Packing Arrays and Packing Designs

Designs, Codes and Cryptography, 27, 165–176, 2002©C 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Packing Arrays and Packing Designs

BRETT STEVENS [email protected] of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6

ERIC MENDELSOHN [email protected] of Mathematics, University of Toronto, 100 St. George St., Toronto, ON M6G 3G3

Communicated by: C. J. Colbourn, D. R. Stinson, G. H. J. van Rees

Received May 16, 2001; Accepted July 16, 2001

Abstract. A packing array is a b × k array, A with entries ai, j from a g-ary alphabet such that given any twocolumns, i and j , and for all ordered pairs of elements from a g-ary alphabet, (g1, g2), there is at most one row,r , such that ar,i = g1 and ar, j = g2. Further, there is a set of at least n rows that pairwise differ in each column:they are disjoint. A central question is to determine, for given g, k and n, the maximum possible b. We examinethe implications when n is close to g. We give a brief analysis of the case n = g and show that 2g rows is alwaysachievable whenever more than g exist. We give an upper bound derived from design packing numbers whenn = g − 1. When g + 1 ≤ k then this bound is always at least as good as the modified Plotkin bound of [12].When the associated packing has as many points as blocks and has reasonably uniform replication numbers, weshow that this bound is tight. In particular, finite geometries imply the existence of a family of optimal or nearoptimal packing arrays. When no projective plane exists we present similarly strong results. This article completelydetermines the packing numbers, D(v, k, 1), when v <

k(k−1)2 .

Keywords: packing array, orthogonal array, packing design, Plotkin bound

AMS Subject Classification: 05B15, 05B40

1. Introduction

It is known that g2 ×k orthogonal arrays on a g-ary alphabet never exist for k > g + 1. For ga non-prime power, orthogonal arrays are only known to have k two larger than MOLS(g).For their applications, it is natural to ask for structures that have similarly useful propertiesas orthogonal arrays but have k larger than these bounds. One generalization is to requirethat all pairwise interactions be covered at least once. These objects are known as coveringarrays or transversal covers and have been extensively studied, see [9,11,13–15] and theirreferences. The other natural generalization of orthogonal array is the packing array.

Definition 1. A packing array (PA(k, g : n)) is a b × k array, A with entries ai, j froma g-ary alphabet such that given any two columns, i and j , and for all ordered pairs ofelements from the g-ary alphabet, (g1, g2), there is at most one row, r , such that ar,i = g1

and ar, j = g2. Further, there is a set of at least n rows that pairwise differ in each column:they are disjoint. The largest number of rows possible in a PA(k, g : n) is denoted bypa(k, g : n).

166 STEVENS AND MENDELSOHN

0 0 0 0 0 0 01 1 1 1 1 1 12 2 2 2 2 2 23 3 3 3 3 3 34 4 4 0 1 2 34 0 1 4 4 3 24 1 0 2 3 4 40 4 3 4 2 4 11 4 2 3 4 0 42 3 4 4 0 1 43 2 4 1 4 4 0

Figure 1. An example of a PA(7, 5 : 4).

Row and column permutations, as well as permuting symbols within each column, leavethe packing conditions intact.

EXAMPLE 1. A packing array with k = 7, g = 5 and n = 4 with eleven rows is givenin Figure 1. Since we can permute rows and symbols within each column we can assumethat the first four rows are the set of disjoint rows and that they use the first four symbols,{0, 1, 2, 3}, one symbol per row. Theorem 4 will establish that this packing arrays is optimal.

We will often and without loss of generality use Zg as the symbol set on each columnand let ri, j equal the number of times symbol i ∈ Zg appears in column j .

Packing arrays are closely related to transversal designs and we include a definition heresince we refer to these in this article.

Definition 2. [4] A transversal design of order or groupsize n, blocksize k, denotedTD(k, n) is a triple (V,G,B), where

1. V is a set of kn elements;

2. G is a partition of V into k classes (the groups), each of size n;

3. B is a collections of k-subsets of V (the blocks);

4. every unordered pair of elements from V is either contained in exactly one group or iscontained in exactly one block, but not both.

If the blocks of a transversal design can be partitioned into resolution classes such that eachresolution class contains each member of V exactly once then the design is called resolvableand we write RTD(k, g).

Packing arrays are also called transversal packings [10] and mutually orthogonal par-tial latin squares [1,2]. Packing array upper bounds and constructions were investigatedpreviously by the authors [12]. Observing that any two rows of the packing array musthave Hamming distance at least k − 1 we see that packing arrays are also error correctingcodes, specifically, Maximal Distance Separating (MDS) codes or Partial MDS codes. This

PACKING ARRAYS AND PACKING DESIGNS 167

motivates the generalization of the Plotkin bound to these objects:

THEOREM 1 [12]. A b × k packing array, PA(k, g : n), must satisfy the following bound forall β ≤ b = pa(k, g : n), β = ug + v where 0 ≤ v < g:

k((g − v)u2 + v(u + 1)2) ≤ β2 − β − n2 + n + kβ. (1)

Abdel-Ghaffar and Abbadi [2] use MDS codes to allocate large database files to multiplehard disk systems so that the retrieval time is optimal. They establish bounds on the size ofpacking arrays, the smallest form of MDS codes, to yield bounds on the efficiency of theirsystem. The MDS or Partial MDS codes that correspond to packing arrays would apply tooptimal disk allocation over a large number of disks, specifically at least gk−2, where, in theirterminology, g is the number of sets into which each attribute of the database is divided. Ifpacking arrays were used for optimal disk allocation in their model, any search with at leasttwo specified attributes would yield search time of one. This is the best possible search time.

Abdel-Ghaffar and Abbadi [2] showed that

pa(k, g : 1) ≥ g + 1 ⇒ k ≤ g2 + g

2, (2)

Abdel-Ghaffar showed that the bound from Theorem 1 is tight when k ≥ 2g − 1 and n = 1[1]. This result was extended by the two authors to all n and a range of parameters withk < 2g − 1 [12].

In Section 2, we examine the upper bounds on the size of packing arrays when the sizeof the set of disjoint rows is either g or g − 1. In the later case we derive a bound in termsof the packing design numbers. We determine the range where this bound is stronger thanthat given in Theorem 1. In Section 3, we discuss conditions which allow us to constructpacking arrays that are tight for this bound. This involves suitably edge colouring the pointset incidence graph of the complementary incidence structure of the packing design inquestion. In Section 4 we examine more closely the connection between packing arraysand the existence of special designs: finite geometries. We show that when a projectiveplane does not exist, the corresponding packing array cannot even have 2g rows, a powerfulnon-existence result.

2. Upper Bounds for Large Sets of Disjoint Rows

In a previous paper the authors presented a number of upper bounds and constructionsyielding lower bounds [12]. One of the upper bounds developed is stated in Theorem 1.This bound is an adaptation of the Plotkin bound from coding theory. The other upper boundarises from consideration of the restrictions necessary if the array is to have more than g rows,and is derived by starting with the set of disjoint rows and extending the array row by row.

THEOREM 2 [12]. The maximum number of columns in a packing arrays with at least g + 1rows is

g(g + 1)

2− n(n − 1)

2. (3)


0 0 · · · 0 01 1 · · · 1 1...

.

.

....

.

.

.

g − 2 g − 2 · · · g − 2 g − 2g − 1 g − 1 · · · g − 1 g − 1

0 1 · · · g − 2 g − 1g − 1 0 · · · g − 3 g − 2

.

.

....

.

.

....

2 3 · · · 0 11 2 · · · g − 1 0

Figure 2. A PA(g, g : g) with 2g rows.

When the packing array has a set of g disjoint rows, Theorem 2 states that if k ≥ g + 1then pa(k, g : g) = g. If g is a prime power, then for k ≤ g, pa(k, g : g) = g2. For any g,the packing array shown in Figure 2 is a PA(g, g : g) with 2g rows. This PA(g, g : g) isjust the dual of an RTD(2, g). In general we get

THEOREM 3. The existence of a RTD(κ, g) implies that pa(g, g : g) ≥ κg. In other words

pa(g, g : g) ≥ (MOLS(g) + 1)g.

When there is a set of g − 1 disjoint rows, the packing array is still considerably con-strained. We can assume that the first g − 1 rows are the set of disjoint rows and these usethe symbols {0, 1, . . . , g −2}, one symbol per row. By the packing conditions, pairs of rowsfrom a packing array can never agree in more than one coordinate position. Thus everysubsequent row can have each of these g − 1 symbols, {0, 1, . . . , g − 2}, at most once. Theremaining symbol, g − 1, must therefore appear at least k − g + 1 times in each row.

We consider the array derived from this packing array in the following manner: Delete thefirst n = g−1 rows (the set of disjoint rows). In the remaining rows, replace all occurrencesof the symbol g −1 with the symbol 1 and all other symbols by 0. This is an (b − g +1)× karray of zeros and ones with at least k −g +1 ones in every row. The collection of subsets ofa k set determined by considering each row as an incidence vector, has the property that notwo of them intersect in more than one point. That is, these subsets from a collection whereno pair of elements from the k set appears in more than one of the subsets. This collectionof sets is called a packing design and has been extensively studied [4,5]. The array of zerosand ones is its incidence matrix. We want to establish bounds on the maximum number ofrows such an array can have, or equivalently, the maximum number of blocks that such apacking design can have. The maximum size of a packing design is denoted D(v, κ, 2),where v is the cardinality of the point set and κ is the block size. From our modification ofthe packing array we get

THEOREM 4. For g + 1 ≤ k,

pa(k, g : g − 1) ≤ g − 1 + D(k, k − g + 1, 2)


For k > 2g − 1, D(k, k − g + 1, 2) ≤ 1 so the interesting range for this bound isg + 1 ≤ k ≤ 2g − 1. The following results about D(v, κ, 2) are known:

THEOREM 5 [5].

D(v, κ, 2) ≤⌊

v

κ

⌊v − 1

κ − 1

⌋⌋

THEOREM 6 [5].

D(v, 3, 2) =⌊

v

3

⌊v − 1

2

⌋⌋− ε

where ε = 1 if v ≡ 5 mod 6 and 0 otherwise.

D(v, 4, 2) =⌊

v

4

⌊v − 1

3

⌋⌋− ε

where ε = 1 if v ≡ 7, 10 mod 12 and v �= 10, 19, or if v = 9, 17, ε = 2 if v = 8, 10, 11,ε = 3 if v = 19 and 0 otherwise.

When k ≥ 5 the known values for D(v, 5, 2) and D(v, 6, 2) that are not covered byLemma 1 are shown in Figure 3 [4,5].

LEMMA 1. For b ≤ κ a 2 − (v, κ, 1) packing design that has at least b blocks must have atleast

v ≥ bκ − b(b − 1)

2

points. A packing on v = bκ − b(b−1)

2 is always achievable. Additionally, D(κ(κ + 1)/

2, κ, 2) ≥ κ + 1.

Proof. We shall prove the result by induction. The base case, that v ≥ 1κ − 1(0)

2 = κ

points are necessary to contain a block of size κ is obvious. To show the result is true forb, we assume that there is a packing design on bκ − b(b−1)

2 − 1 points with b blocks ofsize κ . If we remove one block and all the points that are only on this block and not on anyothers then we will get a 2 − (v, κ, 1) packing design with b − 1 blocks of size κ and v ≤bκ − b(b−1)

2 −1−(κ − b + 1). But bκ − b(b−1)

2 −1−(κ − b + 1) = (b−1)κ − (b−1)(b−2)

2 −1and the inductive hypothesis forbids this packing design.

To see that this bound is tight we also proceed by induction. The case for b = 1 is obvious.By the inductive hypothesis we may assume that we have D((b − 1)κ − (b−1)(b−2)

2 , κ, 2) =b−1. To this packing design we add an additional κ −b+1 points and a block on the unionof these new points and a point on each block chosen to have replication number one (such

v 15 16 17 18 19 20D(v, 5, 2) 6 6 7 9 12 16D(v, 6, 2) 3 3 3 4 4 5

Figure 3. Known values for D(v, 5, 2) and D(v, 6, 2).


a point must exist because b ≤ κ). This yields a 2 − (bκ − b(b−1)

2 , κ, 1) packing designwith b blocks. The dual of complete graph Kκ+1 gives a packing on κ(κ + 1)/2 points withκ + 1 blocks of size κ .

COROLLARY 1. For

k ≥ g +⌈√

1 + 8g − 1

2

⌉,

the bound from Theorem 4 gives

pa(k, g : g − 1) ≤ g − 1 + 3 + 2k − 2g −√

(3 + 2k − 2g)2 − 8k

2(4)

This value is always less than 2g. Substituting b = g + v into Theorem 1 and solvingfor v we get that

COROLLARY 2. When b < 2g then

b ≤ k + 1 −√

(2k − 2g + 1)2 − 4(g + n − 1)(g − n)

2

For n = g − 1 this is identical to equation (4), so we can conclude that for

k ≥ g + �√

1+8g−12 �, Theorems 1 and 4 give an identical upper bound. It will be shown in

Section 3 that this upper-bound is tight.The only range we need consider now is

g + 1 ≤ k < g +⌈√

1 + 8g − 1

2

⌉.

To show that the bound from Theorem 4 is stronger than Theorem 1 we establish somelemmata:

LEMMA 2. The difference between the bound of Theorem 1 and

b2

(k

g− 1

)− b(k − 1) + n(n − 1) ≤ 0

is less than g when n = g − 1.

Proof. The value of

k((g − v)u2 + v(u + 1)2) − b2 + b + n2 − n − kb (5)

from Theorem 1 is always at least

b2

(k

g− 1

)− b(k − 1) + n(n − 1) (6)

and never more than

b2

(k

g− 1

)− b(k − 1) + n(n − 1) + kg

4. (7)


Therefore, as long as expression (5) is not positive on integral values of b less than the smallroot of expression (7), we can bound the difference between Theorem 1 and

b2

(k

g− 1

)− b(k − 1) + n(n − 1) ≤ 0 (8)

by the difference of the two larger roots of expressions (6) and (7).Expression (5) for g ≤ b ≤ 2g and n = g − 1 is

−b2 + b(1 + 2k) + (g − 1)(g − 2) − 2kg.

It is easy to verify that this achieves its maximum at b = k + 1/2 and that the roots differby no more than

√1 + 4k + 4k2 + 4(g − 1)(g − 2) − 8gk,

which for

k ≤ g − 1 +⌈√

8g + 1 − 1

2

⌉

is always less than or equal to one. This establishes that there is no positive integral value ofexpression (5) between g and 2g. Until its large root expression (7) is negative for b ≥ 2g,when g > 2, thus validating the following bound on the difference between Theorem 1 andthe bound obtained from equation (8).

The difference of the roots of expressions (6) and (7) is

1

2(k − g)

(√g(9g + 10kg + gk2 − 4g2k − 8k + 4g3 − 12g2)

−√

g(9g + 10kg − 3g2k + 4g3 − 12g2 − 8k))

< x (9)

for some x . Both these equations have real roots when

10kg + 9g − 3g2k + 4g3 − 12g2 − 8k ≥ 0

which is satisfied for g ≥ 12.We will establish that g is an admissible value for x . Equation (9) can be algebraically

manipulated and rewritten as

x2 − 1/16g4k2/((k − g)2) − 1/2(8g4 − 24g3 + g2k2 − 7kg3 − 16kg

+ 20g2k + 18g2)x/((k − g)2) < 0

We show that when g is substituted for x it satisfies this equation and is thus a valid boundfor equation (9). Substituting g for x gives

−1/16g2

(k − g)2(−208g2 + 192kg − 16k2 + 64g3 + 8gk2

− 56g2k − 128k + 144g + g2k2) < 0 (10)


The part of the numerator in parentheses can be rewritten as

g2(64g − 48k) + 8gk(k − g) + 4k(5g − 4k) + 172g(k − g)

+ g2(k2 − 36) + 16(9g − 8k)

When g > 127 and g + 1 ≤ k < g + �√

1+8g−12 �, this is positive, assuring that the

equation (10) is less than zero. For g ≤ 127 the validity of this lemma can be establishednumerically.

LEMMA 3. For g ≥ 32 the difference between

b2

(k

g− 1

)− b(k − 1) + n(n − 1) ≤ 0

and the bound from Theorem 4 is at least g + 1.

Proof. When we substitute

b = 2g + k(k − 1)

(k − g + 1)(k − g),

which is at least g + 1 bigger than the bound from Theorem 4, into

b2

(k

g− 1

)− b(k − 1) + n(n − 1) (11)

with n = g − 1, we get

−1

g(k − g + 1)2(3g5 − 5g4 − 8kg4 + 7g3k2 + 8kg3 − g3 + 5g2g2k2

− 2g2k3 + kg − 4gk2 − 2g − 3k3 + 2k2 − k − k3). (12)

The numerator of this reduces to

−((k − g)2g(3g2 − 2kg − 5g − 3k) + k(1/6g3 − k2) + 4kg(1/6g2 − k)

+ g3(1/6k − 1) + (5g2 − 2g) + k(g − 1) + 2k2).

When g ≥ 32 and

g + 1 ≤ k < g +⌈√

1 + 8g − 1

2

⌉

Then the following inequalities are true:

2k ≤ 5/2g

5g ≤ 1/4g2

3k < 1/4g2

k < 1/6g2

k2 < 1/6g2,


establishing that the numerator, and thus equation (12) is negative and satisfies Inequality 11.This gives the desired result.

THEOREM 7. The bound from Theorem 4 is always at least as strong as the bound fromTheorem 1.

Proof. For g ≥ 32 we establish this result by using Lemmata 2 and 3. For all remaining

g and g + 1 ≤ k ≤ g − 1 + �√

1+8g−12 �, the result is established numerically. It is clear

from Lemmata 2 and 3, that the bound from Theorem 4 will be strictly less than the boundfrom Theorem 1 in all but a finite number of cases. Calculation determines that equalityonly occurs for k = 5, g = 4, n = 3 and k = 7, g = 5, n = 4.

3. Construction of Packing Arrays from Packing Designs

We do not know, in general, how to construct packing arrays which meet the bound fromTheorem 4 because we have only forbidden the rows from agreeing more than once onthe last symbol. Replacing the 0’s with symbols from the set of the first g − 1 symbolsand maintaining the packing conditions cannot always be done. The simplest exampleis PA(4, 3, 2). The bound gives us pa(4, 3, 2) ≤ 8, but we can only replace the 0’s in theincidence matrix of the affine plane on four points in four of the rows corresponding with onlyfour of the blocks, not all six. This gives pa(4, 3 : 2) ≥ 6. Theorem 1 gives pa(4, 3 : 2) ≤ 7.Exhaustive search shows that seven rows are not possible and the construction from packingdesigns does give the true value, pa(4, 3, 2) = 6.

We can determine some sufficient conditions to construct a packing array from a packingdesign.

THEOREM 8. If a packing design has rx ≥ b − v + k for all points x, then

pa(v, v − k + 1 : v − k) ≥ b. (13)

Proof. The first v − k rows of the packing array are the v − k constant rows with symbols0, 1, . . . , v − k − 1. The remaining b rows have the symbol v − k wherever a 1 appears inthe incidence matrix of the packing design. Now each row is empty in v − k cells and eachcolumn empty in b − rx ≤ v − k cells. Form the bipartite graph with the rows and columnsas the bipartitions and an edge wherever a cell is empty. The maximum degree is v − k andso we can edge colour this graph with colours 0, 1, . . . , v − k − 1. We put symbol k intorow g − 1 + i and column j , when the edge between vertices i and j in the two partitionsof the graph is coloured k.

The packings constructed in Lemma 1 all have rx = 1 or 2 and so satisfy Theorem 8,giving

COROLLARY 3. For k ≥ g + �√

8g+1−12 �, the bounds from Theorem 1 and Theorem 4 are

equal and packing arrays attaining this bound are constructible.


4. Packing Arrays and Finite Geometries

Some known designs and packings satisfy the conditions of Theorem 8.

COROLLARY 4. If there is a projective plane of order p, then

pa(p2 + p + 1, p2 + 1 : p2) = 2p2 + p + 1. (14)

COROLLARY 5. If there is a RBIBD(v, κ, 1), then

2v − κ ≤ pa(v, v − κ + 1 : v − κ) ≤ v − κ + v(v − 1)

κ(κ − 1).

In particular

2ν2 − ν ≤ pa(ν2, ν2 − ν + 1 : ν2 − ν) ≤ 2ν2

when an affine plane of order ν exists.

Proof. Remove all but κ of the resolution classes to produce a packing that satisfiesTheorem 8.

It is known that when ν = 6, 10 or when ν ≡ 1 or 2 mod 4 and there is an odd primeq ≡ 3 mod 4 dividing the square-free part of ν then there is no projective nor affine planeof order ν [3]. This still permits the possibility of infinitely many projective planes of nonprime power order, but it also forbids an infinite number of them. In the case where noprojective plane of order ν exists, recent results on the order of packings with the sameparameters as projective planes let us establish some results about packing arrays.

Dow has shown that the packing number, D(ν2 + ν + 1, ν, 2) ≤ min{ν2 − 2(ν + 3)1/2 +6, ν2 − 3} [6,7] when it is assumed that every pair of blocks intersect in a common pointand no projective plane of order ν exists. When blocks are permitted to be disjoint, Metschhas shown that D(ν2 + ν + 1, ν, 2) ≤ ν2 + 1. Combining this result with Theorem 4 we get

COROLLARY 6. When no projective plane of order ν exists then

pa(ν2 + ν + 1, ν2 + 1 : ν2) ≤ 2ν2 + 1.

What is especially worthy of note here is that this bound is strictly less than twice thegroup size ν2 + 1. In [12] powerful constructions were derived to build packing arrays withno more than 2g rows. It was initially hoped that the structural restrictions on this casewould enable a complete solution: When ever the bound from Theorem 1 is less than 2gthen it is tight. Corollary 6 demonstrates that for an infinite number of g, this conjectureis not true. The only known examples of non-extendible partial projective planes meetingMetsch’s bound are constructed from projective planes of order ν, so when no projectiveplane exists there may be even tighter bounds than this [8].


5. Conclusion

The previously known most powerful bound on the number of rows of a packing array wasa modification of the Plotkin bound stated in Theorem 1 [12]. We introduce a new boundwhen the packing arrays has a set of g − 1 disjoint rows. This bound is derived from theknown values of packing designs [5]. The consideration of sets of disjoint rows is pertinent;as discussed in [12] this consideration permits the use of Wilson’s construction, one of themost powerful construction techniques. Another interest in the consideration of packingarrays with sets of disjoint rows is their equivalence, by duality, to resolvable packings anddesigns with holes [12].

We show that the bounds from Theorems 1 and 4 coincide when

k ≥ g +⌈√

1 + 8g − 1

2

⌉(15)

and that for all but two of the remaining g + 1 ≤ k the bound from Theorem 4 is strongerthan that of Theorem 1. We have given some sufficient conditions for this bound beingconstructible and shown that when equation (15) holds, packing arrays meeting the boundfrom Theorem 4 are constructible.

When the associated packing design is in fact a projective or affine plane then we alsohave a powerful existence result about packing arrays. When projective planes of order ν

do not exist, Theorem 4 gives a strong result showing that Theorem 1 is not constructiblewhenever it implies that pa(k, g : n) ≤ 2g, as was hoped in [12].

We close now with some conjectures motivated by the results from this article. InTheorem 8 we show that the associated packing can be used to construct packing arrayswhen certain conditions on the point replication numbers are satisfied. We conjecture

CONJECTURE 1. A packing on v points with block size κ with v blocks with replicationnumbers as equal as possible always exists when a design with the same v and κ exists

This is certainly true when the design is resolvable but it should be true in general.

CONJECTURE 2. A maximal packing on v ≤ κ2 −κ points with block size κ can be assumedto have replication numbers as equal as possible.

Fully or even partial affirmative answers to these conjectures would supply us with manyconstructible packing arrays for n = g − 1.

Acknowledgments

Eric Mendelsohn was supported by NSERC of Canada Grant No. OGP007681. Brett Stevenswas supported by The Pacific Institute for the Mathematical Sciences (PIMS), IBM T. J.Watson Research Center, Mathematics of Information Technology and Complex Systems(MITACS) and Carleton University.


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