construction of balanced incomplete block designs …

▼ Journal of Research (Science), Bahauddin Zakariya University, Multan, Pakistan. Vol. 22-23, Nos. 1-4 January-October 2011-12, pp. 01-25 ISSN 1021-1012

* Corresponding author 01 ▼J. res. Sci. B. Z. Univ., 2011-12, 22-23(1-4), 01-25

CONSTRUCTION OF BALANCED INCOMPLETE

BLOCK DESIGNS USING CYCLIC SHIFTS

Ijaz Iqbal*, Muhammad Shahzad, Palwasha Fatima Buzdar Department of Statistics, Bahauddin Zakariya University Multan-Pakistan.

email: [email protected], [email protected]

Abstract The construction of balanced incomplete block designs is considered using the method of cyclic shifts. An interesting feature of this method is that there is no need to construct the bocks of the actual design to obtain the properties of a design. One can obtain the off-diagonal elements of the concurrence matrix from the sets of shifts with less effort. A catalogue of balanced incomplete block designs for 3 ≤ v ≤ 18 with k = v -1, 19 ≤ v ≤ 100 with k = 2, 3 and for 19 ≤ v ≤ 25 with k = 4, 5 restricted with r < 60 is compiled.

Keywords: Balanced incomplete block design; Combinatorial

balanced; Cyclic shifts.

INTRODUCTION We address a classical problem of experimental designs, that is, the construction of balanced incomplete block designs (BIBDs). BIBDs are popular as experimental designs for several reasons. Like the randomized complete block designs, they are variance balanced, that is, every normalized contrast is estimated with the same variance. Among the binary block designs with block size less than the number of treatments, that is, k < v they are the only designs which are variance-(efficiency) balanced [Rao, 1958]. There has been an intense combinatorial study of BIBDs since the time of Fisher and Yates [1953], which led to many new designs. Tables of BIBDs are also given in Fisher and Yates [1953], Davis [1956] and Cochran and Cox [1957]. The standard setting for a block design is an integer triple setting with parameters (v, b, k) specifying that it is an arrangement of v treatments in b blocks, each of size k, where k < v. The conditions for a BIBD are that (i) r = b k/v, i.e. each treatment has r replication, (ii) no treatment appears more than once in any block, and (iii) all unordered pair of treatments appears exactly in λ blocks (equi-concurrence), where λ = r (k-1) / (v-1) = b k (k-1) / v(v-1) is often referred to as the concurrence parameter of a BIBD.

Ijaz Iqbal, Muhammad Shahzad and Fatima Palwasha Buzdar 2

Let D (v,b,k) denote the class of all connected block designs for the setting (v,b,k) which are available in an experiment whose contrast are estimable. One crucial problem is to determine the properties of a design

Dd according to some optimality criteria. Let )( dijd nN be the bv

treatment-block incidence matrix associated with any design Dd

(v,b,k) whose elements dijn signify the number of units in block j

allocated to treatment i. The matrix dd NN is referred to as concurrence

matrix of d, and its entries, the concurrences parameters are denoted by

dij . For any block design, dd NN , the treatment concurrence with

diagonal elements equal to r and the off-diagonal elements equal to the number of times any pair of treatment occur together within blocks, in a

balanced design, the off-diagonal entries of dd NN are all equal to a

constant, λ (say) that is, the connon replication for a BIBD is r and the connon pairwise treatment concurrence is λ, i.e.

JIrNN dd )(

Where I is the identity matrix and J is a matrix of ones.

Bose information matrix for estimating treatment contrast using design d is:

ddd NNkrC

Where kδ is the matrix with diagonal elements equal to the corresponding elements of k and off-diagonal elements equal to zero (see for more detail [John 1987] p.8). Cd is also known as C-matrix of the design, which determines the statistical properties of a BIBD. According to John [1987] the information matrix plays a key role in establishing optimality criteria.

Cd is symmetric and non-negative definite, with rank v-1 for Dd .

Shah and Sinha (1989) represented the C-matrix as:

dddvddd NNkrrrdiagC 1

21 )...,,,(

Where rdi is the number of times treatment i is replicated, and diag (rd1, rd2,…, rdv) is a v×v diagonal matrix. This matrix is a positive semi-definite matrix (with zero row and column sums). The information matrix Cd can also be written as:

][ 1 JnIErCd

Where E= λ v / r k, and n is the total number of units in the design. As stated earlier that BIBDs are efficiency-balanced, with all canonical efficiency factors equal to E. These designs are also optimal under all optimality criteria. In this paper, we give a general method of construction of BIBDs, which is based on the cyclical developments of one or more sets of v plots. We refer to this cyclical development as the method of cyclic shifts or sets of

BALANCED INCOMPLETE BLOCK DESIGNS USING CYCLIC SHIFTS 3

shifts. Our paper is organized as follows. Section 2 develops relation between the method of cyclic shifts and the concurrences among off-diagonal of the concurrences matrix, while the method is described in detail in section 3. Section 4 deals with the construction of balanced incomplete block designs. The last section concludes this paper with final remarks.

CONCURRENCES AND CYCLIC SHIFTS We will now describe a systematic method for constructing a design, we call it the method of cyclic shifts (discussed in detail in next section),

which has a dd NN matrix of any pre-specified form. Suppose that we

want to construct a design which has a particular pattern for dd NN , the

entries of dd NN are build up as follows. Define dd NN equal to zero.

Since dd NN is a v×v matrix, having v diagonal elements equal to r and

next to the diagonal there are v -1 off-diagonal elements. Next to these, we have a further v - 2 off-diagonal elements, and so on. We want to systematically convert some or all of the off-diagonal zeros into positive integers. For v=b and k=2, we can convert exactly v zeros into 1’s. Suppose we convert the v-1 zeros next to the main diagonal into 1’s and for the vth element we convert the last zero of the first row into 1. Then for r=2, k=2, the concurrence matrix will be

2...

1...

0...2

0...12

0...012

1...0012

dd NN

The corresponding design is

0...4321

1...3210 v, where columns represent

the blocks. Now, instead of considering the elements next to the main diagonal, let us convert the elements in the next-but-one position to the main diagonal into ones. Since, we have converted v-2 zeros into 1’s, so we also have to convert the second last element of the first row and the last element of the second row into 1’s, i.e.


0

02

102

.........

.........

00......2

00......02

10......102

01......0102

dd NN

The corresponding design is

10...432

12...210 vv .

The conversion of the elements next to the main diagonal into 1 is named as ‘using shift 1’ and the conversion of the elements in the next-but-one position is named as ‘using shift 2’ and so on. In general for any value of k and v=b, we have to convert k (k-1) v/2 zeros into ones. So, when we convert the elements in the ith diagonal position next to the main diagonal we call it ‘using cyclic shift i’. However, we do not need to construct the

dd NN matrix directly because we can find its off-diagonal elements

immediately from the shifts. For further understanding of this method, we again explain this method of cyclic shifts as follows. Let us start with a set of v=b blocks, each containing one plot. The treatment on these plots are respectively, 0,1,2,3,…,v-1. By using a shift q, say, we mean adding a constant q (mod v) to each of the treatments. This then gives a design with k=2. In the previous two designs, we have used shifts q=1 and q=2 respectively. For a design with b=2v blocks, we can use the shifts, say q1 and q2, each separately to two sets of v plots. If q1=1 and q2=2, then

4

14

114

.........

............

000......4

000......14

100......114

010......0114

dd NN

And the corresponding design will be

10...4320...321

12...2101...210 vvv

From the above examples, it is clear that by using certain shifts individually or in combination, one or more sets of b blocks can be constructed.


The underlying problem is to obtain properties of BIBDs. The most appropriate criteria for obtaining optimal designs is to make the off-diagonal elements of the concurrence matrix as close as possible. Thus the properties of a BIBD depend on the number of concurrences between the pair of treatments. A concurrence between two treatments occurs when both treatments are in the same block. Using this method of cyclic shifts, the number of concurrence between any pair of treatments i.e. between any treatment and the remainder can be obtained from the sets of shifts used to construct the designs. It is clear that the concurrence matrix and the concurrences among off-diagonal elements play a vital role for a block design or row-column design to be optimal or not. According to John [1987], the information matrix Cd and concurrence matrix

dd NN of a cyclic design are circulant, so an explicit expression for

the canonical efficiency factors can be obtained. Further, the circulant matrix can be specified by the elements in the first row, since the other rows are obtained from the first row by a cyclic rotation. This is main reason by which we can quickly and easily obtain the properties of a design directly without constructing blocks of the design. Detail of the method of cyclic shifts is described below.

THE METHOD OF CYCLIC SHIFTS The method of cyclic shifts is a particular way of constructing circular balanced incomplete block designs, Here the v treatments are labeled as 0, 1, 2,….1 v-1 and we consider the construction of equireplicate binary designs for v treatments in b = v blocks of size k. The method of construction is to allocate to the first plot in the ith block the treatment i; i = 0, 1, 2, …., v–1. We denote this using the vector u1 = [0,1, 2, ….., v–1]′, which holds the treatments allocated to the first plot in each of the blocks 1, 2, …., v respectively. To obtain the treatment allocation of the remaining plots in each block, we cyclically shift the treatment allocated to the first plot. In order to define a cyclic shift, let ui is the treatment allocated to ith plot in each block. That is, jth element of ui is a treatment allocated to plot i of block j. A cyclic shift of size qi, when applied to plot i, is then such that ui+1 = [ui + qi v, 1 is a vector of ones, 1 ≤ i ≤ k – 1 and 1 ≤ qi ≤ v – 1. Assuming that we always start with u1 as defined above, a design is completely defined by a set of k – 1 shifts, Q , say, where Q = [q1, q2,…,qk-1]. To avoid a treatment occurring more than once in a block we must ensure that the sum of any two successive shifts, the sum of any three successive shits,. . ., the sum of any k – 1 successive shifts, is not equal to zero mod v. Subject to this constraint, Q may consist of any combination of shifts including repeats. To illustrate the above method of construction, let us consider the construction of the design for v = 5and k = 3. The possible shifts are


defined as Q = [q1 , q2], where qi = 1or 2 and i = 1 or 2. Two possible choices are Q1 = [1, 1], and Q2 = [1,2]. Using Q1, we get u2 = [1,2,3,4,0]′ and u3 = [2,3,4,0,1]′. The complete designs obtained from using Q1 and Q2 are given below as Design 3.1 and Design 3.2. The blocks of designs are written vertically.

Design 3.1 Design 3.2 0 1 2 3 4 0 1 2 3 4 1 2 3 4 0 1 2 3 4 0 2 3 4 0 1 3 4 0 1 2 The properties of a design depend on the number of concurrences between the pair of treatments. In Design 3.1, for treatment 0 the concurrences with each of the treatments 1, 2, 3 and 4, respectively are, 2, 1, 1, 2, while in Design 3.2 the corresponding concurrences are 1, 2, 2, 1. Because of the cyclic nature of the construction, the number of the concurrences between any treatment and the remainder can be obtained from the number of concurrences between 0 and the remainder. The number of the occurrences between 0 and the remainder can easily be obtained from Q, the set of shifts used to construct the design. Similarly, the number of concurrences between treatment 1 and treatment 2, 3 and 4 are 2, 1, 1 respectively, that is, they are obtained by cycling the list of concurrences. To show that the number of concurrences between treatments can be obtained from Q, consider the number of concurrences for Design 3.1 and the shifts Q1 = [1, 1] used to construct this design. We note that the number of concurrences are symmetric about [v/2] in the sense that any shift of size q that results in concurrence between 0 and treatment q also results in a concurrence between 0 and v – q. This means that we need to use only shifts such that 1 ≤ qi ≤ [v/2]. If shifts q1 and q2 are applied successively to treatment 0, the result is a concurrence between treatment 0 and treatments q1 and q2, and a concurrence between 0 and treatment q1 + q2. If a third shift q3, say, is then applied after q1 and q2, the following treatments will also concur with treatment 0: q3, q1 + q2 and q1 + q2 + q3,. This adding of shifts to get the treatments which concurs 0 works for the general case and so enables the number of concurrences of a design to be obtained directly from the shifts which defines it. In the general case, if shifts q1, q2,…., qi –1will be applied to treatment 0, then the additional concurrences which result when shift qi is applied are between treatment 0 and treatments q1 + q2 + … + qi, q2 + q3 + … + qi,…, qi – 1 + qi, qi, where addition is mod v. To illustrate the calculation of the number concurrences from the shifts, consider the Design 3.1 again. The shifts are Q1 = [1,1]. The treatment which concur with treatment 0 are therefore 1,1 and 1+1 i.e. treatment 1 concurs twice with treatment 2 concurs once. By the symmetry property, treatment 0 also concurs twice with treatment 4 and once with treatment


3. The concurrences of treatment 1 with treatment 2,1,1 respectively. Similarly, the concurrences of the treatment 2 with 3 and 4 are 2 and 1 respectively, and so on. So far, we have only consider the construction of design for v = b. To construct design for v < b, we can combine two or more designs of v = b to get a new design. For more detail see Iqbal [1991].

FRACTIONAL DESIGNS A feature of the method of construction is that certain sets of shifts produce design that are made up of complete replications of the smaller designs. That is, the v blocks can be divided into s sets of size n = v/s and the n blocks in each set contain the same treatment allocation. For example consider v = 6, k = 4 and Q = [1,2,1]. The design obtained by applying Q to u1 is given below.

Design 3.1.1 0 1 2 3 4 5 1 2 3 4 5 0 3 4 5 0 1 2 4 5 0 1 2 3

It appears from above design that it has been made up from two complete designs as indicated by vertical line. The number of concurrences that treatment makes with each other treatments are respectively 2, 2, 4, 2, 2 in the whole design whereas the concurrences are 1, 1, 2, 1, 1, (half) in the fractional design and is denoted by [1, 2, 1] ½. In order to decide whether a set of shifts will produce a fractional design, we will take the value of v/k and determine the smallest integer z, say, which makes (v × z)/k an integer n, say. If v is also divisible by n then m fractional designs can be obtained. Let v/n = m, then we have m fractional designs within the whole design with z the number of replicates of each fractional design and n the number of blocks of each fractional design. The shifts that will produce the required design must be such that z successive shifts add to n. In the above example, we have

v / k = 6/4 (v × z)/k = (6 × 2)/4= 3 = n

v / n = 6/3 = 2 = m That is m = 2 and there are two fractional designs within the whole design. Each fractional design has n( = 3) blocks, z (= 2) replications and the shifts used are Q = [1, 2, 1] that is, the sum of z (= 2) successive shifts is equal to n (= 3). So far we have only considered the construction of designs for b ≤ v. To construct design for b > v, we can combine two or more full or fractional designs.


ADDING A NEW TREATMENT Sometimes a design for v-1 treatments with block of size k and k-1 can be converted into a design for v treatments, and all blocks of size k by adding additional treatment to each of the smaller block of size k-1. The example of this design for v=8, b= 14 and λ =2 below. The two sets of shifts used to construct the original design for v=7, k1 =4 and k2 =3 were Q1 = [1,2,3] and Q2 = [2,4]. We note that treatment 7 has been added to each block of the smaller design. Design 3.2.1

0 1 2 3 4 5 6 0 1 2 3 4 5 6 1 2 3 4 5 6 0 2 3 4 5 6 0 1 3 4 5 6 0 1 2 6 0 1 2 3 4 5 6 0 1 2 3 4 5 7 7 7 7 7 7 7

CONSTRUCTION OF BALANCED INCOMPLETE

BLOCK DESINGS As noted is section 3 that the set of shifts Q = [q1, q2, …, qk-1] used to construct a design can also be used to determine the treatment concurrences, in fact, if the shifts q1, q2, …, qk-1 are applied in succession to u1, treatment 0 concurs with (i) the treatment listed in set P, defined below and, (ii) the complement of treatments listed in P, where if q is a treatment listed in P its complement is v-q mod v. we use P َ to denote the complement of the set P. P=[q1,q2,q2+q1,q3, q3+q2,q3+q2+q1,…, qk-1,qk-2+qk-1,…, qk-1+qk-2+qk-3+…+q1] It will be noted that any particular treatment may occur more than once in P and P َ the number of times a treatment occur is the number of concurrences it makes with treatment 0. THEOREM 1: If the sets P and P َ , constructed from the set of shifts Q, contain between them an equal number of each of the treatments 1, 2, …, v-1, then the design obtained by using Q is a BIBD. Proof: If P and P َ contain each treatment an equal number of times, the treatment 0 will concur an equal number of times with treatments 1, 2, …, v-1. The cyclic nature of the design further implies that every pair of treatments concur an equal number of times. Given v=b, and k, we can therefore construct a BIBD by searching for a set(s) of shifts which satisfy Theorem 1. However, for values of b > v, Theorem 1 will be of no use. In order to fill this gap, we obtain BIBDs by combining together the blocks of one or more full or fractional designs.


The designs which are combined need not to be balanced. The only restriction is that the final design obtained from the combined set of blocks is balanced i.e. each pair of treatments has the same number of concurrences. We therefore searched for the designs to combined together to produce balanced designs. In table 1, we give a list of the balanced designs so found for 19 ≤ v ≤ 100 with k = 2. The notations used in the Table 1 describe which designs have been combined is best explained by considering a particular set of values of v, k, b and r. for v= 6, k=3 and r=20 the final design described as [1,1]2+[1,2]4+[2,2]2/3. This means that, in the combination of two copies of design obtained from the shifts [1,1], four copies of the design obtained from the shifts [1,2] and two copies of the 1/3 fraction of the design obtained from the shifts [2,2] are added together to give a design containing 40(= 2 × 6 + 4 × 6 + 2 × 2) blocks. Also in Table 1 the designs obtained from Theorem 2 have a t at the end of the list of shifts (i.e. designs) which have been obtained to give the complete design. In order to state Theorem 2, we need to introduce some notations. Let D1 and D2 be the block designs for v-1 treatments such that D1 has b1 blocks of size k1 and replication r1, and D2 has b2 blocks of size k2 and replication r2. Further, let k2 = k1 -1 and let the number of concurrences made between treatment 0 and treatment 1, 2, …, v-1 be λ11,λ12, … λ1v-1 in D1 and λ21,λ22, … ,λ2,v-1 in D2. THEOREM 2: Let D1 and D2 be as defined above. If k2 = k1 -1, b2=r1+r2, λ11+λ21=r2 for i=1,2,…,v-2 and a further treatment is added to each block of D2, then the design which results from combining the blocks of D1 and the augmented blocks of D2 is a BIBD. In this design, each pair of treatments makes r2 concurrences. Proof: By definition, D1 and D2 (before augmentation) are such that when combined they produce an equireplicate design for v-1 treatments in which every pair of treatments concurs an equal number r2 of times. Treatment v-1, when added, makes r2 concurrences with each of the treatments in D2. Hence in the combined design, treatment 0 concurs r2 times with every other treatments, and all block sizes are equal. As an example of Theorem 2, consider designs 5a and 5b below, for v=5. When the blocks of D1 and D2 are combined, treatment 0 makes two concurrences with each of treatments 12,3 and 4. If treatment 5 is added to each block of design 4.1(b), then, because each treatment in design 4.1(b) is replicated twice, treatment 5 will make two concurrences with treatment 0. Hence by combining design 4.1(a) and design 4.1(b) (after augmentation), we obtain a BIBD 4.2 for v=b, b=10, k=3 and r=5.


Design 4.1(a) Design 4.1(b)

0 1 2 3 4 1 2 3 4 0 2 3 4 0 1

0 1 2 3 4 2 3 4 0 1

Design 4.2 after augmentation

0 1 2 3 4 0 1 2 3 4 1 2 3 4 0 2 3 4 0 1 2 3 4 0 1 5 5 5 5 5

The usefulness of Theorem 2 is that it gives yet another way of obtaining balanced designs. The notation used to describe the designs listed in Table 1 that have been constructed using Theorem 2, is well explained in the following example. Consider the design for v=6, b=30 and k=3. The designs corresponding to D2 are enclosed in the parentheses and followed by a t and rest of the description given for the design refers to D1. That is, D1 is [1,1]2+[1,2] and D2 is {[1]+[2]2}t. we added treatment 5 to D2 to obtain the final design. To determine the usefulness of Theorem 1 and Theorem 2 we attempted to construct BIBDs for all values of 19 ≤ v ≤ 100 with k = 2, 3 and 19 ≤ v ≤ 25 with k = 4, 5, restricted v<60, for which BIBDs are known to exist. To find more on the existence of BIBDs, we may refer to Maton and Rose [1985], Takeuchi [1962], Pearce [1967] and Raghavaro [1971], (Table 5.10.1). Using Theorem 1, Theorem 2 and combining designs together, we find all as given in Tables 1 to 14.

REMARKS Bose [1939] described a cyclic method of construction which we referred to as the method of symmetrical repeated differences. In Bose’s method an initial block or blocks of k treatments which satisfy certain symmetry conditions is first obtained. The treatments in blocks 2,3,…,b are obtained as follows: The treatments in blocks i+1 are obtained by adding a constant, q (say), mod v to each treatment in block i, i = 1,2,…, b -1. That is, the blocks are obtained by cyclically changing the treatments in the first block. Bose described a second method which requires an initial set of blocks and a treatment labeled as , which did not change when the constant q is added. A criticism on Bose’s albeit pioneering method is that there is no simple method, in general, for determining the initial set or sets of blocks. A number of series of values of parameters (v, b, k, r, λ) have been developed, however, for which the initial blocks can be determined easily. The two main series are those of Bose [1939, 1942] and Sprott [1954, 1956].


Table 1: Balanced Incomplete Block Designs for 3 ≤ v ≤ 18 and k=v-1

Design # V k r b λ Set of Shifts

1 3 2 2 3 1 [1] 2 3 2 4 6 2 [1]2 3 3 2 6 9 3 [1]3 4 3 2 8 12 4 [4]4 5 3 2 10 15 5 [5] 6 4 2 3 6 1 [1]+[2]1/2 7 4 2 6 12 2 [1]2+[2] 8 4 2 9 18 3 [1]3+[2]3/2 9 4 2 12 24 4 [1]4+[2]2 10 4 3 3 4 2 [1,1] 11 4 3 6 8 4 [1,1]2 12 4 3 9 12 6 [1,1]3 13 5 2 4 10 1 [1]+[2] 14 5 2 8 20 2 [1]2+[2]2 15 5 2 12 30 3 [1]3+[2]3 16 5 3 6 10 3 [1,1]+[1,2] 17 5 3 12 20 6 [1,1]2+[1,2]2 18 5 4 4 5 3 [1,1,1] 19 5 4 8 10 6 [1,1,1,]2 20 5 4 12 15 9 [1,1,1,]3 21 6 2 5 15 1 [1]+[2]+[3]1/2 22 6 2 10 30 2 [1]2+[2]2+[3] 23 6 2 15 45 3 [1]3+[2]3+[3]3/2 24 6 3 5 10 2 [1,1]+[2]t 25 6 3 10 20 4 [1,1]+[1,2]2+[2,2]1/3 26 6 3 15 30 6 [1,1]2+[1,2]+{[1]+[2]2}t 27 6 3 20 40 8 [1,1]2+[1,2]4+[2,2]2/3 28 6 4 10 15 6 [1,1,1]+[1,1,2]+[1,2,1]1/2 29 6 4 20 30 12 [1,1,1]2+[1,1,2]2+[1,2,1] 30 6 5 5 6 4 [1,1,1,1] 31 6 5 10 12 8 [1,1,1,1]2 32 6 5 15 18 12 [1,1,1,1]3 33 7 2 6 21 1 [1]+[2]+[3] 34 7 2 12 42 2 [1]2+[2]2+[3]2 35 7 2 18 63 3 [1]3+[2]3+[3]3 36 7 3 3 7 1 [1,2] 37 7 3 6 14 2 [1,2]2 or [1,2]+[1,4] 38 7 3 9 21 3 [1,2]3 or [1,2]2+[1,4]2 39 7 3 12 28 4 [1,2]2+[1,4,]2 40 7 4 4 7 2 [1,1,2] 41 7 4 8 14 4 [1,1,2]2 42 7 4 12 21 6 [1,1,2]3 43 7 5 15 21 10 [1,1,1,1]+[1,1,12]+[1,1,2,1] 44 7 6 6 7 5 [1,1,1,1,1] 45 7 6 12 14 10 [1,1,1,1,1]2 46 8 2 7 28 1 [1]+[2]+[3]+[4]1/2 47 8 2 14 56 2 [1]2+[2]2+[3]2+[4] 48 8 3 21 56 6 [1,2]4+[1,3]2+[2,2] 49 8 4 7 14 3 [1,1,2]+[1,2]t 50 8 4 14 28 6 [1,1,1]+[1,2,3]+[1,3,1]+[2,2,2]1/2 51 8 5 35 56 20 [1,1,1,1]+[1,1,12]2+[1,1,2,1]2+ [1,1,2,2]+[1,2,1,2] 52 8 6 21 28 15 [1,1,1,1,1]+[1,1,1,1,2]+[1,1,2,1,2]+ [1,1,2,1,1]1/2 53 8 7 7 8 6 [1,11,1,1,1] 54 8 7 14 16 12 [1,11,1,1,1]2 55 9 2 8 36 1 [1]+[2]+[3]+[4] 56 9 2 16 72 2 [1]2+[2]2+[3]2+[4]2 57 9 3 4 12 1 [1,2]+{[4]1/2}t 58 9 3 8 24 2 [1,2]+[1,3]+[2]t 59 9 3 12 36 3 [1,2]2+[1,4]+[2,3] 60 9 3 16 48 4 [1,2]2+[1,3]+[1,4]+[2,2]+[3,3]1/3


61 9 4 8 18 3 [1,1,2]+[1,3,2] 62 9 4 16 36 6 [1,1,2]2+[1,3,2]2 63 9 5 10 18 5 [1,1,1,2]+[1,2,2,1] 64 9 5 20 36 10 [1,1,1,2]2+[1,2,2,1]2



65 9 6 8 12 5 [1,1,2,1,1]1/2+[1,1,12]t 66 9 6 16 24 10 [1,1,1,1,2]+{[1,1,1,2]+[1,1,2,1]}t 67 9 7 28 36 21 [1,1,1,1,1,1]+[1,1,2,1,1,1]+[1,2,1,1,1,1]+[2,1,1,1,1,1] 68 9 8 8 9 7 [1,1,1,1,1,1,1] 69 10 2 9 45 1 [1]+[2]+[3]+[4]+[5]1/2 70 10 2 18 90 2 [1]2+[2]2+[3]2+[4]2+[5] 71 10 3 9 30 2 [1,3]+[2,2]+[3,3]1/3+[1]t 72 10 3 18 60 4 [1,2]2+[1,3]+[1,4]+[2,3]+[2,4] 73 10 4 12 30 4 [1,1,2]+[1,2,4]+[1,4,1]1/2+[2,3,2]1/2 74 10 4 12 30 4 [1,1,2]+[1,2,4]+[1,4,1]1/2+[2,3,2]1/2 75 10 4 18 45 6 [1,1,2]+[1,1,3]+[1,2,4]+[1,3,2]+[2,3,2]1/2 76 10 5 9 18 4 [1,1,1,2]+[1,3,2]t 77 10 5 18 36 8 [1,1,1,2]+[1,2,2,1]+{[1,1,2]+[1,3,2]}t 78 10 6 18 30 10 [1,1,1,2,1]+[1,1,1,2,2]+[1,1,2,1,3] 79 10 6 18 30 10 [1,1,1,2,1]+[1,1,1,2,2]+[1,1,2,1,3] 80 10 7 21 30 14 [1,1,1,1,1,1]+{[1,2,1,1,1]+[2,2,1,1,1]+[1,2,1,2,1]1/3}t 81 10 8 36 45 28 [1,1,1,1,1,1,1]+[1,1,2,1,1,1,1]+ [1,2,1,1,1,1,1]+

[2,1,1,1,1,1,1]+ [1,1,1,2,1,1,1]1/2 82 10 9 9 10 8 [1,1,1,1,1,1,1,1] 83 10 9 18 20 16 [1,1,1,1,1,1,1,1]2 84 11 2 10 55 1 [1]+[2]+[3]+[4]+[5] 85 11 2 20 110 2 [1]2+[2]2+[3]2+[4]2+[5]2 86 11 3 15 55 6 [1,2]+[1,3]+[1,4]+[2,3]+[2,4] 87 11 4 20 55 6 [1,1,30+[1,2,2]+[1,2,4]+[1,2,5]+[1,3,2] 88 11 5 5 11 2 [1,1,4,3] 89 11 5 10 22 4 [1,1,4,3]2 90 11 5 15 33 6 [1,1,4,3]3 91 11 6 6 11 3 [1,1,2,1,2] 92 11 6 12 22 6 [1,1,2,1,2]2 93 11 6 18 33 9 [1,1,2,1,2]3 94 11 7 35 55 22 [1,2,2,1,1,1]+[1,2,1,1,1,1]+[2,2,1,1,1,1]+

[2,1,1,2,1,2]+ [2,1,1,1,2,1] 95 11 8 40 55 28 [1,1,2,1,1,1,1]+[2,1,1,1,1,1,1]+ [2,1,1,2,1,1,1]+

[1,2,1,1,1,1,1]+ [2,1,2,1,1,1,1] 96 11 9 45 55 36 [1,1,1,1,1,1,1,1]+[1,1,1,2,1,1,1,1]+[1,1,2,1,1,1,1,1]+

[1,2,1,1,1,1,1,1]+ [2,1,1,1,1,1,1,1] 97 11 10 10 11 9 [1,1,1,1,1,1,1,1,1] 98 12 2 11 66 1 [1]+[2]+[3]+[4]+[5]+[6]1/2 99 12 3 11 44 2 [1,2]+[1,5]+[2,3]+[4,4]2/3 100 12 4 11 33 3 [1,1,3]+[1,2,5]+[2,4,2]1/2+[3,3,3]1/4 101 12 5 55 132 20 [1,1,1,1]+[1,1,2,3]+[1,1,3,3]+[1,2,2,4]2+[1,1,2,5]+[1,

1,4,3]2+[1,1,4,2]+[1,2,3,2]2 102 12 6 11 22 5 [1,1,1,2,2]+[1,2,3,1,2]1/2+[1,3,1,3,1]1/3 103 12 7 77 132 42 [1,1,1,1,1,1]+[2,1,2,1,1,1]+[1,2,1,2,1,1]+

[2,2,1,1,2,1]2+ [2,1,1,1,2,1]+ [1,1,2,1,2,1]2+ [1,1,2,2,1,1]+ [2,1,2,2,1,1]2

104 12 8 22 33 14 [2,1,1,1,1,1,1]+[2,1,1,1,2,1,1]+ [2,1,1,2,2,1,1]1/2+ [1,2,1,2,1,2,1]3/4

105 12 9 33 44 24 [1,1,1,2,1,1,1,1]+[2,1,1,1,1,1,1,1]+ [2,1,2,1,1,1,1,1]+ [1,1,2,1,1,2,1,1]2/3

106 12 10 55 66 45 [1,1,1,1,1,1,1,1,1]+[2,1,1,1,1,1,1,1,1]+[1,2,1,1,1,1,1,1,1]+[1,1,2,1,1,1,1,1,1]+ [1,1,1,2,1,1,1,1,1]+ [1,1,1,1,2,1,1,1,1]1/2

107 12 11 11 12 10 [1,1,1,1,1,1,1,1,1,1]


108 13 2 12 78 1 [1]+[2]+[3]+[4]+[5]+[6] 109 13 3 6 26 1 [1,3]+[2,5] 110 13 3 12 52 2 [1,3]2+[2,5]2 111 13 3 18 78 3 [1,3]3+[2,5]3 112 13 4 4 13 1 [1,2,6] 113 13 4 8 26 2 [1,2,6]2 114 13 4 12 39 3 [1,2,6]3 115 13 4 16 52 4 [1,2,6]4 116 13 5 15 39 5 [1,1,2,6]+[1,1,3,3]+[1,3,2,2]


Design# V k r b λ Set of Shifts

117 13 6 12 26 7 [1,1,1,3,3]+[1,1,2,2,3] 118 13 7 14 26 7 [1,1,1,1,3,2]+[1,1,2,1,3,2] 119 13 8 24 39 14 [1,2,2,2,1,1,1]+[1,2,1,2,1,1,1]+[2,1,1,1,1,2,1] 120 13 9 9 13 6 [,1,1,1,1,1,2,2,1] 121 13 9 18 26 12 [,1,1,1,1,1,2,2,1]2 122 13 10 20 26 15 [1,1,1,1,2,2,1,1,1]+[1,1,1,1,1,1,1,2,1] 123 13 11 54 78 45 [1,1,1,1,1,1,1,1,1,1]+[2,1,1,1,1,1,1,1,1,1] +

[1,2,1,1,1,1,1,1,1,1]+[1,1,2,1,1,1,1,1,1,1]+ [1,1,1,2,1,1,1,1,1,1]+[1,1,1,1,2,1,1,1,1,1]

124 13 12 12 13 11 [1,1,1,1,1,1,1,1,1,1,1] 125 14 2 13 91 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]1/2 126 14 3 39 182 6 [1,3]5+[2,5]6+[1+3+4]t 127 14 4 26 91 6 [1,2,6]5+{[1,3]+[2,5]}t 128 14 5 65 182 20 [1,1,2,6]3+[1,1,3,3]3+[1,3,2,2]3+ {[1,2,6]5}t 129 14 6 39 91 15 [1,1,1,3,3]2+[1,1,2,2,3]2+{[1,1,2,6]+[1,1,3,3]+[1,3,2,2]}t 130 14 7 13 26 6 [1,1,2,1,3,2]+[1,1,2,2,1,1]t 131 14 8 52 91 28 [1,2,1,2,1,1,1]+[1,2,2,2,1,1,1]+[2,1,1,1,1,2,1]+

{[1,2,1,2,1,1]2+[2,2,12,1,1]2}t 132 14 9 117 182 72 [1,1,1,1,2,1,1,1]5+{[2,1,1,1,1,2,1]3+

[1,2,1,2,1,1,1]3+[2,2,2,1,1,1]3}t 133 14 10 65 91 45 [1,2,1,1,1,1,1,1,1]+[2,1,1,1,2,1,1,1,1]+

{[1,2,1,1,1,1,2,1]5}t 134 14 11 143 182 100 [1,1,1,1,1,1,1,1,1,1]+[1,2,1,1,1,1,1,1,1,1]

+[1,1,2,1,1,1,1,1,1,1]+{[1,2,1,1,1,1,1,1,1]5+ [2,1,1,1,2,1,1,1,1]6}t

135 14 12 78 91 66 [1,1,1,1,1,1,1,1,1,1,1]+[1,1,1,1,2,1,1,1,1,1,1] +[1,1,1,2,1,1,1,1,1,1,1]+[1,1,2,1,1,1,1,1,1,1,1]+ [1,2,1,1,1,1,1,1,1,1,1]+[2,1,1,1,1,1,1,1,1,1,1]+ [1,1,1,1,2,1,1,1,1,1,1]1/2

136 14 13 13 14 12 [1,1,1,1,1,1,1,1,1,1,1,1] 137 15 2 14 105 1 [1]+[2]+[3]+[4]+[5]+[6]+[7] 138 15 3 7 35 1 [1,3]+[2,6]+[5,5]1/3 139 15 3 14 70 2 [1,3]2+[2,6]2+[5,5]2/3 140 15 4 28 105 28 [1,2,4]+[1,2,7]+[1,2,8]+[1,3,2]+[1,4,2]+ [2,3,4]+[1,3,5] 141 15 5 14 42 4 [1,1,4,6]+[1,2,1,5]+{[2,5,2]1/2+ [3,4,3]1/2}t 142 15 6 14 35 5 [1,1,1,3,5]+[1,2,4,1,2]1/2+[1,3,2,2]t 143 15 7 7 15 3 [1,1,2,1,3,2] 144 15 7 14 30 6 [1,1,2,1,3,2]2 145 15 8 8 15 4 [1,1,1,2,2,1,3] 146 15 8 16 30 8 [1,1,1,2,2,1,3]2 147 15 9 21 35 12 [1,2,2,2,1,1,1,1]+{[1,2,1,1,1,2,1]+ [2,1,1,3,2,1,1]1/2}t 148 15 10 28 42 18 [1,1,1,2,2,1,1,1,2]+[1,2,1,1,2,1,2,1,1]1/2+

{[1,1,2,1,1,2,1,1]+ [1,1,1,2,1,1,1,3]}t 149 15 11 77 105 55 [1,1,2,2,1,1,1,1,1,1]+[1,2,1,1,1,2,1,1,1,1]+

[1,2,2,1,1,1,1,1,1,1]+[2,1,1,1,1,1,2,1,1,1]+ [2,1,1,2,1,1,1,1,1,1]+[2,1,1,1,1,1,1,2,1,1]+ [2,1,2,1,1,2,1,1,1,1]

150 15 12 28 35 22 [1,2,1,1,1,1,1,1,1,1,1]+[2,1,1,1,1,2,1,1,1,1,1,] +[1,1,1,2,1,1,1,2,1,1,1]1/3


151 15 13 91 105 78 [1,1,1,1,1,1,1,1,1,1,1,1]+[1,1,1,1,1,2,1,1,1,1,1,1]+ [1,1,1,1,2,1,1,1,1,1,1,1]+[1,1,1,2,1,1,1,1,1,1,1,1]+ [1,1,2,1,1,1,1,1,1,1,1,1]+[1,2,1,1,1,1,1,1,1,1,1,1]+ [2,1,1,1,1,1,1,1,1,1,1,1]

152 15 14 14 15 13 [1,1,1,1,1,1,1,1,1,1,1,1,1] 153 16 2 15 120 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]1/2 154 16 3 15 80 2 [1,3]+[1,6]+[2,4]+[2,5]+[3,5] 155 16 4 5 20 1 [1,2,4]+{[5,5]1/3}t 156 16 4 10 40 2 [1,2,4]+[2,4,1]+[3,5,3]1/2 157 16 4 15 60 3 [1,2,3]+[2,3,7]+[2,4,1]+[1,7,1]1/2+ [4,4,4]1/4 158 16 4 20 80 4 [1,2,4]2+[2,4,1]2+[3,5,3] 159 16 5 15 48 4 [1,1,2,3]+[1,2,2,5]+[1,2,4,4] 160 16 6 12 32 4 [1,1,2,4,3]+[1,1,4,5,3]


Design # V k r B λ Set of Shifts

161 16 6 15 40 5 [1,1,2,2,3]+[1,3,1,2,4]+[1,2,5,1,2]1/2 162 16 6 18 48 6 [1,1,2,2,3]+[1,1,4,2,3]+[1,3,1,2,6] 163 16 8 15 30 7 [1,1,1,2,2,1,3]+[1,1,2,1,3,2]t 164 16 10 20 32 12 [1,1,1,2,1,1,2,1,1]+[1,1,2,1,1,2,1,2,3] 165 16 15 15 16 14 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] 166 17 2 16 136 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8] 167 17 4 16 68 3 [1,3,2]+[1,4,3]+[1,5,2]+[2,4,3] 168 17 5 20 68 5 [1,1,3,3]+[1,1,4,3]+[1,2,4,5]+[2,3,4,6] 169 17 8 16 34 7 [1,1,1,2,3,1,3]+[1,1,2,2,1,2,4] 170 17 9 18 34 9 [1,1,1,1,2,2,1,4]+[1,1,2,3,1,1,3,3] 171 17 16 16 17 15 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] 172 18 2 17 153 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]1/2 173 18 3 17 102 2 [1,3]+[1,6]+[2,3]+[2,7]+[4,5]+[6]t 174 18 6 17 51 5 [1,1,2,3,4]+[1,2,2,3,1]+[1,2,4,5]t 175 18 9 17 34 8 [2,1,1,3,1,2,2,1]+[1,1,1,2,2,1,3]t 176 18 17 17 18 16 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

Table 5: Balanced Incomplete Block Designs for 19 ≤ v ≤ 100 and k=2


1 19 21 81 71 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9] 2 20 2 19 190 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]1/2 3 21 2 20 210 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10] 4 22 2 21 231 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10] +[11]1/2 5 23 2 22 253 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11] 6 24 2 23 276 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]1/2 7 25 2 24 300 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12] 8 26 2 25 325 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+

[13]1/2 9 27 2 26 351 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+ [8]+[9]+[10]+[11]+[12]+[13] 10 28 2 27 378 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]1/2 11 29 2 28 406 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14] 12 30 2 29 435 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]1/2 13 31 2 30 465 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15] 14 32 2 31 496 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]1/2 15 33 2 32 528 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12] +[13]+[14]+[15]+[16] 16 34 2 33 561 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]1/2 17 35 2 34 595 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+


[13]+[14]+[15]+[16]+[17] 18 36 2 35 630 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]1/2 19 37 2 36 666 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] 20 38 2 37 703 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]1/2 21 39 2 38 741 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19] 22 40 2 39 780 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]1/2 23 41 2 40 820 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20] 24 42 2 41 861 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]1/2 25 43 2 42 903 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21] 26 44 2 43 946 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22]1/2



27 45 2 44 990 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

28 46 2 45 1035 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+ [22]+[23]1/2 29 47 2 46 1081 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+

[13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+ [22]+[23]

30 48 2 47 1128 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+ [19]+[20]+[21]+ [22]+ [23]+[24]1/2 31 49 2 48 1176 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+ [22]+[23]+ [24] 32 50 2 49 1225 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22] +[23]+[24]+[25]1/2 33 51 2 50 1275 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22] +[23]+[24]+[25] 34 52 2 51 1326 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]1/2 35 53 2 52 1378 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26] 36 54 2 53 1431 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]1/2 37 55 2 54 1485 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27] 38 56 2 55 1540 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+

[13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28]1/2

39 57 2 56 1596 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28]

40 58 2 57 1653 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+


[13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28]+[29]1/2

41 59 2 58 1711 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28] +[29]

42 60 2 59 1770 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28] +[29]+[30]1/2

43 61 2 60 1830 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30] 44 62 2 61 1891 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28]+ [29]+[30]+[31]1/2 45 63 2 62 1953 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28] +[29]+[30]+[31] 46 64 2 63 2016 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28] +[29]+[30]+[31]+ [32]1/2

47 65 2 64 2080 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28] +[29]+[30]+[31] +[32]



48 66 2 65 2145 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28] +[29]+[30]+[31]+ [32]+[33]1/2

49 67 2 66 2211 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22] +[23]+[24]+[25]+[26]+[27]+[28] +[29]+[30]+[31]+ [32]+[33]

50 68 2 67 2278 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]

+[19]+[20]+[21]+[22]+[23]+[24]+[25]+[26]+[27]+[28] +[29]+[30]+[31]+[32]+[33]+[34]1/2

51 69 2 68 2346 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]

+[19]+[20]+[21]+[22]+[23]+[24]+[25]+[26]+[27]+[28] +[29]+[30]+[31]+[32]+[33]+[34]

52 70 2 69 2415 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+[32]+[33]+[34]+[35]1/2

53 71 2 70 2485 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]

+[19]+[20]+[21]+[22]+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+[32]+[33]+[34]+[35]

54 72 2 71 2556 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]

+[22]+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+ [32]+[33]+[34]+[35]+[36]1/2

55 73 2 72 2628 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+

[22]+ [23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+[32]+ [33]+[34] +[35]+[36]


56 74 2 73 2701 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+ [22]+ [23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+[32]+ [33]+[34]+[35]+[36]+[37]1/2

57 75 2 74 2775 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+

[22]+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+[32]+[33]+[34]+[35]+[36]+[37]

58 76 2 75 2850 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+

[22]+ [23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+[32]+[33]+[34]+[35]+[36]+[37]+[38]1/2

59 77 2 76 2926 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+

[22]+[23]+[24]+[25]+[26]+[27]+[28]+ [29]+[30]+[31]+[32]+[33]+[34]+[35]+[36]+[37]+[38]

60 78 2 77 3003 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+

[22]+[23]+[24]+[25]+[26]+[27]+[28]+ [29]+[30]+ [31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]1/2

61 79 2 78 3081 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+

[22]+[23]+[24]+[25]+[26]+[27]+[28]+ [29]+[30]+ [31]+[32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]

62 80 2 79 3160 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+

[22]+[23]+[24]+[25]+[26]+[27]+[28]+ [29]+[30]+[31] +[32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]1/2

63 81 2 80 3240 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+ [29]+[30]+[31] +[32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]



64 82 2 81 3321 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+[32] +[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]1/2

65 83 2 82 3403 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]

66 84 2 83 3486 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]1/2

67 85 2 84 3570 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]

68 86 2 85 3655 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]1/2

69 87 2 86 3741 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+


[32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]

70 88 2 87 3828 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]1/2

71 89 2 88 3916 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]

72 90 2 89 4005 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]+[45]1/2

73 91 2 90 4095 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]+[45]

74 92 2 91 4186 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]+[45]+[46]1/2

75 93 2 92 4278 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]+[45]+[46]

76 94 2 93 4371 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]+[45]+[46]+[47]1/2

77 95 2 94 4465 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18] +[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+ [30]+[31] +[32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+ [41]+[42]+[43]+[44]+[45]+[46]+[47]



78 96 2 95 4560 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]+ [45]+[46]+[47]+[48]1/2

79 97 2 96 4656 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43] +[44]+[45]+[46]+[47]+[48]

80 98 2 97 4753 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42] +[43]+[44]+[45]+[46]+[47]+[48]+[49]1/2

81 99 2 98 4851 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+


[13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]+[45]+[46]+[47]+[48]+[49]

82 100 2 99 4950 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+[8]+[9]+[10]+[11]+[12]+ [13]+[14]+[15]+[16]+[17]+[18]+[19]+[20]+[21]+[22]

+[23]+[24]+[25]+[26]+[27]+[28]+[29]+[30]+[31]+ [32]+[33]+[34]+[35]+[36]+[37]+[38]+[39]+[40]+[41]+[42]+[43]+[44]+[45]+[46]+[47]+[48]+[49]+[50]1/2



1 19 3 9 57 1 [1,5]+[2,8]+[3,4] 2 20 3 19 190 1 [1]+[2]+[3]+[4]+[5]+[6]+[7]+ [8]+[9]+[10]1/2 3 21 3 10 70 1 [1,2]+[4,8]+[5,6]+[7,7](1/3) 4 22 3 42 308 4 [1,2](4)+[4,6]+[4,7]+[4,8]+[4,9]+[5,6]+[5,7](2)+

[5,8]+[6,7]+[6,8] 5 23 3 33 253 3 [1,2](3)+[4,6]+[4,8]+[4,9]+[5,6]+[5,7]+[5,9]+[6,7]+[7,8] 6 24 3 23 184 2 [1,2](2)+[4,6]+[4,9]+[5,7]+[5,9]+ [6,7]+[8,8](2/3) 7 25 3 12 100 1 [1,2]+[4,9]+[5,6]+[7,8] 8 26 3 150 1300 12 [1,2](12)+[4,6](4)+[4,7](2)+[4,8](2)+[4,10](2)+[4,11](2)

+[5,6](2)+[5,7]+[5,8](4)+[5,9](3)+[5,10](2)+[6,7](2)+ [6,8](3)+[6,9]+[7,7]+[7,8] +[7,9](4)+[8,9](2) 9 27 3 13 117 1 [1,2]+[4,10]+[5,6]+[7,8]+ [9,9](1/3) 10 28 3 27 252 2 [1,2](2)+[4,7]+[4,9]+[5,6]+[5,9]+[6,10]+[7,8]+[8,10] 11 29 3 42 406 3 [1,2](3)+[4,8](2)+[4,10]+[5,6]+[5,9]+[5,11]+[6,7]+

[6,11]+[7,8]+[7,9]+[9,10] 12 30 3 58 580 4 [1,2](4)+[4,5]+[4,8]+[4,11]+[4,13]+[5,7](2)+[5,11]+

[6,8](2)+[6,9]+[6,10]+[7,10]+[7,11] +[8,9]+[9,10]+ [10,10](1/3) 13 31 3 15 155 1 [1,2]+[4,7]+[5,10]+[6,12]+[8,9] 14 32 3 93 992 6 [1,2](6)+[4,9](2)+[4,11](3)+[4,12]+[5,7](3)+[5,8]

+[5,10]+[5,11]+[6,7]+[6,8](2)+[6,9]+[6,10]+[6,12] +[7,8]+[7,11]+[8,10](2)+[9,10](2)+[9,11]

15 33 3 16 176 1 [1,2]+[4,12]+[5,9]+[6,7]+[8,10]+ [11,11](1/3) 16 34 3 66 748 4 [1,2](4)+[4,8]+[4,10]+[4,12](2)+[5,7]+[5,9]+[5,10]+

[5,13]+[6,7]+[6,8]+[6,9]+[6,11]+[7,10]+[7,11]+ [8,11]+[8,13]+[9,10]+[9,11]

17 35 3 51 595 3 [1,2](3)+[4,8]+[4,9]+[4,12]+[5,10]+[5,11](2)+[6,8] +[6,9]+[6,12]+[7,10]+[7,11]+[7,13] +[8,13]+[9,10]

18 36 3 35 420 2 [1,2](2)+[4,10](2)+[5,11](2)+ [6,13](2)+[7,8](2)+ [9,9]+[12,12](2/3) 19 37 3 18 222 1 [1,2]+[4,7]+[5,14]+[6,10]+[8,12] +[9,13]



20 38 3 222 2812 12 [1,2](12)+[4,7]+[4,9](2)+[4,10](3)+[4,12](2)+[4,14](4)+ [5,8](2)+[5,10]+[5,11]+[5,12](4)+[5,13](2)+

[5,14](2)+[6,7]+[6,8]+[6,9](2)+[6,10](3)+[6,11](4) +[6,12]+[7,9](2)+[7,10](2)+[7,11](4)+[7,12](2)+ [8,8](2)+[8,12]+[8,13](2)+[8,15](2)+[9,10](2)+ [9,13](2) +[9,14](2)+[10,13]+[11,12](2)

21 39 3 19 247 1 [1,2]+[4,14]+[5,10]+[6,11]+[7,9] +[8,12]+[13,13](1/3)

22 40 3 39 520 2 [1,2](2)+[4,13]+[4,14]+[5,10]+[5,11]+[6,9]+[6,14]+ [7,12](2)+[8,9]+[8,10]+[11,13] 23 41 3 60 820 3 [1,2](3)+[4,11]+[4,14](2)+[5,8]+[5,14]+[5,15]+[6,9]+

[6,10]+[6,16]+[7,10](2)+[7,13]+[8,9]+[8,11]+[9,12]+[11,12]+[12,13]


24 42 3 82 1148 4 [1,2](4)+[4,14]+[4,15](3)+[5,8]+[5,13](3)+[6,11](3) +[6,14]+[7,8]+[7,9](2)+[7,10]+[8,12]+[8,14]+[9,12](2)+[10,10]+[10,16]+[11,12]+[14,14](1/3)

25 43 3 21 301 1 [1,2]+[4,16]+[5,13]+[6,11]+[7,8] +[9,12]+[10,14] 26 44 3 129 1892 6 [1,2](6)+[4,13](2)+[4,14]+[4,15][4,18](2)+[5,12](4)+

[5,14](2)+[6,10](3)+[6,15](3)+ [7,9](3)+[7,14](3)+ [8,10](3)+ [8,11](3)+[9,13]+ [9,15](2)+

[11,13](3)+[12,12] 27 45 3 22 230 1 [1,2]+[4,17]+[5,11]+[6,12]+[7,13]+[8,14]+[9,10]+ [15,15](1/3) 28 46 3 90 1380 4 [1,2](4)+[4,8]+[4,13](2)+[4,17]+[5,15](2)+[5,18](2)+

[6,13](2)+[6,14](2)+[7,9](2)+[7,15](2)+[8,9]+ [8,10](2)+[9,12]+[10,12](2)+[11,14](2)+[11,16](2)

29 47 3 69 1081 3 [1,2](3)+[4,11]+[4,15]+[4,18]+[5,12]+[5,16]+ [5,17]+[6,7]+[6,14]+[6,17]+[7,11]+[7,12]+[8,10]+ [8,12]+[8,13]+[9,11]+[9,14]+[9,16]+[10,14]+[10,16]+[13,15]

30 48 3 47 752 2 [1,2](2)+[4,15]+[4,18]+[5,7]+[5,17]+[6,11]+[6,14] +[7,12]+[8,13]+[8,15]+[9,11]+[9,18]+ [10,13]+[10,14]+[16,16](2/3)

31 49 3 24 392 1 [1,2]+[4,9]+[5,21]+[6,14]+[7,17] +[8,11]+[10,12]+[15,16]

32 50 3 294 4900 12 [1,2](12)+[4,16](2)+[4,18](5)+[4,20](5)+[5,11]+ [5,12](3)+[5,13](3)+[5,14]+5,16](3)+[5,19]+ [6,8](2)+[6,9]+ [6,13](3)+ [6,14](3) +[6,17](3)+[7,8]+[7,10]+[7,11]+ [7,12](3)+ [7,15](3)+[7,18](3)+[8,11](2)+ [8,14](3)+ [8,15]+ [8,16](3)+ [9,11](2)+ [9,12](6)+[9,15](3)+ [10,11](3)+ [10,13](5)+ [10,17](3)+ [11,14](3)+ [13,15]+[15,16](2)+[16,17]

33 51 3 25 425 1 [1,2]+[4,18]+[5,9]+[6,15]+[7,12]+[8,20]+[10,16] +[11,13]+[17,17](1/3)

34 52 3 51 884 2 [1,2](2)+[4,14]+[4,20]+[5,11]+[5,16]+[6,12]+[6,17]+ [7,13]+[7,15]+[8,11]+[8,17]+[9,19]+[9,21]+[10,13]+ [10,15]+[12,14] 35 53 3 78 1378 3 [1,2](3)+[4,13]+[4,17]+[4,19]+[5,15](2)+[5,19]+

[6,10]+[6,16]+[6,18]+[7,15]+[7,18]+[7,19]+[8,14]+[8,16]+[8,18]+[9,11]+[9,12](2)+[10,13](2)+[11,14]+[11,17]+[12,14]

36 54 3 106 1908 4 [1,2](4)+[4,6]+[4,21](2)+[4,23]+[5,14](2)+[5,15](2)+ [6,16](2)+[6,17]+[7,13](2)+[7,19](2)+[8,14](2)+ [8.16](2)+ [9,12](2)+[9,17](2)+[10,15](2)+[10,17]+ [11,12](2) 37 55 3 27 295 1 [1,2]+[4,13]+[5,15]+[6,18]+ [7,23]+[8,14]+[9,12]+

[10,19]+[11,16] 38 56 3 165 3080 6 [1,2](6)+[4,18](3)+[4,23](3)+[6,9](3)+[6,19](3)+

[7,13](3)+[7,14]+[7,20]+[7,23]+[8,9]+[8,14](2)+[8, 17](3)+[9,17](2)+[10,11](2)+[10,12]+[10,14](3)+[11,12](2)+[11,16](2)+[12,18](3)+[13,15](3)+[16,20]

39 57 3 28 532 1 [1,2]+[4,16]+[5,8]+[6,17]+[7,22]+[9,15]+[10,21] +[11,14]+[12,18]+[19,19](1/3)

40 58 3 114 2204 4 [1,2](4)+[4,20](3)+[4,24]+[5,21](4)+[6,16](4)+ [7,11](4)+[8,12]+[8,19](3)+[9,14](4)+ [10,15](3)+ [10,19]+[12,13]+[12,15]+[12,17]+[13,17](3)

41 59 3 87 1711 3 [1,2](3)+[4,19]+[4,20](2)+[5,13]+[5,17]+[5,21]+ [6,8]+[6,21](2)+[7,9]+[7,17]+[7,18]+[8,14]+[8,15]+[9,13]+[9,19]+[10,15]+[10,18]+[10,20]+[11,15]+[11,16]+[11,23]+[12,14]+[12,17]+[12,19]+[13,16]

42 60 3 59 1180 2 [1,2](2)+[4,24]+[4,26]+[5,17]+[5,18]+[6,17]+[6,22]+ [7,11]+[7,12]+[8,13]+[8,21]+[9,15] +[9,16]+[10,16]+ [10,19]+[11,14]+[12,15]+[13,14]+[20,20](2/3)




43 61 3 30 610 1 [1,2]+[4,25]+[5,17]+[6,14]+[7,21]+[8,16]+ [9,18]+ [10,13]+[11,15]+[12,19]

44 62 3 66 7564 12 [1,2](12)+ [4,19](12)+[5,7]+[5,12](5)+[5,24](5)+ [5,28]+[6,7](5)+[6,24](6)+[6,28]+[7,13](6)+[6,14](6)+[8,21](6)+[9,18](6)+[9,22](6)+[10,15](6)+[10,16](4)+[10,18](2)+[11,13]+[11,6]+[11,17](5)+[12,20](4)+[12,25](2)+[14,14](2)+[14,16](2)+[15,21](6)+[17,20](2)+[18,18](2)

45 63 3 31 651 1 [1,2]+[4,22]+[5,18]+[6,28]+[7,20]+[8,9]+[10,14]+ [11,19]+[12,13]+[15,16]+[21,21](1/3)

46 64 3 63 1344 2 [1,2](2)+[4,24]+[4,25]+[5,14](2)+[6,20]+[6,25]+ [7,15]+[7,21]+[8,16]+[8,21]+[9,18](2)+[10,12]+ [10,16]+[11,12]+[11,23]+[13,17]+[13,20] +[15,17]

47 65 3 96 2080 3 [1,2]3+[4,7]+[4,18](2)+[5,10]+[5,26](2)+[6,19](2)+ [6,23]+[7,11]+[7,20]+[8,14]+[8,25]+[8,26]+[9,21](3)+[10,17](2)+[11,13]+[12,16](3)+[13,20](2)+[14,15](2)+[17,24]+[19,23]

48 66 3 130 2860 4 [1,2](4)+[4,17](2)+[4,20](2)+[5,20]+[5,24](2)+[5,27] +[6,25](2)+[6,26](2)+[7,8]+[7,11]+[7,21](2)+[ 8,11]+ [8,15](2)+[9,11]+[9,19](2)+[9,25]+[10,17](2)+ [10,23](2)+[11,19]+[12,14](2)+[12,15]+[12,18]+[13,16](2)+[13,18](2)+[14,16](2)+[22,22]+[22,22](1/3)

49 67 3 33 737 1 [1,2]+[4,23]+[5,21]+[6,9]+[7,25] +[11,17]+[12,18]+ [13,20]+[14,22]

50 68 3 201 4556 6 [1,2](6)+[4,21](3)+[4,23](3)+[5,10]+[5,16](2)+ [5,23](3)+[6,24](3)+[6,26](3)+[7,15](3)+[7,26](3)+ [8,18]+[8,14](3)+[8,27]+[9,16]+[9,18]+[9,19](3)+[9,21]+[10,15](2)+[10,24](3)+[11,16]+[11,18](5)+[12,19](3)+[12,20](3)+[13,16]+[13,17](3)+[13,20](2)+ [14,17](3)

51 69 3 34 782 1 [1,2]+[4,29]+[5,26]+[6,12]+[7,21]+[8,17]+[9,11]+ [10,22]+[13,14]+[15,24]+[16,19]+[23,23](1/3)

52 70 3 138 3220 4 [1,2](4)+[4,20](3)+[4,25]+[5,14]+[5,25]+[5,26]+ [5,30]+[6,21](2)+[6,26](2)+[7,13]+[7,18]+[7,22]+[7,23]+[8,11](3)+[8,29]+[9,22](3)+[9,23]+[10,17](2)+[10,24]+[10,26]+[11,21]+[12,16](4)+[13,17]+[13,21]+[13,23]+[14,15](2)+[14,23]+[15,18](2)+[17,18]+

53 71 3 105 2485 3 [1,2](3)+[4,19]+[4,29](2)+[5,24]+[5,26](2)+ [6,15](2)+[6,22]+[7,10]+[7,27](2)+[8,18]+[8,24]+[8,27]+[9,16](2)+[9,25]+[10,13](2)+[11,13]+[11,17](2)+[12,18](2)+[12,20]+[14,19]+[14,22](2)+[15,16]+[19,20]+[20,21]

54 72 3 71 1704 2 [1,2](2)+[4,18]+[4,30]+[5,14]+[5,26]+[6,21]+[6,26]+ [7,16]+[7,27]+[8,12]+[8,28]+[9,21]+[9,23]+[10,19]+[10,25]+[11,20]+[11,22]+[12,13]+[13,15]+[14,15]+[16,17]+[17,18]+ [24,24](2/3)

55 73 3 35 876 1 [1,2]+[4,17]+[5,27]+[6,18]+[7,26]+[8,15]+[9,30]+ [10,19]+[11,20]+[12,16]+[13,25]+ [14,22]

56 74 3 438 10804 12 [1,2](12)+[4,31](12)+[5,19](10)+[5,20](2)+ [6,15]+ [6,21](11)+[7,25](2)+[7,26](10)+[8,28](12)+[9,9]+

[9,14]+[9,15](2)+[9,18]+[9,25](6)+[10,15](2)+ [10,20](10)+[11,12](11)+[11,15]+[12,14]+[13,16] (12)+[14,18](10)+[15,22](6)+[17,17](6)+[19,22](2)+[22,22](2)

57 75 3 37 925 1 [1,2]+[4,31]+[5,19]+[6,28]+[7,13]+[8,22]+[9,27]+ [10,23]+[11,18]+[12,14]+[15,17]+ [16,21]+ [25,25](1/3)

58 76 3 75 1900 2 [1,2](2)+[4,26](2)+[5,24]+[5,29]+[6,8]+[6,25]+[7,8]+ [7,24]+[9,18]+[9,23]+[10,22]+[10,27]+[11,17]+[11,2


2]+ [12,16]+[12,21]+[13,21]+ [13,23]+[14,25]+ [15,20]+ [16,19]+[17,19]+[18,20]

59 77 3 114 2926 3 [1,2](3)+[4,21]+[4,35](2)+[5,27]+[5,31](2)+ [6,18](2)+[6,21]+[7,15]+[7,30](2)+[8,20]+[8,26](2)+[9,14](2)+[9,19]+[10,16]+[10,22](2)+[11,16]+[11,19]+11,28+[12,17](2)+[12,24]+[13,20](2)+[13,21]+ [14,17]+[15,18]+[15,25]+ [16,19]+[23,25]



60 78 3 154 4004 4 [1,2](2)+[4,6](2)+[4,3](2)+[5,28](2)+[5,30](2)+ [6,18](2)+[7,13](2)+[7,16](2)+[8,24]+[8,28](2)+ [8,30]+[9,22](2)+[9,25](2)+[10,21](2)+[11,19]+ [11,23](2)+[11,27]+12,17](2)+[12,25](2)+[13,14]+ [13,27]+[14,15](2)+[14,24]+[15,17](2)+[16,19](2)+ [18,21](2)+[19,27]+[20,22](2)+[26,26]+[26,26](1/3)

61 79 3 39 1027 1 [1,2]+[4,20]+[5,21]+[6,35]+[7,27]+[8,22]+[9,25]+ [10,32]+[11,17]+[12,19]+[13,14]+ [15,18]+[16,23]

62 80 3 237 6320 6 [1,2](6)+[4,18]+[4,23]+[4,27](2)+[4,28](2)+[5,20]+ [5,30](3)+[5,32](2)+[6,15](3)+[6,27](3)+[7,18](3)+ [7,23]+[7,24]+[7,25]+[8,18](2)+[8,28]+[8,29](3)+ [9,10](3)+[9,29](3)+[10,13]+[10,20](2)+[11,23](3)+ [11,24](2)+[11,26]+[12,13]+[12,24](2)+[12,28](3)+ [13,21](3)+[13,32]+[14,17](3)+[14,19](3)+[15,24]+ [15,26](2)+[16,20](3)+[16,22](2)+[16,26]+ [17,22](3)

63 81 3 40 1080 1 [1,2]+[4,35]+[5,25]+[6,8]+[7,17]+[11,33]+[12,26]+ [13,15]+[16,20]+[18,29]+[19,22]+ [27,27](1/3)

64 82 3 162 4428 4 [1,2](4)+[4,12]+[4,29]+[4,32](2)+[5,11](2)+ [5,30](2)+[6,13](2)+[6,27](2)+[7,25](2)+[7,31](2)+ [8,21]+[8,26](2)+[8,29]+[9,19](2)+[9,22](2)+[10,20] (2)+[10,24](2)+[11,17](2)+ [12,21]+[12,26](2)+ [13,23](2)+[4,23](2)+[14,27](2)+[15,20](2)+[15,25](2)+[16,29]+[17,22](2)+[18,21](2)+ [18,24](2)

65 83 3 123 3403 3 [1,2](3)+[4,17](2)+[4,36]+[5,20]+[5,26](2)+[6,29]+ [6,33](2)+[7,17]+[7,23](2)+[8,29]+[8,32](2)+[9,18]+ [9,19]+[9,22]+[10,23]+[10,24](2)+[11,15]+

[11,27](2)+[12,18]+[12,25](2)+[13,21]+[13,22](2)+ [14,18]+[14,28](2)+[15,16]+[15,29]+[16,20](2)+ [19,19]

66 84 3 83 2324 2 [1,2](2)+[4,21]+[4,32]+[5,15]+[5,35]+[6,31]+[6,33]+ [7,10]+[7,33]+[8,19]+[8,23]+[9,20]+[9,25]+[10,24]+ [11,19]+[11,30]+[12,26]+[12,27]+[13,16]+[13,23]+ [14,18]+[14,24]+[15,22]+[16,26]+[17,18]+[21,22]+ [28,28](2/3)

67 85 3 42 1190 1 [1,2]+[4,32]+[5,29]+[6,20]+[7,28]+[8,17]+[9,33]+ [10,14]+[11,27]+[12,18]+[13,31]+[15,22]+[16,23]+ [19,21]

68 86 3 510 14620 12 [1,2](12)+[4,21](6)+[4,37](6)+[5,31](6)+[5,32](6)+ [6,33](12)+[7,17](6)+[7,28](6)+[8,18](5)+[8,20]+ [8,26](6)+[9,13](2)+[9,14](3)+[9,18]+[9,21](6)+ [10,10](2)+[10,20]+[10,24](6)+[10,26]+[11,17](6)+ [11,18](6)+[12,20](6)+[12,31](6)+[12,23](3)+ [13,30]+[13,29](6)+[14,27](5)+[14,30](2)+[14,31]+ [14,36]+ [15,23](6)+[15,25](6)+ [16,19](6)+ [16,22](6)+ [19,27](6)+[20,22](3)

69 87 3 43 1247 1 [1,2]+[4,37]+[5,8]+[6,25]+[7,27]+[9,33]+[10,11]+ [12,16]+[14,22]+[15,24]+[17,23]+18,26]+[19,30]+ [20,32]+[29,29](1/3)

70 88 3 87 2552 2 [1,2](2)+[4,26]+[4,38]+[5,10]+[5,33]+[6,21]+[6,22]+ [7,30]+[7,35]+[8,19]+[8,311]+[9,32]+[9,35]+[10,24]


+[11,21]+[11,25]+[12,24]+[12,28]+[13,18]+[13,20]+[14,25]+[14,29]+[15,26]+[16,18]+[16,29]+[17,23]+ [17,20]+ [19,22]

71 89 3 132 3916 3 [1,2](3)+[4,15]+[4,30](2)+[5,17]+[5,35]+[5,19]+ [6,27](2)+[6,29]+[7,25]+[7,32]+[7,35]+[8,14]+[8,31](2)+[9,11](2)+[9,22]+[10,28](2)+[10,30]+[11,22]+ [12,16]+[12,25]+[12,32]+[123,24](2)+[13,27]+ [14,29](2)+[15,21](2)+[16,20]+[16,26]+[17,21]+ [17,25]+[18,23](2)+[18,26]+[19,26]+[23,23]

72 90 3 178 5340 4 [1,2](4)+[4,24](3)+[4,42]+[5,18]+[5,32](2)+[5,38]+ [6,10]+[6,15]+[6,27]+[6,39]+[7,17]+[7,35]+[7,36]+[7,41]+[8,21](2)+[8,33](2)+[9,31](2)+[9,35]+[9,36]+[10,17]+[10,19](2)+[11,20]+[11,26](2)+[11,39]+[12,22](4)+[13,14]+[13,25](2)+[13,27]+[14,25](2)+[14,28]+[15,17](2)+[15,21]+[16,19](2)+[16,33]+[18,20]+[18,26](2)+[20,23](2)+[23,31]+[30,30]+(1/3)

73 91 3 45 1365 1 [1,2]+[4,27]+[5,42]+[6,33]+[7,17]+[8,28]+[9,32]+ [10,16]+[11,29]+[12,22]+[13,15]+[14,21]+[15,30]+ [18,19]+[20,23]

Table 14: Balanced Incomplete Block Designs for 19 ≤ v ≤ 100 and k=3 and for 19 ≤ v ≤ 25 with k=4, 5 restricted with r ≤ 60.


74 92 3 273 8372 6 [1,2](6)+[4,18]+[4,20]+[4,24](2)+[4,39](2)+[5,16](2) +[5,27]+[5,38](3)+[6,32]+[6,35](2)+[6,40](3)+[7,20](2)+[7,21]+[7,33](3)+[8,29](3)+[8,34](3)+[9,21](3)+ [9,30](3)+[10,22](2)+[10,34](3)+[10,39]+[11,17]+ [11,18](2)+[11,25](3)+[12,19](3)+[12,23]+[12,26] (2)+[13,16]+[13,22](3)+[13,23](2)+[14,23](3)+ [14,31](30)+[15,17](2)+[15,26](4)+[16,17](3)+ [18,24](3)+[19,25](3)+[20,27](3)+[28,28]

75 93 3 46 1426 5 [1,2]+[4,37]+[5,22]+[6,26]+[7,35]+[8,39]+[9,10]+ [11,25]+[12,28]+[13,21]+[14,29]+[15,23]+[16,17]+[18,30]+[20,24]+[31,31](1/3)

76 94 3 186 5828 4 [1,2](4)+[4,21]+[4,24]+[4,30](2)+[5,30](2)+[5,39](2) +[6,35](2)+6,37](2)+[7,26]+[7,32]+[7,36](2)+[8,10] (2)+[8,20](2)+[9,29](2)+[9,33]+[9,39]+[10,11](2)+ [11,20](2)+[12,19](2)+[12,34](2)+[13,19](2)+ [13,23](2)+[14,15]+[14,26]+[14,33](2)+[15,26](2)+ [15,29]+[16,22](2)+[16,24](2)+[17,25](2)+[17,27]+ [17,33]+[18,27](2)+[21,27]+[22,23](2)+[24,28]+ [25,32]

77 95 3 141 4465 3 [1,2](3)+[4,17+[4,35]+[4,41+[5,28]+[5,31]+[5,34+ [6,21](2)+[6,31]+[7,20]+[7,37](2)+[8,23]+[8,40](2)+[9,17]+[9,36](2)+[10,23](2)+[10,34]+[11,19](3)+[12,22]+[12,29](2)+[13,22](2)+[13,29]+[14,24]+[14,25]+[14,32]+[15,28](2)+[15,32]+[16,24]+[16,26](2)+[17,32]+[18,20](2)+[18,25]+[24,25]

78 96 3 95 3040 2 [1,2](2)+[4,19]+[4,24]+[5,6]+[5,31]+[6,38]+[7,22]+ [7,43]+[8,29]+[8,33]+[9,38]+[9,39]+[10,23]+[10,31]+[11,25]+[12,18]+[12,27]+[13,21]+[13,34]+[14,26] (2)+[15,20]+[15,30]+[16,28]+[16,35]+[17,20]+ [17,25]+[18,24]+[19,27]+[21,22]+[32,32](2/3)

79 97 3 48 1552 1 [1,2]+[4,17]+[5,26]+[6,38]+[7,12]+[8,37]+[9,25]+ [10,32]+[11,29]+[13,22]+[14,33]+15,28]+[16,23]+[18,30]+[20,36]+[24,27]

80 98 3 582 19012 12 [1,2](12)+[4,26](6)+[4,46](6)+[5,36](5)+[5,37](6)+ [5,40]+[6,23](6)+[6,34](6)+[7,25](5)+[7,28](6)+ [7,32]+[8,38](6)+[8,39](6)+[9,33](6)+[9,35](6)+ [10,20](5)+[10,31](6)+[10,38]+[11,11](2)+[11,15]+ [11,18](6)+[11,25]+[12,21](6)+[12,24]+[12,38](5)+


[13,15](6)+[13,17]+[13,21](5)+[14,20]+[14,22](5)+ [14,23](6)+[15,20](5)+[16,27](6)+[16,31](6)+ [17,22](5)+[17,32](6)+[18,27](6)+[19,24](6)+ [19,25]+[19,26](5)+[20,21]+ [20,24](5)

81 99 3 49 1693 1 [1,2]+[2,27]+[5,42]+[6,5]+[7,30]+[8,32]+[9,20]+ [10,16]+[11,38]+[12,13]+[14,22]+[15,39]+[17,34]+ [18,28]+[19,24]+[21,23]+[33,33](1/3)

82 100 3 99 3300 1 [1,2]+[4,22]+[4,33]+[5,22]+[5,35]+[6,23]+[6,24]+ [7,13]+[7,35]+[8,24]+[8,33]+[9,38]+[9,43]+[10,31]+ [10,39]+[11,18]+[11,34]+[12,34]+[12,38]+[13,31]+ [14,23]+[14,28]+[15,25]+[15,36]+[16,27]+[16,28]+ [17,19]+[17,30]+[18,21]+[19,26]+[20,32]+[21,25]

83 19 4 12 57 2 [1,2,5}+[1,3,5]+[2,4,6] 84 19 5 45 171 10 [1,1,3,5]+[1,1,3,9]+[1,2,2,8]+[1,2,3,6]+[1,2,4,4]+

[1,2,4,5]+[1,2,6,6]+[2,3,4,4] 85 20 4 19 95 3 [1,1,6]+[1,2,4]+[3,4,4]+[8,9,9]+ [5,5,5](2/3) 86 20 5 19 76 4 [1,2,5,9,]+[1,6,9,9]+[2,2,4,8]+ [1,6,4]t 87 21 5 5 21 1 [1,5,2,10] 88 25 5 6 30 1 [2,1,4,9]+{[6,6,6](1/2)}t

These series involve a Galois field of p elements, where p is a prime

and α is some positive integer. It should be noted that the designs developed in Table 1 cannot be constructed through Bose’s method of differences. In contrast to Bose’s method, the suggested method of cyclic shifts is rather a simpler and general for the construction of BIBDs. The purpose of constructing BIBDs in Table 1 is to show that the proposed method could, despite of its apparent simplicity, be capable of finding a set of known designs that have previously been obtained through a number of different existing methods. Of course, we did not expect to find any, as yet, undiscovered BIBDs. The distinguishing feature of our proposed method is that it can be used to search other types of designs, not only the BIBDs.

References Bose, RC. (1939) “On the construction of balanced incomplete block design”, Ann. Eugen. 9, 353-399.

Bose, RC. (1942) “On some new series of balanced incomplete block design”, Bull. Calcutta. Math. Soc. 34, 17-31.

Cochran, WG. and Cox, GM. (1953) “Experimental Designs”, Second Edition, John Wiley, New York.

Davies, OL. (1956) “Design and Analysis of Industrial Experiments”, Second Edition, Hafner Publishing Company, New York.

Fisher, RA. and Yates, F. (1953) “Statistical Tables for Biological, Agriculture, and Medical Research”, Fourth Edition, Oliver and Boyd, Edinburg.


Iqbal, I. (1991) “Construction of Experimental Designs Using Cyclic Shifts”, PhD thesis, University of Kent at Canterbury, UK.

John, JA. (1987) “Cyclic Designs”, Chapman and Hall, London.

Mathon, RA. and Rosa, A. (1985) “Tables of parameters of BIBDs with r ≤ 41”, Ann. Disc. Math. 26, 275-308.

Preece, DA. (1967) “Nested balanced incomplete block designs”, Biometrika 54, 279-286.

Raghavarao, D. (1971) “Construction and Combinatorial Problems in Designs of Experiments”, John Wiley and Sons, New York.

Rao, VA. (1958) “A note on balanced designs”, Ann. Math, Statist. 29, 290-294.

Shah, SK. and Sinha, BK. (1989) “Theory of Optimal Designs”, Springer-Verlag, New York.

Sprott, DA. (1954) “A note on balanced incomplete block designs”, Canad. J. Math. 6, 341-345.

Sprott, DA. (1956) “Some series of balanced incomplete block designs”, Sankhya 17, 185-192.

Takeuchi, K. (1962) “A table of differences sets generating balanced incomplete block designs”, Rev. Inst. Int. Statist. 30, 361-366

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