8, 9} {1, 4, 0, 10} {1, 5, 8, 7} {2, 1, 6, 9} {2, 5, 10, 0}{3, 5, 9, 1} {3, 6, 8, 0} {4, 9, 2, 8} {6, 7, 3, 10} {7, 6, 4, 1} {8, 5, 4, 3} {8, 10, 6, 2}{3, 2, 4, 7} {9, 10, 7, 5}For v = 17, the required design is constructed on Z15 ∪ {∞0,∞1}, and obtained by

developing the elements of Z15 in the following base blocks +5 (mod 15), where ∞0,∞1

keep fixed under the action of the automorphism group.

{0, 2, 4, 6} {0, 1, 2, 3} {0, 9, 12,∞1} {0, 3, 6, 11} {12, 5,∞0, 3} {5, 3, 4,∞1}{13, 7,∞1, 6} {0, 1, 4, 5} {3, 7, 9, 14} {5, 12, 2, 13} {3, 9, 1, 13} {13, 14,∞0, 5}{1, 4, 7, 11} {∞0, 11, 9, 2}.

�For v ≡ 1, 2 (mod 3), v ≥ 7 and v �∈ {7, 8, 11, 17}, we obtain optimal super-simple

(v, 4, 2)-packings by Lemma 6.1. For v ∈ {4, 5}, it is obvious that the design has just oneblock. So we have the following result.

Lemma 6.3 D′2(v, 4, 2) = U2(v, 4, 2) for v ≡ 1, 2 (mod 3), v ≥ 7; D′

2(v, 4, 2) =U2(v, 4, 2) − 1 for v ∈ {4, 5}.

6.2 The case v ≡ 0 (mod 3)

Lemma 6.4 D′2(v, 4, 2) = U2(v, 4, 2) − 1 for each v ∈ {6, 9}.

Proof For v = 6, it is obvious that: for any point in an optimal super-simple (6, 4, 2)-pack-ing, there are at most two blocks containing it. So the design has at most 2×6

4 = 3 blocks.The design reaching the upper bound is constructed on {0, 1, 2, 3, 4, 5} with 3 blocks listedbelow:

{0, 1, 2, 3} {2, 3, 4, 5} {0, 1, 4, 5}For v = 9, by Lemma 2.16, D2(9, 4, 2) = U2(9, 4, 2)−1. Hence for a super-simple (9, 4, 2)-packing, D′

2(9, 4, 2) ≤ U2(9, 4, 2) − 1. The design reaching the upper bound is constructedon {0, 1, 2, 3, 4, 5, 6, 7, 8} with 10 blocks listed below:

{0, 1, 2, 3} {0, 4, 5, 6} {1, 0, 7, 8} {2, 1, 5, 4} {2, 6, 7, 0}{3, 5, 7, 1} {5, 8, 3, 0} {6, 8, 4, 1} {8, 6, 5, 2} {3, 4, 2, 8}.

�Lemma 6.5 There exists an optimal super-simple (v, 4, 2)-packing for eachv ∈ {12, 24, 36}.Proof The required designs are constructed on Zv and obtained by developing the elementsof Zv in the following base blocks +4 (mod v).v = 12 :

{0, 1, 2, 3} {0, 4, 10, 9} {0, 7, 11, 8} {3, 9, 6, 2} {3, 5, 1, 11} {6, 10, 3, 8} {2, 5, 9, 8}v = 24 :{0, 1, 2, 3} {19, 7, 17, 15} {0, 6, 10, 13} {1, 5, 10, 16} {12, 10, 5, 21} {21, 17, 20, 11}{0, 7, 11, 14} {12, 16, 11, 8} {1, 7, 13, 20} {6, 11, 16, 4} {19, 18, 10, 9} {20, 17, 1, 14}{6, 18, 2, 15} {3, 19, 21, 14} {2, 19, 20, 4}

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Completely reducible super-simple designs 339

v = 36 :{10, 23, 30, 35} {0, 19, 16, 20} {35, 34, 28, 26} {20, 5, 6, 32} {23, 31, 18, 9}{14, 29, 35, 18} {16, 10, 29, 19} {3, 16, 24, 33} {32, 21, 31, 19} {8, 9, 22, 18}{5, 34, 16, 22} {19, 22, 23, 15} {15, 30, 25, 4} {34, 14, 0, 31} {20, 25, 10, 22}{20, 2, 16, 8} {5, 28, 19, 35} {33, 15, 13, 32} {7, 13, 18, 16} {6, 14, 13, 17}{13, 15, 20, 35} {27, 9, 17, 29} {1, 32, 29, 25}.

Lemma 6.6 There exists an optimal super-simple (v, 4, 2)-packing for eachv ∈ {15, 27, 39}.

Proof The required designs are constructed on Zv−3 ∪ ({a} × Z3) with a hole {{a} × Z3}by developing the elements of Zv−3 in the following base blocks +4 (mod v − 3), wherethe subscripts on the elements x0 ∈ {x} × Zn are developed modulo the unique subgroup inZv−3 of order n.v = 15 :

{9, 3, 7, 0} {0, 4, 9, a0} {2, 7, a1, 5} {a1, 4, 3, 11} {a2, 6, 1, 5} {0, a1, 10, 8}{0, 1, 3, 2} {5, 2, 4, 10} {2, 6, a0, 3} {3, a2, 10, 4} {3, 9, 5, a1}

v = 27 :{2, 23, 4, 10} {9, 2, 16, 5} {23, 13, 8, a1} {11, 9, 7, 19} {20, 14, 10, a0} {10, 19, a1, 20}{0, 16, 23, 2} {21, 5, 8, 9} {14, 21, 13, 2} {14, 3, 18, 9} {5, 23, a0, 20} {12, a0, 15, 17}{0, 8, 20, 17} {10, 2, 8, 3} {a2, 5, 19, 14} {19, 8, 12, 23} {17, a1, 18, 23} {a0, 13, 12, 10}{3, 19, 2, 13}v = 39 :{2, 5, 25, 7} {10, 11, 19, 1} {6, 31, 28, 27} {30, 8, 0, 10} {22, 24, 29, 5} {8, 30, 22, 20}{1, 0, 30, 29} {28, 3, 12, 25} {19, 29, 17, 7} {0, 34, 6, 18} {1, 11, 18, 31} {16, 12, a1, 19}{a2, 9, 17, 6} {17, a2, 0, 26} {6, 35, 10, a2} {1, 33, 2, a2} {13, 29, 35, 0} {2, 19, 35, 34}{3, 6, 17, 30} {16, 7, 11, a0} {0, 19, 21, 25} {a0, 26, 5, 0} {21, 19, 6, 24} {a1, 12, 35, 27}{29, 4, 20, 8} {3, 24, 32, 33} {3, 15, 34, 28}.

We notice that the optimal super-simple (v, 4, 2)-packings for v ∈ {15, 27, 39} constructedin Lemma 6.6 are also super-simple (v, 3; 4, 2)-MIPDs.

Lemma 6.7 There exists an optimal super-simple (v, 4, 2)-packing for each v ≡ 0, 3(mod 12) and v ≥ 12.

Proof For 12 ≤ v ≤ 39, the required designs are constructed in Lemmas 6.5 and 6.6.For each v ≥ 48, take a (4, 2)-CRSSGDD of type 12t for t ≥ 4 from Lemma 3.2. By Con-

structions 2.14 and 2.15, if we fill in the holes with optimal super-simple (12, 4, 2)-packings,the result is an optimal super-simple (12t, 4, 2)-packing for t ≥ 4; if we adjoin three idealpoints and fill in the holes with super-simple (15, 3; 4, 2)-MIPDs, the result is an optimalsuper-simple (12t + 3, 4, 2)-packing for t ≥ 4. �

123

340 H. Zhang, G. Ge

Lemma 6.8 There exists an optimal super-simple (v, 4, 2)-packing for eachv ∈ {18, 21, 30}.

Proof For v = 18, the design is constructed on {0, 1, . . . , 17} with blocks listed below.

{1, 6, 5, 9} {6, 8, 7, 15} {9, 8, 14, 12} {15, 2, 13, 0} {8, 16, 10, 17} {10, 7, 1, 11}{5, 2, 7, 3} {7, 0, 5, 10} {14, 17, 7, 9} {12, 7, 6, 17} {17, 3, 12, 10} {15, 16, 9, 3}{3, 8, 4, 1} {0, 14, 6, 3} {14, 1, 2, 10} {17, 8, 0, 11} {13, 1, 15, 12} {16, 7, 12, 4}{0, 2, 8, 9} {2, 17, 1, 4} {13, 3, 11, 7} {11, 12, 8, 5} {10, 13, 4, 14} {16, 11, 6, 1}{4, 3, 6, 16} {0, 4, 15, 7} {12, 0, 1, 16} {15, 4, 10, 8} {2, 12, 15, 11} {5, 15, 1, 14}{4, 5, 0, 12} {9, 5, 4, 11} {10, 3, 15, 5} {5, 17, 6, 13} {12, 3, 14, 13} {14, 5, 16, 8}{1, 7, 13, 8} {11, 3, 9, 2} {6, 14, 11, 4} {17, 16, 2, 5} {13, 9, 10, 16} {4, 9, 13, 17}{1, 3, 0, 17} {10, 9, 6, 0} {6, 10, 12, 2} {7, 2, 16, 14} {11, 16, 0, 13} {11, 17, 14, 15}{8, 13, 2, 6}

For v = 21, the required design is constructed on Z20 ∪ {∞} and obtained by developing theelements of Z20 in the following base blocks +5 (mod 20), where ∞ keeps fixed under theaction of the automorphism group.

{0, 2, 5, 7} {4, 3, 14, 5} {1, 17, 9, 13} {11, 16, 0, 19} {10, 0, 11, 4} {3, 18, 7, 17}{6, 3,∞, 9} {6, 18, 16, 7} {4, 7, 10, 19} {5, 18, 12,∞} {0, 12, 13, 6} {18, 10, 15, 13}{3, 13, 6, 4} {17, 4, 19, 2} {13, 11, 5, 9} {11, 16, 7, 15} {1, 7, 14,∞}

For v = 30, the required design is constructed on Z30 and obtained by developing theelements of Z30 in the following base blocks +15 (mod 30).

{19, 8, 0, 11} {19, 20, 0, 21} {15, 12, 0, 11} {23, 6, 19, 24} {17, 26, 12, 9}{16, 29, 23, 28} {20, 1, 24, 5} {16, 24, 1, 27} {21, 22, 27, 6} {20, 11, 4, 13}{21, 5, 29, 13} {18, 20, 15, 21} {17, 8, 11, 6} {22, 10, 8, 12} {16, 13, 0, 17}{21, 3, 26, 10} {22, 21, 29, 7} {14, 24, 26, 11} {18, 8, 2, 13} {21, 5, 27, 14}{21, 15, 9, 24} {18, 17, 29, 0} {16, 12, 0, 18} {16, 15, 17, 18} {23, 9, 3, 13}{18, 27, 6, 26} {18, 5, 25, 28} {24, 28, 12, 2} {22, 26, 17, 28} {20, 18, 23, 22}{22, 4, 24, 8} {17, 1, 23, 20} {23, 11, 5, 29} {10, 21, 18, 4} {19, 17, 22, 14}{20, 22, 26, 16} {20, 8, 3, 25} {22, 11, 28, 3} {17, 10, 27, 5} {19, 29, 27, 8}{22, 15, 13, 25} {25, 12, 13, 27} {18, 12, 7, 1} {25, 14, 16, 7} {20, 27, 2, 29}{24, 19, 3, 12} {14, 16, 13, 10} {21, 13, 28, 19} {17, 10, 6, 2} {18, 19, 22, 1}{17, 7, 19, 15} {21, 0, 14, 28} {20, 15, 11, 10} {22, 12, 23, 15} {18, 14, 3, 9}{24, 7, 10, 26} {22, 13, 5, 24} {23, 0, 25, 10} {10, 14, 29, 24} {21, 24, 25, 16}{16, 8, 23, 6} {21, 1, 11, 17} {20, 17, 7, 24} {26, 19, 1, 13} {16, 15, 29, 19}{14, 27, 11, 23} {2, 9, 10, 19} {15, 8, 9, 28} {18, 14, 2, 19} {19, 4, 20, 27}{19, 10, 16, 11}.

Lemma 6.9 There exists an optimal super-simple (v, 4, 2)-packing for each v ∈ {54, 57}.

Proof We use the method as in [6] to construct a (v, 4, 2)-packing. Firstly, we construct a{4}-GDD of type 228 on the point set Z56 with group set {{0, 28} + i : 0 ≤ i ≤ 27}, blockset D1 obtained by developing the elements in the following base blocks +2 (mod 56).

{0, 3, 33, 52} {0, 35, 43, 55} {0, 8, 19, 22} {0, 9, 15, 20} {1, 15, 19, 44}{0, 2, 12, 18} {0, 26, 47, 49} {0, 1, 17, 32} {0, 5, 29, 39}

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Completely reducible super-simple designs 341

For v = 54, we construct a (52, 4, 1)-BIBD on Z52 with the following base blocks +4(mod 52). Name the block set as D2.

{7, 29, 50, 30} {0, 12, 16, 6} {0, 50, 41, 24} {7, 13, 10, 38} {7, 11, 1, 24} {14, 49, 25, 39}{0, 15, 43, 44} {21, 8, 26, 40} {21, 28, 6, 13} {7, 28, 39, 47} {7, 4, 18, 26} {21, 12, 33, 17}{0, 49, 47, 10} {7, 14, 32, 33} {14, 15, 50, 2} {7, 21, 43, 41} {0, 7, 25, 2}

We can see that there is a block {15, 30, 54, 55} in D1, which is a translate of {0, 1, 17, 32}.Replace 15, 30, 52, 53 in the blocks of D1 with 52, 53, 15, 30 respectively, to get a new blockset D′

1 with {52, 53, 54, 55} as one of the blocks. Drop this block and in the remaining blocksof D′

1, replace 55 with 52, replace 54 with 53 to obtain D′′1. Take the following permutation:

(0 48 31 6 21 7 28 27 12 5 44 45 13 23 11 34 32 15)(1 8)(2)(3 43 20 4 30)

(9)(10)(14)(16)(17)(18)(19)(22)(24)(25)(26)(29)(33)(35)(36)(37)(38)(39)(40)(41)(42)

(46 50)(47)(49)(51)(52)(53)

to act on the points of all the blocks in D2 to get D′2. Finally, combine D′′

1 and D′2 together

to obtain the required design.For v = 57, we construct a {4}-GDD of type 15171 on the point set {0, 1, . . . , 57} with

group set {{{i} : 0 ≤ i ≤ 50}∪{51, 52, . . . , 57}}. The block set D3 is generated from the fol-lowing base blocks with the automorphism group G = 〈(0 3 . . . 48)(1 4 . . . 49)(2 5 . . .

50) (51)(52)(53)(54)(55)(56)(57)〉.

{1, 4, 34, 46} {2, 3, 23, 26} {1, 20, 26, 35} {0, 6, 10, 18} {0, 26, 36, 38} {0, 21, 34, 48}{0, 8, 19, 57} {2, 22, 35, 37} {2, 16, 21, 54} {2, 7, 42, 51} {1, 30, 44, 56} {0, 25, 47, 55}{0, 9, 40, 44} {2, 36, 43, 53} {2, 46, 48, 52} {0, 1, 28, 29}

For each point x in D1, if x < 28, take the map x �→ 2x ; otherwise take the mapx �→ 2(x − 28) + 1, to obtain D′′′

1. In D3, replace 57 with 56, and add three blocks{51, 52, 53, 56}, {51, 54, 55, 56}, {52, 53, 54, 55} to get D′

3. Take the following permuta-tion:

(0 10 29)(1 11)(2 31 48 12)(3)(4 26 22 27 50 35 14 37 51 44 47 33 6)(5 21)

(7 8)(13)(23)(28)(9 43)(15 49 20 19 18 17 16)(24 25)(30 36)(32)(34)(38)(39)(40)

(41)(42)(45)(46)(52)(53)(54)(55)(56)

to act on the points of all the blocks in D′3 to get D′′

3. Finally, combine D′′′1 and D′′

3 togetherto get the required design. �

Lemma 6.10 There exists a super-simple (v,w; 4, 2)-packing for each ordered pair of(v,w) ∈ {(33, 7), (42, 10), (45, 13)} with 166, 268, 300 blocks, respectively.

Proof A super-simple (33, 7; 4, 2)-packing with 166 blocks is constructed on {0, 1, . . . , 32}with a hole {26, 27, . . . , 32}. The desired blocks are generated from the following base blocks,which are developed under the automorphism group G = 〈(0 13)(1 14)(2 15) . . . (12 25)

(26 27)(28 29)(30 31)(32)〉.

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342 H. Zhang, G. Ge

{1, 25, 9, 5} {18, 20, 8, 1} {32, 5, 24, 2} {18, 27, 14, 9} {28, 19, 3, 21} {22, 15, 24, 23}{2, 28, 8, 7} {19, 17, 2, 7} {5, 17, 27, 4} {2, 15, 26, 23} {28, 25, 4, 11} {22, 17, 32, 16}{3, 8, 30, 5} {2, 11, 3, 20} {5, 26, 7, 18} {2, 20, 25, 21} {29, 10, 14, 1} {23, 30, 13, 20}{7, 1, 17, 6} {2, 31, 24, 4} {6, 5, 32, 21} {21, 10, 17, 0} {3, 32, 12, 23} {24, 13, 18, 25}{0, 13, 26, 6} {20, 8, 0, 31} {8, 2, 31, 22} {22, 29, 13, 2} {5, 10, 19, 23} {25, 10, 14, 12}{0, 22, 7, 32} {21, 4, 1, 32} {9, 11, 7, 27} {23, 14, 31, 6} {5, 14, 22, 13} {27, 14, 19, 24}{0, 6, 25, 27} {22, 6, 8, 19} {1, 19, 22, 11} {23, 17, 2, 18} {10, 20, 31, 16} {28, 13, 10, 21}{11, 31, 3, 5} {24, 21, 1, 8} {10, 30, 6, 17} {24, 20, 28, 9} {10, 22, 27, 20} {29, 13, 18, 10}{11, 4, 0, 24} {27, 2, 13, 1} {12, 18, 28, 3} {25, 19, 3, 24} {12, 26, 10, 24} {29, 15, 19, 25}{12, 27, 8, 4} {28, 6, 7, 20} {13, 12, 31, 9} {25, 3, 20, 26} {12, 32, 15, 19} {30, 11, 18, 19}{13, 3, 15, 6} {3, 10, 22, 9} {14, 30, 16, 2} {25, 31, 14, 7} {15, 17, 28, 14} {32, 14, 20, 11}{13, 7, 4, 25} {3, 2, 14, 26} {16, 14, 3, 13} {26, 16, 4, 21} {16, 13, 28, 11} {25, 18, 15, 31}{14, 4, 30, 0} {3, 6, 18, 29} {17, 9, 19, 31} {26, 8, 11, 10} {16, 17, 20, 23} {17, 25, 28, 22}{18, 0, 21, 2} {30, 25, 9, 8} {18, 22, 28, 4} {27, 21, 22, 3} {21, 29, 11, 23}

A super-simple (42, 10; 4, 2)-packing with 268 blocks is constructed on {0, 1, . . . , 41} withthe hole {32, 33, . . . , 41}. The design is generated from the following base blocks, which aredeveloped under the automorphism group G = 〈(0 8 . . . 24)(1 9 . . . 25) . . . (7 15 . . .

31)(32 33 34 35)(36 37 38 39)(40 41)〉.

{5, 8, 30, 1} {9, 39, 1, 14} {30, 38, 26, 4} {15, 35, 22, 12} {32, 26, 21, 8}{21, 15, 30, 41} {12, 3, 4, 32} {28, 14, 40, 2} {4, 15, 27, 16} {15, 26, 25, 33}{22, 24, 2, 35} {36, 21, 15, 18} {32, 8, 28, 9} {17, 16, 2, 10} {3, 15, 31, 39}{13, 30, 29, 34} {31, 20, 40, 8} {37, 11, 25, 28} {18, 13, 2, 3} {27, 24, 1, 30}{14, 13, 0, 38} {24, 15, 23, 33} {25, 36, 9, 11} {40, 25, 16, 14} {3, 11, 5, 15}{31, 25, 2, 6} {2, 17, 21, 40} {17, 11, 40, 22} {12, 29, 7, 31} {11, 31, 18, 34}{11, 1, 34, 9} {4, 6, 18, 14} {19, 24, 3, 32} {13, 21, 25, 36} {18, 38, 17, 29}{28, 39, 16, 24} {19, 16, 5, 0} {0, 28, 5, 32} {19, 29, 9, 12} {31, 39, 24, 14}{24, 21, 18, 38} {15, 14, 35, 25} {28, 4, 5, 36} {2, 30, 7, 12} {15, 34, 9, 28}{33, 26, 17, 28} {15, 1, 41, 13} {38, 31, 22, 10} {33, 24, 9, 7} {11, 22, 32, 6}{8, 36, 27, 14} {30, 39, 16, 11} {35, 16, 26, 18} {22, 29, 33, 14} {0, 41, 8, 4}{4, 32, 5, 29} {23, 41, 5, 19} {28, 38, 17, 12} {10, 39, 23, 8} {10, 32, 19, 11}{5, 16, 23, 9} {18, 3, 5, 37} {38, 28, 3, 10} {41, 12, 27, 18} {12, 10, 15, 17}{23, 27, 22, 39} {3, 4, 29, 6}

A super-simple (45, 13; 4, 2)-packing with 300 blocks is constructed on {0, 1, . . . , 44} withthe hole {32, 33, 34, . . . , 44}. The design is generated from the following base blocks, whichare developed under the automorphism group G = 〈(0 8 16 24)(1 9 17 25) . . . (7 15 2331)(32 33 34 35)(36 37 38 39)(40 41 42 43)(44)〉.

{7, 11, 4, 9} {5, 4, 44, 19} {2, 21, 38, 31} {35, 19, 11, 6} {17, 31, 18, 34}{32, 26, 10, 31} {0, 32, 9, 27} {8, 19, 18, 7} {20, 2, 43, 31} {36, 8, 23, 20}{18, 30, 39, 24} {33, 15, 24, 16} {0, 4, 20, 34} {9, 38, 24, 5} {20, 7, 24, 41}{39, 21, 6, 26} {19, 42, 30, 15} {35, 28, 18, 10} {12, 9, 2, 40} {1, 17, 37, 21}{22, 15, 34, 2} {42, 6, 24, 11} {21, 28, 32, 23} {36, 12, 10, 16} {13, 3, 4, 39}{1, 31, 23, 42} {23, 40, 1, 30} {44, 14, 9, 18} {22, 23, 11, 38} {37, 31, 24, 25}{16, 5, 2, 39} {10, 29, 2, 27} {24, 40, 21, 4} {44, 15, 8, 11} {23, 38, 21, 14}{38, 18, 20, 12} {2, 9, 42, 24} {11, 37, 17, 2} {25, 44, 14, 4} {44, 29, 0, 23}

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Completely reducible super-simple designs 343

{24, 13, 29, 32} {40, 21, 10, 13} {21, 42, 8, 1} {11, 5, 40, 29} {26, 8, 24, 34}{5, 26, 35, 30} {24, 17, 36, 29} {40, 30, 22, 26} {3, 1, 15, 35} {13, 12, 4, 23}{27, 2, 38, 19} {7, 23, 19, 39} {25, 17, 28, 32} {43, 12, 18, 19} {34, 28, 1, 6}{13, 7, 31, 42} {3, 13, 19, 34} {8, 43, 17, 11} {27, 17, 29, 35} {43, 30, 24, 12}{36, 14, 0, 1} {16, 14, 6, 34} {3, 35, 25, 12} {11, 30, 43, 28} {28, 22, 36, 19}{30, 13, 17, 14} {37, 7, 12, 6} {17, 41, 26, 9} {32, 12, 14, 5} {12, 11, 25, 39}{29, 30, 42, 12} {14, 11, 24, 16} {40, 0, 2, 11} {18, 23, 6, 17} {32, 30, 7, 21}.

�In [66, Lemmas 5.3 and 5.5], the authors have established the following result about

(4, 2)-CRSSGDDs.

Lemma 6.11 ([66]) There exists a (4, 2)-CRSSGDD of type 12u181 for each integer u ≥ 24or u ∈ [4, 8] ∪ {16} ∪ [20, 22]. There exist also (4, 2)-CRSSGDDs of types 24um1 foreach (u, m) ∈ {(4, 30), (5, 18), (5, 30), (6, 18), (6, 30), (7, 18), (7, 30)}and 36um1 for each(u, m) ∈ {(5, 42), (6, 18), (6, 30), (6, 78)}.Lemma 6.12 There exists an optimal super-simple (v, 4, 2)-packing for each v ≡ 6, 9(mod 12) and v ≥ 18.

Proof For v ∈ {18, 21, 30, 54, 57}, the required designs are constructed in Lemmas 6.8 and6.9.

For v ∈ {33, 42, 45}, take the super-simple (v,w; 4, 2)-packings constructed in Lemma6.10 for (v,w) ∈ {(33, 7), (42, 10), (45, 13)}, then fill in the holes with optimal super-simple(w, 4, 2)-packings for w ∈ {7, 10, 13} (see Lemma 6.3), respectively, to obtain the requireddesigns.

By Constructions 2.14 and 2.15, if we fill in the holes of the CRSSGDDs in Lemma 6.11with suitable optimal super-simple (v, 4, 2)-packings for v ∈ {12, 18, 24, 30, 36, 42, 78} (seeLemma 6.7), the results are optimal super-simple (12t +18, 4, 2)-packings for all t ≥ 4; if weadjoin three ideal points to these CRSSGDDs, and fill in the holes with suitable super-simple(v, 3; 4, 2)-MIPDs for v ∈ {15, 27, 39} (see Lemma 6.6) and optimal super-simple (v, 4, 2)-packings for v ∈ {21, 33, 45, 81}, the results are optimal super-simple (12t + 21, 4, 2)-packings for all t ≥ 4. The proof is complete. �Combining Lemmas 6.3, 6.4, 6.7 and 6.12, we have the following result.

Theorem 6.13 D′3(v, 4, 2) = U (v, 4, 2) for each v ≥ 4 and v �∈ {4, 5, 6, 9}; D′

3(v, 4, 2) =U (v, 4, 2) − 1 for v ∈ {4, 5, 6, 9}.Acknowledgments Research supported by the National Outstanding Youth Science Foundation of Chinaunder Grant No. 10825103, National Natural Science Foundation of China under Grant No. 10771193, Special-ized Research Fund for the Doctoral Program of Higher Education, Program for New Century Excellent Talentsin University, and Zhejiang Provincial Natural Science Foundation of China under Grant No. D7080064. Theauthors express their gratitude to the two anonymous reviewers for their detailed and constructive commentswhich are very helpful to the improvement of the technical presentation of this paper.

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