STRONG FORMS OF ORTHOGONALITY FOR SETS OF HYPERCUBES

Transcription

1 The Pennsylvania State University The Graduate School Department of Mathematics STRONG FORMS OF ORTHOGONALITY FOR SETS OF HYPERCUBES A Dissertation in Mathematics by John T. Ethier c 008 John T. Ethier Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 008

2 The dissertation of John T. Ethier was reviewed and approved* by the following: Gary L. Mullen Professor of Mathematics Dissertation Adviser Chair of Committee George E. Andrews Evan Pugh Professor of Mathematics James A. Sellers Associate Professor of Mathematics James L. Rosenberger Professor of Statistics John Roe Professor of Mathematics Head of Department of Mathematics *Signatures are on file in the Graduate School.

3 iii Abstract The main goal of this work is to examine strong forms of orthogonality for various types of hypercubes. For integers n and d, we prove that there are at most n + d 1 hypercubes of dimension d and order n which are orthogonal d at a time, or d-orthogonal. We then provide a useful connection between d-orthogonality and MDS codes; this connection to MDS codes leads us to examine a stricter form of d-orthogonality, which we call strong d-orthogonality. We show that there are at most n 1 mutually strongly d-orthogonal hypercubes of dimension d and order n. Furthermore, we modify d-orthogonality to provide a bound on the number of hypercubes of dimension d and order n which are orthogonal k < d at a time. We will then use these stronger forms of orthogonality to construct a new type of (t, m, s)-net, which we call a (t, m, s)-strong net. We also examine hypercubes of order n which are based on n symbols, provide a definition of orthogonality, and a bound on the number of these hypercubes which can be mutually orthogonal. Finally, we generalize the notion of equiorthogonality and define orthogonality of strength s for frequency hypercubes and frequency hyperrectangles and provide a subsequent bound for the maximum number of frequency hypercubes which are orthogonal of strength s. For all of these forms of orthogonality, we provide construction techniques for prime power orders using linear permutation polynomials over finite fields.

4 iv Table of Contents List of Figures vi Acknowledgments vii Chapter 1. Introduction Chapter. Dimensional Orthogonality Definitions and Preliminaries Mutually d-orthogonal Hypercubes and MDS codes MDS Codes and Higher Class Hypercubes Chapter 3. Constructions of Sets of Mutually d-orthogonal and Mutually Strong d-orthogonal Hypercubes Definitions and Preliminaries Constructions in Prime Power Orders Constructions in Non-Prime Power Orders k-orthogonality Chapter 4. Strong Nets Definitions and Preliminaries Sets of Mutually Strong d-orthogonal Hypercubes and (0, d, s) Strong Nets Sets of Mutually k-orthogonal Hypercubes and (t, t + k, s) Strong Nets 53

5 v Chapter 5. Strong Orthogonality for Frequency Hypercubes Definitions Connections Between Strong Orthogonality and Equiorthogonality Mutually s-strong Orthogonal Frequency Hypercubes Constructions of Sets of Mutually s-strong Orthogonal Frequency Hypercubes Frequency Hyperrectangles Chapter 6. Latin Hypercubes of Class r Definitions Orthogonal Hypercubes of Class Chapter 7. Conclusion References

6 vi List of Figures 1.1 Mutually orthogonal latin cubes of order Example of a Kronecker product construction Three types of elementary intervals with b = 3, s =, and volume 1/ A (0,, )-net in base Three types of general intervals with b = 3, s =, and volume 1/ A single general interval is depicted by the nine 1/9 by 1/9 squares A (0,, )-strong net in base A (0,, )-net in base 3 with a lattice structure Construction of a pair of MOLS of order 3 from a (0,, )-strong-net in base Frequency squares illustrating orthogonality of strengths 1 and A complete set of MS 1 OFS F () (4; ) A latin hypercube of dimension 3, order 3, and class

7 vii Acknowledgments First and foremost, I would like to thank my parents, Thomas and Chong, for their eternal love and support. Without them, I would not be where I am today and I am truly grateful. I would also like to thank Dr. Peter Shiue and Dr. Michelle Schultz, both of whom instilled in me a greater understanding and appreciation of the beauty of mathematics at the undergraduate level. I am not sure that I would have continued in mathematics if it was not for their inspired teaching and advice. Next, I must thank my advisor Dr. Gary Mullen. Not only has he been a great source of help and direction for my dissertation, but he has always been completely supportive and has provided invaluable advice in all aspects of life, not only mathematics. I also need to thank Dr. James Sellers for his useful comments on this work. Finally, I must thank my fiancée Christina, who provides me with both balance and support. I could not have finished my last two years at Penn State, nor can I picture any type of future, without her and her love.

8 1 Chapter 1 Introduction A latin square of order n is an n n array based on n distinct symbols such that each symbol appears exactly once in each row and each column. We say that two latin squares are orthogonal if each of the n ordered pairs occurs exactly once when the two squares are superimposed. A set of mutually orthogonal latin squares (MOLS) is a set of latin squares of the same orders in which any pair of distinct squares is orthogonal. It is easy to show that the maximum number of MOLS of order n, N(n), is at most n 1, and a set of MOLS is called complete if it attains this bound. If n = p e where p is prime and e is a positive integer, then a complete set of p e 1 MOLS of order p e can be constructed [1]. While it is known that N(n) < n 1 for infinitely many orders n [], there is no known general formula for N(n). Dénes and Keedwell provide good surveys of the work done on latin squares and their applications in [4] and [5]. In this work we are interested in generalizations of latin squares and of concepts of orthogonality for these generalizations. Latin squares can be generalized in a number of ways. One way is to increase the number of dimensions. Typically we say that a latin hypercube of order n and dimension d is an n n (d times) array based on n distinct symbols such that each symbol occurs exactly n d times in each (d 1)- dimensional subarray. Note that in the case of latin squares, this definition is equivalent to fixing a row or column. Here the usual definition of orthogonality states that two

9 latin hypercubes are orthogonal if each ordered pair occurs exactly n d times upon superimposition. Kishen [1] was able to show that the number of mutually orthogonal latin hypercubes of order n and dimension d is at most (n d 1)/(n 1) d. Note that this form of orthogonality does not take into consideration the location of the n d ordered pairs. We can further generalize latin hypercubes by changing the restrictions on the locations of the symbols, or the type of the hypercube. We say that a latin hypercube of order n, dimension d, and type t, 0 t d 1, is an n n (d times) array based on n distinct symbols such that each symbol occurs exactly n d t 1 times in each (d t)-dimensional subarray. Another way to generalize latin squares is to change the number of symbols which occur in the array. A frequency square F (n; λ 1,..., λ m ) of order n is an n n array consisting of m distinct symbols with the property that for each i = 1,..., m, the symbol i occurs exactly λ i times in each row and in each column. Clearly n = λ λ m and an F (n; 1,..., 1) frequency square is a latin square of order n. Frequency squares can also be generalized into dimensions other than d =, with the property that each 1-subarray contains the symbol i exactly λ i times. We denote a frequency hypercube of dimension d with frequency vector (λ 1,..., λ m ) as F (d) (n; λ 1,..., λ m ). In particular we are interested in the case where λ 1 = = λ m and in this case we write F (d) (n; λ) where m = n/λ. While frequency hypercubes reduce the number of symbols to be less than the order of the array, in [1] Kishen increases the number of symbols beyond the order of the array. Kishen used the term order in a different way than is used throughout this

10 3 work. In this work, we will reserve the term order to denote the size of the array and we will use the term class in place of Kishen s definition of order found in [1]. A d-dimensional latin hypercube of order n and class r, with r d, is an n n (d times) array based on n r distinct symbols, each repeated n d r times in the hypercube, such that each symbol occurs exactly once in each r-subarray. If r d/, we say that two such hypercubes are orthogonal if when superimposed, each of the n r ordered pairs occurs exactly n d r times. The most commonly used definition of orthogonality for hypercubes simply ensures that each ordered pair occurs the same number of times upon superimposition of the hypercubes; thus in the generalized version of hypercubes we would say that two hypercubes of dimension d, order n, based on s symbols are orthogonal if upon superimposition each ordered pair occurs exactly n d /s times. However this concept does not yield any information on the positions of the ordered pairs. In Figure 1, we have a set of three mutually orthogonal latin cubes C 1, C, and C 3 of order 4, where each 4 4 subarray represents a level of the cube. Notice that when C 1 and C are superimposed the corresponding levels do not contain every ordered pair, that is corresponding levels are not orthogonal. However, when C 1 and C 3 are superimposed their corresponding levels are orthogonal. Understanding the added structure implicit in the orthogonality of C 1 and C 3 has some interesting and useful applications, and modifying the definition of orthogonality to account for extra structure is a major focus of this work. In [19] Morgan defines the notion of equiorthogonal frequency hypercubes. Unlike the standard definition of orthogonal frequency hypercubes, equiorthogonality takes into

11 4 C 1 = C = C 3 = Fig Mutually orthogonal latin cubes of order 4 account the positions in which ordered pairs occur. We will generalize Morgan s definition to define strong orthogonality for frequency hypercubes. We will also extend the definition of equiorthogonality and strong orthogonality to frequency hyperrectangles. So far we have examined orthogonality for pairs of hypercubes; however another direction we can look is to define orthogonality by superimposing more than two hypercubes at once. For example, if we superimposed two three-dimensional cubes of order n, then each of the n ordered pairs occurs exactly n times; however if we superimpose three cubes at once, then each ordered triplet could occur exactly once, and in a sense this provides a more natural operation. In [9] the authors introduce the concept of a variational cube to account for this type of orthogonality; here we will use the terminology of d-wise orthogonality. For d, a set of d latin hypercubes of order n and dimension d are said to be d-wise orthogonal (d-orthogonal) if when superimposed, each of the n d ordered d-tuples occurs exactly once. We say that a set of j d hypercubes

12 5 of dimension d are mutually d-orthogonal if given any d hypercubes from the set, they are d-orthogonal. These new definitions of orthogonality, while interesting in their own right due to their increased structure, will also prove to be useful by providing a link between latin hypercubes and maximum distance separable linear codes as well as (t, m, s)-nets, which are used in certain numerical integration problems. We will first study the properties of these new definitions of orthogonality for frequency and latin hypercubes and hyperrectangles. We will also examine some forms of orthogonality for hypercubes of varying classes.

13 6 Chapter Dimensional Orthogonality In this chapter we will focus on sets of mutually d-orthogonal hypercubes. We begin with definitions and known results, and then prove a connection between mutually d-orthogonal hypercubes and maximum distance separable codes. We then modify the definition of d-orthogonality to define strong d-orthogonality and examine its connection to coding theory as well..1 Definitions and Preliminaries Throughout this chapter we will let n be a positive integer and d, k, l, and t be non-negative integers. We will reserve q to denote a prime power, and we denote the finite field of order q by F q and the non-zero elements of the field by F q. A hypercube of order n and dimension d based on m symbols is an n n array, repeated d times, in which each of m distinct symbols occurs exactly n d /m times. In order to work with a hypercube combinatorially, it is helpful to use a coordinate system. Here we distinguish one corner as an origin and identify each edge incident with the origin with a coordinate x 1,..., x d. We then label the n cells along each of these edges 0, 1,..., n 1. It is now possible to identify each cell in the hypercube with a set of coordinates (x 1,..., x d ).

14 7 For 1 t d, a t-subarray is a subset of a hypercube of dimension d which is obtained by fixing d t of the coordinates and allowing the other t coordinates to vary. Hence, a t-subarray is made up of n t cells. We say that two t-subarrays in the same hypercube are parallel if they have the same fixed coordinates. Two t-subarrays of distinct hypercubes are said to be corresponding if the fixed coordinates of the two subarrays are exactly the same. Recall that for d, a set of d hypercubes of order n and dimension d is said to be d-orthogonal if when superimposed, each of the n d ordered d-tuples occurs exactly once. We say that a set of j d hypercubes of dimension d is mutually d-orthogonal if given any d hypercubes from the set, they are d-orthogonal. We will use the notation MdOH as an abbreviation for mutually d-orthogonal hypercubes. Observe that if d =, then this is the standard definition for orthogonality of squares. In the next section, we will prove an important connection between sets of MdOH and maximum distance separable codes. A codeword x = x 1 x... x l is a block of l symbols, where a fixed number d, d l, of the symbols are message or information symbols, and the remaining l d positions are check symbols. A collection of codewords form a code. Here l is called the length of the code while d is called the dimension of the code. The code is called n-ary if it is based on n distinct symbols. In this work, we will consider only codes with N = n d codewords. The distance between two vectors x and y, denoted dist( x, y), is the number of places where the two vectors differ. The minimum distance of a code, D, is the minimum distance between any of its codewords. An n-ary code of length l and minimum distance D which contains n d codewords will be denoted in this work as an (l, n d, D) code.

15 8 Remark.1.1. The standard notation used in coding theory uses the variables n, k, and d for length, dimension, and minimum distance respectively. We use different variables in order to maintain consistency with the rest of this work. A code of length l with symbols from F q is called linear if the codewords are distinct and the code forms a vector subspace of F l. We will denote a q-ary linear code q of length l, dimension d, thus with q d codewords, and minimum distance D as an [l, d, D] linear code. The following relationship between the parameters of a (l, n d, D) code is in [6]: Theorem.1.. (The Singleton bound) For any n-ary (l, n d, D) code, D l d + 1. Codes which reach this bound, that is if D = l d + 1, are called maximum distance separable or MDS codes. The following theorem, also shown in [6] about MDS codes will be useful to us: Theorem.1.3. For an (l, n d, l d + 1) MDS code any d coordinates can be regarded as information (message) positions.. Mutually d-orthogonal Hypercubes and MDS codes In this section we will focus on d-orthogonality and its connections to MDS codes. We will begin by proving some preliminary results for sets of mutually d-orthogonal hypercubes. Lemma..1. Given a set of d hypercubes of dimension d and order n which are d- orthogonal, then upon superimposition of any 1 i d hypercubes from the set each ordered i-tuple appears exactly n d i times.

16 9 Proof. Let H 1,..., H d be a set of MdOH of dimension d and order n. Now suppose for a contradiction that upon superimposition of i of the hypercubes, 1 i d, that some ordered i-tuple does not appear exactly n d i times. If this is the case, then at least one ordered i-tuple appears less than n d i times. Without loss of generality, suppose that upon the superimposition of H 1,..., H i the i-tuple (0,..., 0) appears less than n d i times. Now, notice that there are n d i ordered d-tuples which begin with the i-tuple (0,..., 0). Since upon the superimposition of H 1,..., H i the i-tuple (0,..., 0) appears less than n d i times, it follows that in the superimposition of H 1,..., H i, H i+1,..., H d at least one of the ordered d-tuples which begins with the i-tuple (0,..., 0) will not occur, a contradiction. In [4, pg. 186] a bound, first proved in [11], for the maximum number of mutually d-orthogonal hypercubes of order n and dimension d is shown to be (d 1)(n 1). Note that in the result found in [4], a set of MdOH is called a d-dimensional variational set. Here we significantly improve on this bound with the following result: Theorem... Let d. The maximum number of mutually d-orthogonal hypercubes of order n and dimension d is at most n + d 1. Proof. Distinguish a corner of the hypercubes and permute the symbols of all the hypercubes so that the same symbol, say 0, appears in that corner for each hypercube. Now, distinguish d hypercubes H 1,..., H d. By Lemma..1, when these hypercubes are superimposed, the (d )-tuple (0,..., 0) must appear exactly n d (d ) = n times.

17 Hence the (d )-tuple (0,..., 0) appears in exactly n 1 entries e 1,..., e n 1 other than the distinguished corner. 10 Now, for any additional hypercube H t, upon superimposition of H 1,..., H d, H t the (d 1)-tuple (0,..., 0) appears exactly n times. We know that one (d 1)-tuple (0,..., 0) appears in the distinguished corner and furthermore that the other n 1, (d 1)-tuples appear in n 1 of the n 1 entries e 1,..., e n. Also it is easy to see 1 that any two additional hypercubes to H 1,..., H d cannot both contain the symbol 0 in any e 1,..., e n 1. Thus we can have at most (n 1)/(n 1) = n + 1 additional hypercubes. Hence we can have at most (d ) +n + 1 = n+d 1 mutually d-orthogonal hypercubes of order n. Here if we let d = we have as a corollary the known result in latin square theory: Corollary..3. There are at most n + 1 mutually orthogonal squares of order n and type 0. Remark..4. The previous corollary is equivalent to having at most n 1 MOLS of order n. A set of MdOH is called complete if its cardinality is n + d 1, that is if the set reaches the bound defined in Theorem... In the two-dimensional case it is known that this bound is attainable in every prime power order; in fact, in each prime power order a complete set can be constructed using permutation polynomials over the finite field of the same order. We would like to extend these constructions to dimensions greater than and show that the bound in Theorem.. is attainable in higher dimensions for all prime

18 11 power orders. On the surface this does not seem unreasonable as similar constructions can be done to create maximal sets of mutually orthogonal latin hypercubes when we use the standard definition of pairwise orthogonality. To understand the complexity of the current problem it is helpful to look at a connection between d-orthogonality and coding theory. We begin by proving a relationship between d-orthogonal hypercubes and MDS codes: Theorem..5. A set of l, d-orthogonal hypercubes of order n, dimension d, and type 0 is equivalent to an n-ary MDS code (l, n d, l d + 1). Proof. Suppose we are given an n-ary MDS code (l, n d, l d + 1) with n d codewords, each by definition, of length l. Take each of the n d codewords and place it in one of the n d distinct entries of a hypercube H of order n and dimension d. Now we can construct a set of l, d-orthogonal hypercubes of order n, dimension d, and type 0 from H by making the entries of the j-th hypercube to be exactly the j-th symbol of the entries of H. To show that the hypercubes are mutually d-orthogonal, first take any d of these hypercubes. These d hypercubes were generated by some d positions of all the codewords. By Theorem.1.3, these d positions may be used as information positions and hence each d-tuple must appear exactly once in these d positions of the codewords. Therefore each d-tuple appears exactly once in the superimposition of those d hypercubes, and so the set of hypercubes is mutually d-orthogonal. Conversely, suppose we are given a set of l, mutually d-orthogonal hypercubes of order n, dimension d, and type 0. We can construct an n-ary MDS code (l, n d, l d+1) by

19 superimposing all l hypercubes and letting the n d, l-tuple entries be the n d codewords. 1 We must show that this is an MDS code. Notice that if any two codewords agree in d positions, then the d hypercubes which generated these positions would not be d- orthogonal, a contradiction. Hence any two codewords differ in at least l (d 1) = l d+1 positions; therefore the code is MDS. As a corollary to Theorems.. and..5 we obtain: Corollary..6. Given an n-ary MDS code (l, n d, l d+1) it follows that l n+d 1. The equivalence between MDS codes and orthogonal hypercubes has been alluded to previously in the literature, but not shown explicitly when d >. Golomb [7] pointed out the following equivalence between MDS codes and mutually orthogonal latin squares: Theorem..7. An n-ary MDS code (l, n, l 1) is equivalent to a set of l mutually orthogonal latin squares of order n. In [6], Singleton states that a (l, n d, l d + 1) n-ary code with d > can be used to form a set of l d, d-dimensional latin hypercubes of order n which will have some orthogonality relationship as a result of the properties of the code. Furthermore, sets of hypercubes which exhibit this orthogonality relationship can be used to form a MDS code. While Theorem..5 defines an equivalence between MDS codes and certain d- orthogonal hypercubes, d-orthogonality is not the orthogonality relationship alluded to in Singleton s work, since Theorem..5 uses a set of l hypercubes and Singleton requires only l d hypercubes. However, by modifying d-orthogonality to account for the position

20 13 of ordered j-tuples, 1 j d, in the superimposition of hypercubes we can explicitly define the orthogonality relationship Singleton alludes to. We say that a set of r hypercubes of order n and dimension d is mutually strongly d-orthogonal (MSdOH) if upon superimposition of corresponding j-subarrays of any j hypercubes in the set, 1 j min(d, r), each ordered j-tuple appears exactly once. Note that letting j = 1 implies that each hypercube in the set is of type d 1. Also, if d = and r, then this definition is equivalent to the definition of mutually orthogonal latin squares. Furthermore, it should be noted that if r d, then strong d-orthogonality implies d-orthogonality. Theorem..8. If l > d, a set of l d mutually strong d-orthogonal hypercubes of order n and dimension d is equivalent to an n-ary MDS code (l, n d, l d + 1). Proof. Given an n-ary MDS code (l, n d, l d + 1) we can construct l d hypercubes of order n and dimension d. We use the first d positions of the codewords to define a coordinate of a hypercube and construct l d hypercubes from the last l d positions of the codewords by placing the d + j-th entry of a codeword in the j-th hypercube in the coordinate defined by the first d positions. Suppose for a contradiction that there are j hypercubes, 1 j min(d, r), in our set which upon superimposition of some corresponding j-subarray, an ordered j-tuple appears twice. Then consider the two codewords which generate these two j-tuples. The entries are in the same j-subarray so they agree in d j of the first d positions. They also agree in j positions of the repeated j-tuple. Hence these codewords agree in d positions, a contradiction.

21 14 Conversely, given a set of l d mutually d-strong orthogonal hypercubes of order n and dimension d, we will construct an MDS code. We generate codewords by letting the first d positions of each codeword corresponding to each of the n d positions of a hypercube and the last l d letters of the codeword corresponding to the entry in that position in each of the l d hypercubes. It remains to show that this forms an MDS code. Pick any two codewords, which are generated by two positions in the hypercubes. Let j be the smallest value such that these positions lie in the same j-subarray. Then these two codewords differ in exactly j of the first d entries and must differ in at least (l d) (j 1) of the final l d positions. Hence any two codewords must differ in at least l d + 1 positions and therefore the code is MDS. Notice that Theorem..7 is a corollary of Theorem..8 obtained by letting d =. We will now provide a bound for the maximum number of mutually strong d-orthogonal hypercubes. Theorem..9. There are at most n 1 mutually strong d-orthogonal hypercubes of order n and dimension d. Proof. Let H 1,..., H t be a set of MSdOH, and let S 1,..., S t be the set of corresponding -subarrays from each of H 1,..., H t. Now notice that by definition S 1,..., S t forms a set of mutually orthogonal latin squares. Hence t is at most n 1. We conclude this section by generalizing a connection between codes and hypercubes first shown in [6]. As a corollary to Theorem..8 we obtain the following result found in [6].

22 Corollary..10. A n-ary MDS code (l, n l 1, ) is equivalent to a latin hypercube cube of dimension l 1, order n, and type l. 15 Note, that in [6] the authors use the terminology permutation cube of dimension d and order n in place of latin hypercube of dimension d, order n and maximal type n 1. The paper [6] also contains the following result which links the number of distinct MDS codes of dimension d 1 to the number of distinct latin hypercubes of dimension d. Theorem..11. The number of (d 1)-dimensional latin hypercubes of order n and type d is equal to the number of n-ary (d, n d 1, ) MDS codes. Using the construction outlined in Theorem..8, we can generalize Theorem..11 to the following: Theorem..1. Let L(n, l, d) be the number of MDS n-ary codes (l, n d, l d + 1) and let S(n, l, d) be the number of distinct sets of l d, mutually dimensionally strongorthogonal latin hypercubes of order n, dimension d, and type d 1. Then L(n, l, d) = (l d)!s(n, l, d). Proof. The equivalence defined in Theorem..8 is a one-to-one and onto function from the set of MDS n-ary codes (l, n d, l d + 1) to the set of ordered sets of l d mutually dimensionally strong-orthogonal latin hypercubes of order n, dimension d, and type d 1. Hence, there are (l d)! MDS codes for each distinct set in S(n, l, d). Notice that Theorem..11 is simply the case when l d = 1 in Theorem..1.

23 16.3 MDS Codes and Higher Class Hypercubes Recall that for d r, a d-dimensional latin hypercube of order n, class r, and type j, 0 j d r, is an n n array (d times) based on n r distinct symbols, such that each symbol occurs exactly n d r j times in each (d j)-subarray. When d r, we say that two such hypercubes are orthogonal if when superimposed, each of the n r ordered pairs occurs exactly n d r times. We now give a generalization of d-orthogonality for hypercubes of higher classes. We say that a set of t, d-dimensional hypercubes H 1,..., H t, each of order n and with respective classes c 1, c,..., c t where C = c c t k, is d-orthogonal if upon superimposition of all hypercubes each ordered t-tuple appears exactly n d C times. Theorem.3.1. Given a pair of orthogonal r-dimensional hypercubes of order n and class r, we can construct a set of r, r-dimensional hypercubes each of order n and class 1 which are d-orthogonal. Proof. Each of the n r symbols in a hypercube of class r can be considered as a block of length r of n distinct symbols. Given a r-dimensional hypercube of order n and class r, we can construct r hypercubes by letting the entries in the j-th hypercube be the j-th symbol of the block in the corresponding entry in the hypercube of class r. Then, given a pair of orthogonal r-dimensional hypercubes of order n and class r we can construct a set of r, r-dimensional hypercubes each of order n and class 1. If the r hypercubes are superimposed then each of the n r ordered r-tuples must occur exactly once by the orthogonality of the original pair. Hence, the r hypercubes are d-orthogonal.

24 17 Corollary.3.. Given a set of T mutually orthogonal r-dimensional hypercubes of order n and class r, we can construct T sets, each containing r, r-dimensional hypercubes of order n and class 1 such that the union of any two sets is d-orthogonal. Theorem.3.3. Given an n-ary MDS code (l, n d, l d + 1) we can construct a set of t, d-orthogonal hypercubes of order n, dimension d and classes c 1, c,..., c t each of type 0, where c c t d. Proof. Suppose we are given an n-ary MDS code (l, n d, l d + 1). Take each of the n d codewords and place it in one of the n d distinct entries of a hypercube H of order n and dimension d. Now we can construct a set of t, d-orthogonal hypercubes of order n, dimension d, and type 0 and classes c 1, c,..., c t from H by making the entries of the first hypercube to be the ordered c 1 -tuple given by the first c 1 symbols of the entries of H. We construct the second hypercube from the next c symbols and so on. Thus the j-th hypercube is based on n cj symbols as required. It follows from Theorem.1.3 that each c j -tuple will occur exactly n d cj times in the j-th hypercube. Orthogonality also follows by Theorem.1.3. Theorem.3.4. Given an n-ary MDS code (l, n d, l d + 1) we can construct T orthogonal hypercubes of order n, dimension d and classes c 1, c,..., c t with respective types d c i where c c t = l d. Proof. Similar to the proof of Theorem.3.3 using the first d coordinates as labels as in the proof of Theorem..8.

25 18 Chapter 3 Constructions of Sets of Mutually d-orthogonal and Mutually Strong d-orthogonal Hypercubes In this chapter, we will provide a method for constructing sets of mutually d- orthogonal hypercubes (MdOH) and mutually strong d-orthogonal hypercubes (MSdOH) in prime power orders q using permutation polynomials over the finite field F q. Furthermore, we will examine whether the bound provided by Theorem..9 is attainable in prime power orders; that is whether or not we can construct a complete set of MSdOH for a given prime power order. By modifying the equivalence in Theorem..8 to link linear codes and sets of MSdOH generated by linear polynomials, we will see that the problem of constructing complete sets of MSdOH is a equivalent to a long list of open problems from coding theory and projective geometry. We will also include constructions of sets of MdOH and MSdOH in non-prime power orders. We will conclude the chapter, by giving a generalization of d-orthogonality. 3.1 Definitions and Preliminaries As denoted in [15, ch. 7], a polynomial f(x 1,..., x d ) F q [x 1,..., x d ] is a permutation polynomial if the equation f(x 1,..., x d ) = α has exactly q d 1 solutions in F d q for every α F q. It is easy to see that a linear polynomial is a permutation polynomial over F q if and only if it has at least one non-zero coefficient. Also, notice that addition of a constant does not affect the property of being a permutation polynomial.

26 We say that the polynomials f 1 (x 1,..., x d ),..., f r (x 1,..., x d ), with 1 r d, form an orthogonal system of polynomials in F q if the system of equations f i (x 1,..., x d ) = α i, i = 1,..., r, has exactly q d r solutions in F r q for each r-tuple (α 1,..., α r ) in Fr q. As before, addition of a constant to one or more of the equations does not affect the property of orthogonality. As shown by Niederreiter [3], the system f 1 (x 1,..., x d ),..., f r (x 1,..., x d ) is orthogonal if and only if for all (b 1,..., b r ) F r q with (b 1,..., b r ) (0,..., 0), the polynomial b 1 f 1 (x 1,..., x d ) + + b r f r (x 1,..., x d ) is a permutation polynomial over F q. It is easy to see that this is equivalent to the following result: 19 Lemma Let f i (x 1,..., x d ) = a i,1 x a i,d x d, for 1 i r, be a set of r, linear polynomials over F q, and let M = (a i,j ) be the matrix formed by the coefficients of the functions f i, 1 i r. Then f 1 (x 1,..., x d ),..., f r (x 1,..., x d ) forms an orthogonal system if and only if M is nonsingular. 3. Constructions in Prime Power Orders For q a prime power, we can construct a d-dimensional hypercube of order q from f(x 1,..., x d ) = a 1 x a d x d over F q as follows. We use the elements of F q as the q distinct symbols and as the coordinate labels for each coordinate x i, 1 i d. Then the symbol in entry (x 1,..., x d ) is given by a 1 x a d x d. Lemma Let f(x 1,..., x d ) = a 1 x a d x d be a linear polynomial over F q. If (a 1,..., a d ) (0,..., 0), then the above construction yields a d-dimensional hypercube of order q. Furthermore, if a i 0 for all 1 i d, then the hypercube has type d 1.

27 0 Proof. Let f(x 1,..., x d ) = a 1 x a d x d, with (a 1,..., a d ) (0,..., 0). Then f(x 1,..., x d ) is a permutation polynomial and hence f(x 1,..., x d ) = α has exactly q d 1 solutions in F d q for every α F q. It follows that every symbol α F q appears exactly q d 1 in a hypercube constructed from f(x 1,..., x d ). Now, let a i 0 for all 1 i d. Notice that any 1-subarray is generated by fixing d 1 coordinates and allowing the final coordinate to be free. Without loss of generality, suppose that x,..., x d are fixed as b,..., b d respectively. Then the 1-subarray in question is generated by the polynomial f(x 1 ) = a 1 x 1 + a b a d b d (3.1) = a 1 x 1 + C (3.) where C F q is the constant a b + + a d b d. Since a 1 0, this is a permutation polynomial and hence f(x 1 ) = α has exactly one solution for each α F q. Therefore, each symbol appears exactly once in this 1-subarray and in general any 1-subarray. Hence the hypercube has type d 1. We will now show how a system of permutation polynomials can be used to construct a set of MSdOH. Theorem 3... Let f i (x 1,..., x d ) = a i,1 x a i,d x d, for 1 i r, be a set of r, linear polynomials over F q. Then the hypercubes generated by f 1,..., f r form a set of mutually strong d-orthogonal hypercubes of order q and dimension d if and only if every square submatrix of the matrix M = (a i,j ) is nonsingular.

28 1 Proof. Let f i (x 1,..., x d ) = a i,1 x a i,d x d, for 1 i r, be a set of r, linear polynomials over F q and M = (a i,j ) be the matrix formed by the coefficients of the functions f i, 1 i r. Let N be any t t square submatrix of M. The t rows of N are formed by t of the r functions f i ; hence these rows correspond to the t hypercubes generated by these functions. Similarly, the t columns of N are used to build t-subarrays of hypercubes with free coordinates corresponding to these coefficients. Now, consider any system, S, of t equations in t free variables which has the t coefficients of the t equations corresponding to the matrix N and fixes the other d t variables. Notice that S generates corresponding t-subarrays of the t hypercubes corresponding to the equations of S. Now, N is nonsingular if and only if S is an orthogonal system by Lemma Also, S is an orthogonal system if and only if each of the q t, ordered t-tuples is a solution to the system S exactly once. This is true if and only if each ordered t-tuple appears exactly once in the superimposition of the t-subarrays of the t hypercubes corresponding to the equations of S. Since S and N were chosen arbitrarily, it follows that every square submatrix of the matrix M is nonsingular if and only if the hypercubes generated by f 1,..., f r form a set of MSdOH. well: We can adapt the requirements of Theorem 3.. to construct sets of MdOH as Theorem Let f i (x 1,..., x d ) = a i,1 x a i,d x d, for 1 i t, be a set of t d, linear polynomials over F q. Then the hypercubes generated by f 1,..., f t form a

29 set of mutually d-orthogonal hypercubes of order q and dimension d if and only if every d rows of the matrix M = (a i,j ) are linearly independent. Proof. Let f i (x 1,..., x d ) = a i,1 x a i,d x d, for 1 i t, be a set of t d, linear polynomials over F q and M = (a i,j ) be the matrix formed by the coefficients of the functions f i, 1 i t. Now, any d rows of M correspond to d distinct functions and hence correspond to d hypercubes. These rows are linearly independent if and only if the corresponding functions form an orthogonal system by Lemma As in the proof of Theorem 3.., this is equivalent to the generated hypercubes being d-orthogonal. We will now use linear permutation polynomials to construct linear codes. Let f i (x 1,..., x d ) = a i,1 x a i,d x d, with 1 i r, be linear permutation polynomials over F q. Then for each of the q d ordered d-tuples x = (x 1,..., x d ), we use these polynomials to generate a codeword by: C(x 1,..., x d ) = x 1 x... x d f 1 ( x)... f r ( x) (3.3) The next lemma follows from the linearity of f 1,..., f r. Lemma The code defined by Equation 3.3 is a linear code [d + r, d, D]. Corollary Let f i (x 1,..., x d ) = a i,1 x a i,d x d, for 1 i r, be a set of r, linear polynomials over F q with the property that every square submatrix of the matrix M = (a i,j ) is nonsingular. Then the code formed by the codewords C(x 1,..., x d ) = x 1 x... x d f 1 ( x)... f t ( x), is a linear MDS code [d + r, d, r + 1] with q d codewords.

30 3 Proof. Follows by Lemma 3..4, and Theorems 3.. and..8. Notice that the construction of MDS codes used in Corollary 3..5 requires polynomials which can also be used to generate sets of MSdOH. We already saw, in Theorem..8, an equivalence between sets of MSdOH and MDS codes, and here it is shown that constructing sets of MSdOH using linear polynomials over F q is equivalent to generating a linear MDS code. MDS codes are also equivalent to sets of MdOH, as pointed out in Theorem..5. It follows that we should be able to generate MDS codes using sets of polynomials with the less restrictive conditions needed to generate sets of MdOH. Similar to before, we let f i (x 1,..., x d ) = a i,1 x a i,d x d, for 1 i l, be linear permutation polynomials F q with l d. Then for each of the q d ordered d-tuples x = (x 1,..., x d ), we can use these polynomials to generate a codeword by: C( x) = f 1 ( x)... f l ( x) (3.4) As before, this code is also linear since each of f 1,..., f l is linear. Lemma The code defined by Equation 3.4 is a linear code [l, d, D]. Also, the code will be MDS if the polynomials generate a set of MdOH. Corollary Let f i (x 1,..., x d ) = a i,1 x a i,d x d, for 1 i l, be a set of l, linear polynomials over F q with the property that every d rows of the matrix M = (a i,j ) are linearly independent. Then the code formed by the codewords C( x) = f 1 ( x)... f l ( x), is a linear MDS code [l, d, l d + 1] with q d codewords.

31 4 Proof. Follows by Lemma 3..6, and Theorems 3..3 and..5. Corollaries 3..5 and 3..7 show us that the problem of constructing complete sets of d-orthogonal hypercubes of order q using linear polynomials over F q is equivalent to finding maximal length q-ary MDS linear codes. We have shown that this problem is equivalent to the following three open problems, also listed in [17, ch. 11]: Problem Given d and q, find the largest l for which there is a d l matrix over F q having every d columns linearly independent? Problem Given a d-dimensional vector space over F q, what is the largest number of vectors with the property that any d of them form a basis for the space? Problem Given d and q, find the largest l such that there exists a d l matrix having entries from F q with the property that every square submatrix is nonsingular. In [17, ch. 11], the authors also show that the problem of finding maximal length q-ary MDS linear codes, and thus finding maximal sets of MSdOH or MdOH, is equivalent the following problems as well. An n-arc is a set of n points in the projective geometry P G(k 1, q) such that no k points lie in a subspace of dimension k, where n k 3. Problem Given d and q, find the largest value of n for which there exists an n-arc in P G(d 1, q). An M n matrix A with entries from a set S, where S = q, is called an orthogonal array (λq k, n, q, k) of size M with n constraints, q levels, strength k, and index λ if any set of k columns of A contains all q k possible row vectors exactly λ times.

32 Problem Find the largest possible n in a (q k, n, q, k) orthogonal array of index unity. 5 Corollaries 3..7 and 3..5 show us that questions of whether or not we can construct complete sets of MdOH or MSdOH using linear polynomials is equivalent to many open problems; therefore constructing complete sets using this method may not be feasible. However, by examining the known results in coding theory, the two corollaries lead us to the many results for these types of hypercubes. Theorem.. tells us that the maximum number of mutually d-orthogonal hypercubes of order q and dimension d is at most q + d 1. Hence, we have for a q-ary MDS code (l, d, l d + 1) that l q + k 1. However, in [17], we see that if d 3 and q is odd, then a q-ary linear MDS code [l, d, l d + 1] has the bound l q + k. Using the construction given in Corollary 3..7, we now know the following: Theorem Let d 3. When q is odd, the maximum number of MdOH of order q and dimension d which can be constructed using linear polynomials over F q is at most q + d. Similarly, we obtain: Theorem Let d 3. When q is odd, the maximum number of MSdOH of order q and dimension d which can be constructed using linear polynomials over F q is at most q. The following conjecture [17, ch. 11] has been made for q-ary MDS linear codes [l, d, l d + 1]:

33 Conjecture Given a q-ary linear MDS code [l, d, l d + 1] with q > d, then: 6 q + l q + 1 for k = 3 and k = q 1 both with q even in all other cases. This leads to the following conjectures for MdOH and MSdOH: Conjecture The maximum number, M, of mutually d-orthogonal hypercubes of order n and dimension d, n > d, must satisfy: n + M n + 1 for k = 3 and k = n 1 both with n even in all other cases. Conjecture The maximum number, N, of mutually d-orthogonal hypercubes of order n and dimension d, n > d, must satisfy: n d + N n d + 1 for k = 3 and k = n 1 both with n even in all other cases. In prime power orders, these bounds are attainable [17, ch. 11]. We now list the polynomials which can be used to generate q + 1 MdOH of order q and dimension d, q > d. The coefficients of these polynomials follow from the entries of the parity check matrix used to generate the equivalent MDS code. Let α 1,..., α q 1 be the non-zero elements of F q, then the polynomials:

34 7 p 1 = x 1 + α 1 x + α 1 x αd 1 x 1 d p = x 1 + α x + α x αd 1 x d. p q 1 = x 1 + α q 1 x + α q 1 x αd 1 q 1 x d p q = x 1 + 0x + + 0x d p q+1 = 0x x d 1 + x d (3.5) generate a set of q +1 MdOH of order q and dimension d. These can be used to create an equivalent MDS code, which in turn can be used to construct a set of q d + 1 MSdOH of order q and dimension d. It is known that the bounds found in the previous three conjectures are attainable in prime power orders; however it is still unknown as to whether this bound is correct in all cases. Hirschfeld [10] provides a good survey as to when Conjecture holds. 3.3 Constructions in Non-Prime Power Orders So far we have used permutation polynomials over finite fields to construct sets of MdOH and MSdOH. Since we used a finite field with q elements to construct hypercubes of order q, it follows that this construction will only work in prime power orders. In [13, ch. 3], a method, first used by MacNeish [16] in the construction of latin squares, is outlined to construct sets of pairwise orthogonal hypercubes in non-prime power orders.

35 8 In this section, we will extend this method to construct sets of MdOH and MSdOH in non-prime power orders. Suppose we have two hypercubes A and B of dimension d and orders m and n respectively. Let the coordinates of A range from 0 to m 1, and let the entry in position i 1,..., i d be denoted by a i1,...,i d. Similarly, let the coordinates of B range from 0 to n 1, and let the entry in position j 1,..., j d be denoted by b j1,...,j. The d Kronecker product A B is defined as follows. The coordinates of A B range from 0 to mn 1, and the entry in position (mj 1 + i 1,..., mj d + i d ) is given by the ordered pair (a i1,...,i, b d j1,...,j ). It is not hard to see that if A and B are latin hypercubes of d orders m and n, then A B is a latin hypercube of order mn. In Figure 3.3, we depict the Kronecker product construction of a latin square of order 6 using latin squares, A, and B, of orders and 3 respectively. Let A = and B = 1 0, then we have: 0 1 A B = A, 0 A, 1 A, A, 1 A, A, 0 A, A, 0 A, 1 = (0, 0) (1, 0) (0, 1) (1, 1) (0, ) (1, ) (1, 0) (0, 0) (1, 1) (0, 1) (1, ) (0, ) (0, 1) (1, 1) (0, ) (1, ) (0, 0) (1, 0) (1, 1) (0, 1) (1, ) (0, ) (1, 0) (0, 0) (0, ) (1, ) (0, 0) (1, 0) (0, 1) (1, 1) (1, ) (0, ) (1, 0) (0, 0) (1, 1) (0, 1) Fig Example of a Kronecker product construction

36 In [13, pg. 4], we find the following the result on the construction of orthogonal latin squares using the Kronecker product. 9 Theorem Let A 1, A be orthogonal latin squares of order m, and let B 1, B be orthogonal latin squares of order n. Then A 1 B1 and A B are orthogonal latin squares of order mn. It follows that if A 1,..., A t is a set of mutually orthogonal latin squares of order m and B 1,..., B t is a set of mutually orthogonal latin squares of order n, then A 1 B1,..., A t Bt forms a set of mutually orthogonal latin squares of order mn. Using this result recursively for each prime power in the prime factorization of a given order n, along with the known prime power constructions, it is easy to prove the following: Theorem Let n = q 1 q... q r, where q i is a prime power for each 1 i r, be the prime power factorization of n with q 1 < q <... < q r. Then the maximum number of mutually orthogonal latin squares of order n, N(n), is at least q 1 1. Similar results are available, [13], for the construction of mutually pairwise orthogonal hypercubes. We will now show that if the usual notion of orthogonality is replaced with dimensional orthogonality or strong orthogonality, then the Kronecker product constructions will continue to be useful. Theorem Let A 1,..., A d be a set of dimensionally orthogonal hypercubes of dimension d and order m, and let B 1,..., B d be a set of dimensionally orthogonal hypercubes of dimension d and order n. Then A 1 B1,..., A d Bd is a set of dimensionally orthogonal hypercubes of dimension d and order mn.

37 30 Proof. Recall that dimensional orthogonality requires that in the superimposition of A 1 B1, A B,..., A d Bd each ordered d-tuple ((a 1, b 1 ), (a, b ),..., (a d, b d )) occurs exactly once. This follows from the fact that (a 1,..., a d ) occurs exactly once in the superimposition of A 1,..., A d as does (b 1,..., b d ) in the the superimposition of B 1,..., B d. This result leads immediately to the following corollary for sets of mutually d- orthogonal hypercubes. Corollary For t d, let A 1,..., A t be a set of MdOH of dimension d and order m, and let B 1,..., B t be a set of MdOH of dimension d and order n. Then A 1 B1,..., A t Bt is a set of MdOH of dimension d and order mn. Now, by noting that t-subarrays of A B are constructed using corresponding t-subarrays of A and B, and using a method similar to the proof of Theorem 3.3.3, we also can obtain a similar result for strong d-orthogonality. Theorem Let A 1,..., A t be a set of MSdOH of dimension d and order m, and let B 1,..., B t be a set of MSdOH of dimension d and order n. Then A 1 B1,..., A t Bt is a set of MSdOH of dimension d and order mn. Now, similar to Theorem 3.3., we can obtain a lower bound for the maximum number of MdOH of order n, when n is a non-prime power. Theorem Let n = q 1 q... q r, where q i is a prime power for each 1 i r, be the prime power factorization of n with q 1 < q < < q r. If d, then the maximum number of MdOH of order n is at least q