Covering arrays from m-sequences and character sums

Transcription

1 Covering arrays from m-sequences and character sums Georgios Tzanakis a,, Lucia Moura b, Daniel Panario a, Brett Stevens a a School of Mathematics and Statistics, Carleton University 1125 Colonel By Dr., Ottawa, ON K1S 5B6 b School of Electrical Engineering and Computer Science, University of Ottawa 800 King Edward Ave., Ottawa, ON K1K 6N5 Abstract A covering array of strength t on v symbols is an array with the property that, for every t-set of column vectors, every one of the v t possible t-tuples of symbols appears as a row at least once in the sub-array defined by these column vectors. Arrays constructed using m-sequences over a finite field possess many combinatorial properties and have been used to construct various combinatorial objects; see the recent survey [23]. In this paper we construct covering arrays whose elements are the remainder of the division by some integer of the discrete logarithm applied to selected m-sequence elements. Inspired by the work of Colbourn [8], we prove our results by connecting the covering array property to a character sum, and we evaluate this sum by taking advantage of the balanced way in which the m-sequence elements are distributed. Our results include new infinite families of covering arrays of arbitrary strength. Keywords: covering arrays, linear feedback shift register sequences, primitive polynomials over finite fields, characters over finite fields, character sums 2010 MSC: 94A55, 05B15, 05B40 1. Introduction Let M be an N k array with entries from an alphabet A of cardinality v. A t-set of columns of M is covered if each of the v t possible t-set in A t appears in at least one row of the N t sub-array defined by these columns. If each of the ( k t) possible t-tuples of columns of M is covered, then M is a covering array of strength t and size N, denoted CA(N; t, k, v) Covering arrays are used in areas such as software development and manufacturing to test systems for which exhaustive testing is infeasible. In particular, the rows of a CA(N; t, k, v) provide a collection of configurations to be tested, for a system of k factors represented by the columns, where each factor admits v possible values. Performing experimental runs for all N configurations, guarantees that every combination of t factors and their values Corresponding author addresses: gtzanaki@math.carleton.ca (Georgios Tzanakis), lucia@eecs.uottawa.ca (Lucia Moura), daniel@math.carleton.ca (Daniel Panario), brett@math.carleton.ca (Brett Stevens) Preprint submitted to Designs, Codes and Cryptography October 17, 2016

2 is tested. This is known as t-way combinatorial testing and it has been shown to be an extremely efficient and effective alternative to exhaustive testing [6, 21]. In this context, finding covering arrays with a small number of rows is important. For fixed t, k, v, the smallest size N such that a CA(N; t, k, v) exists is the covering array number CAN(t, k, v). A trivial lower bound for CAN(t, k, v) is v t. Arrays that attain this minimum number of rows are orthogonal arrays of index 1 (also discussed later), and only exist for k max {v + 2, t + 1}; see [16] for a textbook on the subject. The exact value for covering array numbers has also been determined for the case when t = 2 and v = 2 [18, 20, 29]. Other than that, only a finite number of sporadic covering array numbers have been determined, while most of the existing research provides upper bounds. Notably, for any fixed t and v, a CA(N; t, k, v) with N = O(log k) can be constructed using a greedy heuristic algorithm [4]. Furthermore, explicit upper bounds for covering array numbers follow from a wide variety of constructions, and there also exist bounds of a probabilistic and asymptotic nature; we refer to [7, 9, 23] for surveys on the subject. Colbourn as of this date actively maintains an online database [10] of the best known upper bounds for covering array numbers, for strengths up to 6, and alphabet sizes up to 25. A linear recurrence sequence over F q of maximum period with respect to the order of the linear recurrence relation is an m-sequence. Among a large number of applications of m-sequences, there exist constructions for various combinatorial arrays [23], most of them focusing on orthogonal arrays. An orthogonal array of index λ, denoted OA λ (t, k, v), is a λv t k array over an alphabet of size v, with the property that the λv t t sub-array defined by any t columns contains each t-tuple exactly λ times. An orthogonal array is also a covering array, which has the minimum possible number of rows when λ = 1. A straightforward construction of an OA q m 2 (2, (q m 1)/(q 1), q), m 2, follows from well known properties of m-sequences (see [23, Proposition 2]). Using m-sequences corresponding to primitive trinomials of degree m, Munemasa [25] constructs OA 2 m 1(2, k, 2) with k up to 2m + 1, which are close to having strength 3, by exploiting connections among divisibility of polynomials over finite fields, linear codes, and orthogonal arrays. Munemasa s work has been generalized by Dewar et al. [12], who provide OA 2 m 1(3, k, 2) with k up to 2m; Panario et al. [26], whose work implies the construction of OA 3 m 1(3, k, 3) with k up to 3m; and Kim et al. [19], who provide the construction of OA 2 m 1(4, k, 2) with k up to 2m. The first construction that uses m-sequences which yields covering arrays that are not orthogonal arrays was proposed by Raaphorst et al. [28]. They employ m-sequences to construct CA(2q 3 1; 3, q 2 +q +1, q) by exploiting connections between m-sequences, projective geometry and combinatorial designs. This work has been generalized by Tzanakis et al. [32], who give a construction for a CA(l(q t 1) + 1; t, k, q), t, l 2 where k depends on q, t, and l. Finding the maximum k is an optimization problem to which the authors provide an algorithmic solution. In this paper we construct covering arrays from m-sequences combining previous work and ideas from Colbourn [8]. Let q be a prime power, v 2 be a divisor of q 1, ω be a 2

3 primitive element of F q, and set { 0, if i = j; a ij = log ω (j i) mod v, otherwise. Using character theory Colbourn proves that if q > t 2 v 2t, then the q q array a ij, 0 i, j, q 1, is a CA(q; t, q, v). In particular, he provides a character sum which, if large enough, guarantees that the array is a covering array. He then uses a Weil-type argument to give a lower bound for the sum, and comparing that bound with the target value, yields the result. We adapt this method by constructing similar arrays, built using discrete logarithms of selected elements from m-sequences, and finding lower bounds for character sums that imply the covering array property. Using properties of m-sequences, we are able to compute exact values of such sums, without resorting to Weil bounds. This results in constructions of covering arrays of strength 3 and 4, given in Corollaries 2.4 and 2.5, respectively, as well as an infinite family of general strength covering arrays, given in Corollary 2.7, that comes from a connection with linear codes which we also discuss. To the best of our knowledge, this and the construction by Colbourn [8] are the only direct constructions that yield a CA(N; t, k, v) for arbitrary t, k, v. Our construction does not produce new covering arrays for the parameter ranges kept in [10]; however, Table 2 shows a large number of new covering arrays. The structure of this paper is as follows. In Section 2, we give an overview of our results, which consist of a main theorem and constructions for covering arrays that this implies. In Section 3, we give the background on m-sequences and finite field characters that is necessary for the proof of the main theorem, to which Section 4 is dedicated. In Section 5 we conclude with some final remarks, providing a comparison with [8] as well as some experimental results. 2. Main result and covering array constructions Throughout this paper, we use the following notation. For integers a < b, we write {a, a + 1, a + 2,..., b} as [a, b]. We reserve t to be an integer with t 2, q to be a prime power, v 2 to be a divisor of q 1, and α to be a primitive element of F q t. We set k = (q t 1)/(q 1), ω = α k, and we note that ω is a primitive element of F q. Furthermore, we denote Tr to be the trace Tr Fq t/f q of F q t over F q, i.e. Tr(x) = x + x q + x q2 + + x qt 1, x F q t. For integers i and j, we define { 0, if Tr(α i+j ) = 0; M ij (α, t, v) = (1) log ω (Tr(α i+j )) mod v, otherwise. Finally, we use the following notion extensively from now on. Definition 2.1. A set with cardinality n in a vector space with the property that every subset of size t (or smaller) is linearly independent, is an (n, t)-set. 3

4 The main results of our paper are the following theorem and its corollaries; the proof of the main theorem is given in Section 4. Theorem 2.2 (Main theorem). Let q be a prime power, v be a positive integer such that v q 1, t be an integer with t 2, and set k = (q t 1)/(q 1). Let C [0, q t 2] such that {α i i C} is a ( C, t 1)-set and M be the vk C array whose cell indexed by (i, j) [0, vk 1] C is M ij (α, t, v). If then M is a CA (vk; t, C, v). q t 2 2 (q tv) v t 1 (2) Remark 2.3. The bound on q improves with respect to the number of columns as t increases. Indeed, we observe that in Equation (2) it is required that q > tv, so a sufficient condition for Equation (2) to hold is q t 2 2 v t 1, which is equivalent to q v 2+ 6 t 4 }. We conclude that a sufficient condition for Theorem 2.2 is q > max {tv, v 2+ 6 t 4, which means that q needs to be larger than tv. We now focus on special cases of t in the main theorem. It follows from [15, Theorem 5.7] that Tr(α i ) and Tr(α j ) are linearly dependent if and only if i j (mod k). Therefore, setting C = [0, k 1] for t = 3 yields the following result. Corollary 2.4 (Covering arrays of strength 3). Assume the notation from Theorem 2.2 for t = 3 so that k = q 2 + q + 1, and let M be the vk k array whose (i, j)-th element is M ij (α, 3, v), for (i, j) [0, vk 1] [0, k 1]. If q v 4 + 6v 3 + 9v 2 then M is a CA (vk; 3, k, v). We note that the bound in Corollary 2.4 can be improved to q v 4 + 6v 3 using the better but more complicated conditions used in the proofs in Section 4. For the case t = 4, and C = {i(q + 1) i [0, q 2 ]}, it follows from [13, Theorem 3] that {α j j C} is a (q 2, 3)-set. Thus, we have the following result. Corollary 2.5 (Covering arrays of strength 4). Assume the notation from Theorem 2.2 for t = 4 so that k = q 3 +q 2 +q+1, and set C = {i(q + 1) i [0, q 2 ]}. Let M be the vk (q 2 +1) array whose cell indexed by (i, j) [0, vk 1] C is M ij (α, 4, v). If q v 3 + 4v then M is a CA (vk; 4, q 2 + 1, v). There is a connection between linear codes and (n, t)-sets that we use to derive a construction for covering arrays of general strength. A q-ary linear code C of length n, dimension k, and minimum distance d, referred to as an [n, k, d] q code, is a subspace of F n q with dimension k, such that the minimum Hamming distance between distinct elements is d. An (n k) n matrix whose nullspace is C is a parity check matrix for C. Theorem 2.6 (See [11, Theorem 1]). A linear code has parameters [n, k, d] q if and only if the set of columns of its parity check matrix is an (n, d 1)-set in F n k q. 4

5 An [n, n d, d] q code is an almost maximum distance separable (AMDS) code. In the following corollary, we show a construction of covering arrays using AMDS codes. Corollary 2.7 (Covering arrays from AMDS codes). Let q be a prime power, t, n integers with 2 t < n, and C be an [n, n t, t] q code (AMDS) with parity check matrix H = [H] ij, i [0, t 1], j [0, n 1]. Let α be a primitive element of F q t and { ( t 1 ) } C = log α [H] is α i s [0, n 1]. i=0 Let v 2 such that v q 1 and q t 2 2 (q tv) v t 1. Then the vk n array M whose cell indexed by (i, j) [0, vk 1] C is M ij (α, t, v), is a CA(vk; t, n, v). Proof. In view of Theorem 2.2 it is sufficient to show that S = { α i i C } { t 1 } = [H] is α i s [0, n 1] i=0 is a (n, t 1)-set of elements of F q t. Consider the mapping ϕ : F t q F q t such that ϕ(v 0,..., v t 1 ) = t 1 i=0 v iα i. Denoting H s F t q to be the s-th column of H we have that S = {ϕ(h 1 ),..., ϕ(h t )} and, furthermore, by Theorem 2.6 we have that {H 1,..., H t } is a (n, t 1)-set of vectors in F t q. From the fact that ϕ is a vector space isomorphism, it follows that S is also an (n, t 1)-set of F q t. Considering Corollary 2.7, we are interested in AMDS codes of maximum length. Let µ(t 1, q) be the maximum length n for which there exists an [n, n t, t] q code. It is well known that µ(2, q) = q 2 + q + 1 [3] and that µ(3, q) = q for q 2 [3, 27]. Thus the covering arrays in Corollaries 2.4 and 2.5 have the maximum number of columns that can be obtained from our type of construction. However, finding exact values for µ(t 1, q) for t 5 is a difficult problem. Some exact values and upper bounds for the case q 19 exist; see for example [1, 3]. Other than that, we have an infinite class of AMDS codes from elliptic curves. A near maximum distance separable (NMDS) code is an AMDS code whose dual code is also AMDS. The original, rather challenging proof of the following theorem appears in [31]; see [14] for a more accessible proof. Theorem 2.8 ([14, Theorem 1.1]). Let q be a prime power and suppose that there exists an elliptic curve with n rational points over F q. Then, for every k [2, n 1], there exists an [n, k, d] q NMDS code. It is known, see for example [31, Theorem ], that if q = p r then there exists an elliptic curve with N q rational points over F q, where { q + 2 q, if p 2 q and r 3, r odd; N q = q + 2 (3) q + 1, otherwise. 5

6 Hence, from Theorem 2.8 there exists an AMDS [N q, N q t, t] q Corollary 2.7, gives the following result. code which, along with Corollary 2.9. Let q, t, v, k be as in Theorem 2.2, such that q t 2 2 (q tv) v t 1. Then there exists a CA(vk; t, N q, v), with N q as in Equation (3). A small number of AMDS codes for q 19 of length larger than N q are known [1, 3] but the values for q are too small to produce interesting results when used in Corollary Preliminaries and necessary background 3.1. Balance properties of m-sequences Let q be a prime power and α, β F q t, with α primitive, and recall that we denote Tr = Tr Fq t/f q. The sequence Tr(βα i ), i N is a q-ary m-sequence, and it is periodic, with period q t 1 [15, Theorem 4.11]. Such sequences have a rich algebraic structure and many useful properties. The following proposition is a generalization of the well known two-tuple balance property (see [15, Theorem 5.7]). Proposition 3.1. Let n, t be positive integers such that n t, and γ 1,..., γ n F q t be linearly independent over F q. Then, for every b 1,..., b n F q, there exist exactly q t n elements x F q t that satisfy Proof. Consider the mapping (Tr(γ 1 x), Tr(γ 2 x),..., Tr(γ n x)) = (b 1, b 2,..., b n ). (4) ϕ : F q t F n q x (Tr(γ 1 x), Tr(γ 2 x),..., Tr(γ n x)). From the linearity of the trace over F q, it follows that ϕ is a linear transformation between vector spaces over F q. We show that ϕ is surjective; assume by contradiction that this is not the case. Then Im(ϕ) is a proper subspace of F n q, and thus Im(ϕ) {0}. Let a = (a 1,..., a n ) Im(ϕ) \ {0}. Then, for every x F q t, we have that (( n n ) ) 0 = a ϕ(x) = a i Tr(γ i x) = Tr a i γ i x. (5) i=1 We know that Tr(βx) = 0 for all x if and only if β = 0, so Equation (5) implies that n i=1 a iγ i = 0. Since a 0, this contradicts the assumption that γ 1,..., γ n are linearly independent and completes the proof that φ is surjective. It follows that the rank of ϕ is n and, hence, for every (b 1,..., b n ) F n q there exist exactly q t n elements x F q t such that ϕ(x) = (b 1,..., b n ). i=1 6

7 Corollary 3.2. Let n, t be positive integers such that n t, and suppose that {γ 1,..., γ n } F q is a linearly dependent (n, n 1)-set over F t q. Let y 1,..., y n F q, not all them zero, such that y 1 γ y n γ n = 0. Then for every b 1,..., b n F q we have that the number of elements x F q t satisfying (Tr(γ 1 x), Tr(γ 2 x),..., Tr(γ n x)) = (b 1, b 2,..., b n ) (6) is q t n+1 if y 1 b y n b n = 0, and 0 otherwise. Proof. First, we observe that {γ 1,..., γ n } being an (n, n 1)-set, implies that y i 0, for all i [1, t]. In particular, y n 0 and thus we can write γ n = n 1 y i i=1 y n γ i. Now, let b 1,..., b n F q. We have that {γ 1,..., γ n 1 } is linearly independent as a proper subset of an (n, n 1)-set. Then, by Proposition 3.1, there exist exactly q t n+1 elements x F q t so that Tr(γ i x) = b i, for all i [1, t 1]. For any such x, we have that b n = Tr(γ n x) if and only if ( n 1 b n = Tr i=1 y i y n γ i x ) n 1 = which is equivalent to y 1 b y n b n = 0. i=1 y i n 1 y i Tr(γ i x). = b i, y n y i=1 n 3.2. Characters A character of a finite Abelian group G is a homomorphism from G to the multiplicative group C of complex numbers with absolute value 1. The next lemma is a well known fact for characters; see for example [22, Section 5]. Lemma 3.3 (Orthogonality relations for characters). Let ψ be a character of a group G. Then { G if ψ is trivial; ψ(g) = 0 otherwise. g G Furthermore, if g G and Ĝ is the group of all characters of G, we have { G if g = 1 G ; ψ(g) = 0 otherwise. ψ Ĝ In this paper we use characters of the multiplicative group F q; such characters are called multiplicative characters of F q. The set of all multiplicative characters of F q forms a cyclic group of order q 1 under multiplication. That group is generated by the character ψ, defined by ψ(ω j ) = e 2πi q 1 j, where ω is a primitive element of F q. In particular, any multiplicative character is of the form ψ k, where k [0, q 2], and ψ k (ω j ) = e 2πik q 1 j. The character ψ 0 that maps every element to 1 is the trivial multiplicative character of F q. Finally, we extend the domain of a multiplicative character ψ of F q by zero, by setting { 0 if ψ is non-trivial; ψ(0) = 1 otherwise. 7

8 We reserve the letter χ to denote from now on the multiplicative character F q defined by χ(ω j ) = e 2πi v j. We note that the order of χ is v. Corollary 3.4. Let y F q and l be an integer. Then { v 1 χ j (ω v l )χ j v if log (y) = ω (y) l (mod v); 0 otherwise. j=0 Proof. We have χ j (ω v l )χ j (y) = χ j (ω v l )χ j (ω log ω (y) ) = χ j (ω v l+log ω (y) ). Let s = v l + log ω (y) and S = v 1 j=0 χj (ω s ) = v 1 j=0 e 2πi v sj. If s 0 (mod v) then e 2πi v sj = 1 and S = v; otherwise, applying the formula for a finite geometric sum we have that S = 0. Lemma 3.5. Let j 1, j 2,..., j n [1, v 1].Then the mapping X : ( F q) n C defined by X(x 1,..., x n ) = n i=1 χj i (x i ) is a nontrivial character of ( F q) n, and x 1,...,x n F q i=1 n χ j i (x i ) = 0. (7) Proof. It is trivial to prove X is a homomorphism whose order is lcm(ord(χ j 1 ),..., ord(χ jn )), which is greater than 1 since the orders of χ j i, i [1, n] are all greater than 1. Thus X is a non-trivial character, and Equation (7) follows from Lemma 3.3. We close with the definition of Jacobi sums and a theorem that is the cornerstone of our results later on. Definition 3.6. For χ 1,..., χ n multiplicative characters on F q, the Jacobi sum J(χ 1,..., χ n ) over F q is defined by J (χ 1,..., χ n ) = χ 1 (a 1 )... χ n (a n ). a 1,...,a n F q a 1 + +a n=1 Theorem 3.7 ([24, Theorem ]). Let χ 1,..., χ n be characters of F q such that χ 1,..., χ n, and n i=1 χ i are all nontrivial. Then J (χ 1,..., χ n ) = q n Proof of the main theorem The proof of the main theorem relies on two propositions that we give here, namely Propositions 4.1 and 4.4; while the first is fairly straightforward, the latter is longer and also requires some lemmas. As before, we let q be a prime power, v be a positive integer such that v q 1, α be a primitive element of F q t, k = qt 1 q 1, and ω = αk, which is a primitive element of F q. Proposition 4.1. Consider M i,j = M i,j (α, t, v) as defined in Equation (1), and let c 1,..., c t be integers such that {α c 1,..., α ct } is linearly independent. Then, for every l 1,..., l t [0, v 1], there exists r [0, q t 2] such that (M r,c1,..., M r,ct ) = (l 1,..., l t ). 8

9 Proof. Since v q 1, for every l [0, v 1] there exists b F q such that log ω (b) l (mod v). In particular, there exist b 1,..., b t F q such that log ω (b i ) l i (mod v), for all i [1, t]. From Proposition 3.1, and since α is primitive, there exists r [0, q t 2] such that (Tr(α c 1 α r ),..., Tr(α ct α r )) = (b 1,..., b t ). We note that b i 0 for all i [1, t], which means that Tr (α c i α r ) 0. Therefore, (M r,c1,..., M r,ct ) = (log ω (Tr(α r α c 1 ) mod v,..., log ω (Tr(α r α ct ) mod v) which completes the proof. = (log ω (b 1 ) mod v,..., log ω (b t ) mod v) = (l 1,..., l t ), In Proposition 4.4 we establish a result similar to Proposition 4.1, for the case when α c 1,..., α ct are linearly dependent. To prove Proposition 4.4 we need a couple of auxiliary lemmas. For l 1,..., l t [0, v 1], integers c 1,..., c t, and r [0, q t 2], we define ( t v 1 ) h(r) = χ j (ω v l i )χ j (Tr(α c i α r )), and i=1 j=0 N r = {i i [1, t], Tr(α c i α r ) 0}. Lemma 4.2. We have that { v Nr if log h(r) = ω (Tr(α c i α r )) l i (mod v), for all i N r ; 0 otherwise. In particular, if N r = then h(r) = 1. Proof. We show that v 1 v if Tr(α c i α r ) 0 and log ω (Tr(α c i α r )) l i (mod v); χ j (ω v l i )χ j (Tr(α c i α r )) = 0 if Tr(α j=0 c i α r ) 0 and log ω (Tr(α c i α r )) l i (mod v); 1 if Tr(α c i α r ) = 0. (8) In fact, the first two cases are a straightforward application of Corollary 3.4. For the case when Tr(α c i α r ) = 0, we recall that { χ j 0 if j 0 (mod v); (0) = 1 otherwise, and thus we have v 1 j=0 χj (ω v l i )χ j (0) = χ 0 (ω v l i )χ 0 (0) = 1. From Equation (8) we observe that all the product terms in h(r) corresponding to i N r are equal to 1. Therefore, the only terms contributing to the product correspond to i N r, and each of those is equal to v or 0, according to Equation (8). 9

10 Lemma 4.3. Consider M i,j = M i,j (α, t, v) as defined in Equation (1), and let c 1,..., c t be integers such that {α c 1,..., α ct } is a linearly dependent (t, t 1)-set. Let l 1,..., l t [0, v 1], and set σ = h(r). We have that if σ tvq t 1 1 (9) then there exists r [0, q t 2] such that (M r,c1,..., M r,ct ) = (l 1,..., l t ). Proof. For every n [0, t] we denote σ n = N r =n h(r), so we have that σ = t n=0 σ n. We observe that for all r such that N r = t, we have that N r = [1, t] and thus Tr(α c i α r ) 0 for all i [1, t]. Hence, by Lemma 4.2, we have that h(r) = v t or h(r) = 0. Therefore, if σ t > 0, then there exists an r such that N r = t and h(r) = v t ; by Lemma 4.2, it follows that, for this r, we have (M r,c1,..., M r,ct ) = (l 1,..., l t ). In conclusion, to complete the proof it is sufficient to show that if Equation (9) holds, then σ t > 0. We begin by evaluating σ 0. For every r such that N r = 0, we have that N r = and thus h(r) = 1, by Lemma 4.2. This implies that σ 0 is equal to the number of r such that N r = or, equivalently, the number of r such that (Tr(α c 1 α r ),..., Tr(α ct α r )) = (0,..., 0). According to Corollary 3.2, the number of x F q t such that (Tr(α c 1 x),..., Tr(α ct x)) = (0,..., 0) is q, an obvious solution being x = 0. Since α is a primitive element of F q t, we have that α r runs through all the elements of F q t except zero as r runs through [0, q t 2], including all the q 1 nonzero solutions. Hence, we have that there exist exactly q 1 such r and σ 0 = q 1. Next we examine σ n for n [1, t]. We rewrite σ n as σ n = N r =n h(r) = I [1,t] I =n N r=i h(r). (10) We focus on the inner sum of the right hand side; for clarity, we assume without loss of generality that I = [1, n]. Then we have that N r=i h(r) = Tr(α c iα r ) 0, all i I Tr(α c iα r )=0, all i I h(r) = b 1,...,b n F q Tr(α c iα r )=b i, all i I Tr(α c iα r )=0, all i I h(r). Since {α c 1,..., α ct } is a linearly dependent (t, t 1)-set, there exist y i F q such that t i=1 y iα c i = 0. Setting b i = 0 for i [n + 1, t], we have from Corollary 3.2 that there are no 10

11 r satisfying the inner sum, unless t i=1 y ib i = n i=1 y ib i = 0. This does not hold when n = 1, since y 1 and b 1 are both nonzero, and thus σ 1 = 0. We continue assuming that n [2, t]. In that case, we have that N r=i h(r) = b 1,...,b n F q n i=1 y ib i =0 We can break the sum on the right hand side as b 1,...,b n F q n i=1 y ib i =0 and log ω (b i ) l i (mod v) for all i [1,n] Tr(α c iα r )=b i, all i I Tr(α c iα r )=0, all i I h(r) + Tr(α c iα r )=b i, all i I Tr(α c iα r )=0, all i I b 1,...,b n F q n i=1 y ib i =0 and log ω (b i ) l i (mod v) for some i [1,n] h(r). Tr(α c iα r )=b i, all i I Tr(α c iα r )=0, all i I h(r). From Lemma 4.2, we have that h(r) = v n for all r in the first double sum, and h(r) = 0 for all r in the second double sum. Therefore N r=i h(r) = b 1,...,b n F q n i=1 y ib i =0 and log ω (b i ) l i (mod v) for all i [1,n] Tr(α c iα r )=b i, all i I Tr(α c iα r )=0, all i I It follows by Corollary 3.2, that there exist exactly q elements r satisfying the condition of the inner sum, thus N r=i h(r) = b 1,...,b n F q n i=1 y ib i =0 and log ω (b i ) l i (mod v) for all i [1,n] Now, since v q 1, for every l [0, v 1] there exist q 1 choices of elements b F v q such that log ω (b) l (mod v). Thus there exist ( ) q 1 n 1 v choices of (n 1)-tuples (b1,..., b n 1 ) ( ) F n 1, q such that logω (b i ) l i (mod v) for all i [1, n 1]. For every such choice of b 1,..., b n 1, let b n F q such that n i=1 y ib i = 0 or equivalently b n = n 1 y i i=1 y n b i. Then we ( have that log ω (b n ) l n (mod v) if and only if l n log ω ) n 1 y i i=1 y n b i (mod v), which does not necessarily hold for all b 1,..., b n such that the sum runs over. In conclusion, there are at most ( ) q 1 n 1 n-tuples (b1,..., b n ) satisfying the conditions of the last sum, and therefore v we have that N r=i qv n. ( ) n 1 q 1 h(r) qv n. v 11 v n.

12 From this and Equation (10) we conclude that, for all n [2, t], we have σ n I [1,t] I =n ( ) n 1 q 1 qv n = Now, solving for σ t in σ = t n=0 σ n, yields Thus, σ t > 0 if v t 1 σ t = σ σ 0 σ 1 = σ (q 1) 0 σ n n=2 t 1 σ n n=2 n=2 ( ) t (q 1) n 1 qv. n t 1 ( ) t σ (q 1) (q 1) n 1 qv. n t 1 ( ) t σ > q 1 + (q 1) n 1 qv n n=2 ( ) q t 1 = q 1 + qv q 1 (q 1)t 1 t. ( ) We have that tvq t 1 q 1 > q 1 + qv t 1 (q q 1 1)t 1 t, so a less complicated sufficient condition for σ t to be positive is σ tvq t 1 1 as given in Equation (9). From the discussion in the beginning of this proof, this completes the proof. We now have everything we need in order to prove the analogue of Proposition 4.1 for the case when {α c 1,..., α ct } is a linearly dependent (t, t 1)-set. Proposition 4.4. Let M i,j = M i,j (α, t, v) as defined in Equation (1), c 1,..., c t be integers such that {α c 1,..., α ct } is a linearly dependent (t, t 1)-set, and l 1,..., l t [0, v 1]. We have that if q t 2 2 (q tv) > v t 1, (11) then there exists r [0, q t 2] such that (M r,c1,..., M r,ct ) = (l 1,..., l t ). Proof. For convenience, from now on we write ω i instead of ω v l i. To prove our proposition, it is sufficient to show that if Equation (11) holds, then Equation (9) holds as well; the proof then is complete by Lemma

13 We rewrite σ as follows. σ = h(r) = = t ( v 1 ) χ j (ω i )χ j (Tr(α c i α r )) j=0 ( ) t v χ j (ω i )χ j (Tr(α c i α r )) = q t 1 + = q t 1 + = q t 1 + I [1,t] I t n=1 t n=1 j=1 v 1 χ j (ω i ) χ j (Tr(α c i α r )) {i 1,...,i n} [1,t] {i 1,...,i n} [1,t] i I j=1 j 1,...,j n [1,v 1] n j 1,...,j n [1,v 1] n (ω ik ) (ω ik ) n (Tr (α c i k α r )) n (Tr (α c i k α r )). (12) We first examine the terms of the sum in Equation (12) corresponding to n < t. For such n, we have that {α c i 1,..., α c in } is linearly independent, as a proper subset of the (t, t 1)-set {α c 1,..., α ct }. Therefore, from Proposition 3.1 we have that, for every b 1,..., b n F q, there exist exactly q t n values of r [0, q t 2] such that (Tr(α c i 1 α r ),..., Tr(α c in α r )) = (b 1,..., b n ). Considering also Lemma 3.5 we conclude that, for all n < t we have n (Tr(α c i k α r )) = q t n b 1,...,b n F q n (b k ) = 0. (13) Equation (13) shows that the terms corresponding to n < t of the sum in Equation (12) vanish, and thus σ q t + 1 = t j 1,...,j t [1,v 1] (ω k ) t (Tr (α c k α r )). Now, since {α c 1,..., α ct } is a linearly dependent set, there exist y 1,..., y t F q, not all zero, such that t y kα c k = 0. Since it is also a (t, t 1)-set, we observe that moreover none of y 1,..., y t can be zero. Multiplying the equation by α r we have t y kα c k α r = 0, and applying the trace, which is linear over F q, yields t y ktr(α c k α r ) = 0. Solving for the term corresponding to k = t, we get t 1 Tr(α ct α r y k ) = Tr(α c k α r ). y t 13

14 Hence, σ q t + 1 = t j 1,...,j t [1,v 1] = t j 1,...,j t [1,v 1] (ω k ) χ jt (Tr (α ct α r )) (ω k ) χ ( jt t 1 t 1 (Tr (α c k α r )) y k y t Tr (α c k α r ) ) t 1 (Tr (α c k α r )). We note that {α c 1,..., α c t 1 } is linearly independent as a proper subset of the (t, t 1)-set {α c 1,..., α ct }, so it follows from Proposition 3.1 that, for every b 1,..., b t 1 F q, there exist exactly q values of r [0, q t 2] such that Thus, we have that (Tr(α c 1 α r ),..., Tr(α c t 1 α r )) = (b 1,..., b t 1 ). σ q t + 1 = j 1,...,j t [1,v 1] = j 1,...,j t [1,v 1] t (ω k ) q t χ jk (ω k ) q z F q b 1,...,b t 1 F q χ jt (z) t 1 y χ ( jt k b k y t b 1,...,b t 1 F q z= t 1 y k yt b k ) t 1 (b k ) t 1 (b k ). We normalize the last sum by substituting b k with zyt y k b k, which yields σ q t + 1 = j 1,...,j t [1,v 1] t χ jk (ω k ) q z F q χ jt (z) Since the characters are multiplicative, we have that t 1 ( zy ) t 1 t b k = y k (z) t 1 b 1,...,b t 1 F q b 1 + +b t 1 =1 ( yt t 1 = χ j 1+ +j t 1 (z) y k ) (b k ) ( yt y k ) (b k ). ( zy ) t b k. y k 14

15 Considering Definition 3.6, we rewrite our equation as σ q t + 1 = q t 1 ( ) yt ω χ jt k (ω t ) χ j 1+ +j t (z) j 1,...,j t [1,v 1] = q j 1,...,j t [1,v 1] t 1 χ jt (ω t ) y k z F q b 1,...,b t 1 F q b 1 + +b t 1 =1 t 1 (b k ) ( ) yt ω k χ j 1+ +j t (z)j (χ j 1,..., χ j t 1 ). Since χ is a character of order v, from Lemma 3.3 we have that { χ j 1+ +j t q 1 if j j t 0 (mod v); (z) = 0 otherwise. z F q Therefore our equation becomes σ q t + 1 = q(q 1) j 1,...,j t [1,v 1] j 1 + +j t 0 (mod v) y k t 1 χ jt (ω t ) z F q ( yt ω k y k ) J (χ j 1,..., χ j t 1 ). (14) For all 1 j 1,..., j t v 1 such that j 1 + +j t 0 (mod v), we have j 1 + +j t 1 j t 0 (mod v), since j t [1, v 1]. Hence t i=1 χj i is a nontrivial character, and by Theorem 3.7 we have that J (χ j 1,..., χ j t 1 ) = q t 2 1. Furthermore, characters are absolutely bounded by 1, thus t 1 χjt (ω t ) ( yt ω k and applying absolute values to Equation (14) yields σ q t + 1 q(q 1) y k ) 1, j 1,...,j n [1,v 1] j 1 + +j n 0 (mod v) q t 2 1. Using inclusion-exclusion we deduce that the number of j 1,..., j t satisfying the conditions of the sum above is t 1 S = ( 1) t n 1 (v 1) n. Thus n=1 σ q t + 1 q(q 1)q t 2 1 S t 1 q t 2 (q 1) ( 1) t n 1 (v 1) n n=1 = q t (v 1) t + ( 1) t (v 1) 2 (q 1), v 15

16 which implies that σ q t 1 q t (v 1) t + ( 1) t (v 1) 2 (q 1). (15) v From the lower bound for σ in Equation (15), it follows that Equation (9) is satisfied if q t 1 q t 2 (q 1) (v 1) t + ( 1) t (v 1) v tvq t 1 1. Using the fact that v t 1 (v 1)t +( 1) t (v 1), we get that if q t 2 1 (q tv) v t 1, then Equation (9) is v satisfied. In the proof of Proposition 4.4 we take the absolute value of the Jacobi sums, and use Theorem 3.7 to evaluate them and then invoke the triangle inequality to derive bounds on σ. When t is small, the exact theoretical values of Jacobi sums are known in some cases [2]. Additionally, for any particular set of parameters, Jacobi sums can be directly computed [30]. Using these facts offers a more exact computation of σ. In Table 1, we compare these two methods for a sample of values of v, t and q. The column σ min is the worst case (smallest) value of σ taken over all possible t-sets of columns and all t-tuples. The column RHS of Eq. 15 is the theoretical lower bound on σ in the right hand side of Equation (15). The last column is the value that σ must exceed to guarantee a covering array from Lemma 4.3. From the table we can see, as expected, that using explicit values for Jacobi sums is always at least as good as bounding with absolute values and additionally the rows marked with a star show that these entries construct covering arrays which are not found by Proposition 4.4. We remark that the sharp bounds on σ obtained by exact values of Jacobi sums do not change the asymptotic behaviour of the construction. The next final lemma is used to show why the array in Theorem 2.2 can be truncated to vk rows, instead of q t 1. Lemma 4.5. Let r, r [0, ] so that r r (mod vk). Then M r,j (α, t, v) = M r,j(α, t, v), for all j. Proof. Suppose that r = r + nvk for some integer n. Then Tr(α r +j ) = Tr (( α k) nv α r+j ) = Tr(ω nv α r+j ). We recall that ω is a primitive element in F q, so ω nv F q and from the linearity of the trace over F q we have Tr(α r +j ) = ω nv Tr(α r+j ). (16) Hence Tr(α r +j ) = 0 if and only if Tr(α r+j ) = 0. (17) If Tr(α r +j ) and Tr(α r+j ) are nonzero, then by Equation (16) we have that ) ( log ω (Tr(α r +j ) mod v = log ω ω nv Tr(α r+j ) ) mod v = ( nv + log ω ( Tr(α r+j ) )) mod v = log ω ( Tr(α r+j ) ) mod v. (18) From Equations (17) and (18), we conclude that M r,j (α, t, v) = M r,j(α, t, v). 16

17 v t q σ min RHS of Eq. 15 tvq t * * * * Table 1: Comparison for a sample of values of v, t and q of the theoretical lower bound for σ in the right hand side of Equation (15), with the sharp lower bound σ min obtained using explicit values of Jacobi sums. Stars indicate parameters for which the sharp bound guarantees a covering array in Lemma 4.3, whereas the theoretical bound does not. We now put together the results of this section, to prove our main theorem. Proof of Theorem 2.2. Let C [0, ] such that {α c c C} is a ( C, t 1)-set. Consider the vk C array M with elements M ij = M ij (α, t, v), as in Equation (1), for (i, j) [0, vk 1] C. We want to prove that if Equation (2) holds, then M is a CA (vk; t, C, v). Let c 1,..., c t C, and l 1,..., l t [0, v 1]. It follows from Propositions 4.1 and 4.4 that there exists r [0, q t 2] such that (M r,c 1,..., M r,c t ) = (l 1,..., l t ). Let r [0, vk 1] such that r r (mod vk); then, by Lemma 4.5, we have that which completes the proof. (M r,c1,..., M r,ct ) = (M r,c 1,..., M r,c t ) = (l 1,..., l t ), 17

18 5. Evaluating our construction 5.1. Comparison with covering arrays from cyclotomy Colbourn [8] constructs covering arrays using cyclotomic generators over finite fields. These are CA (q; t, q, v) for t, q, v satisfying v q 1 and q > t 2 v 2t, and are referred to as covering arrays from cyclotomy. The techniques used to provide covering arrays from cyclotomy and covering arrays from m-sequences are similar and, furthermore, they are the only direct constructions that provide a CA(N; t, k, v) for any arbitrary t, k and v. The main similarity of the two constructions is that they start with a function whose values are uniformly distributed over the field, and take the logarithm. Furthermore, in both constructions, the conditions that guarantee the covering array property rely on the evaluation of similar character sums. On the other hand, the resulting arrays have different dimensions and existence conditions. In what follows, we consider v and t fixed and analyze when m-sequences yield better covering arrays than covering arrays from cyclotomy. Let k t,v (q) and N t,v (q) be the number of columns and number of rows in the m-sequence construction, respectively, that is We define q 2 + q + 1 if t = 3, k t,v (q) = q if t = 4, q + 2q if t 5, and N t,v (q) = v qt 1 q 1. q min (t, v) = min {q a CA(N t,v (q); t, k t,v (q), v) can be constructed from m-sequences}, q cycl (t, v) = min {q a CA(q; t, q, v) can be constructed from cyclotomy}. If N t,v (q min (t, v)) < q cycl (t, v), we further define q max (t, v) = max {q q q min (t, v) and N t,v (q) < q cycl (t, v)}. and note that for all prime powers q such that q min (t, v) q q max (t, v), covering arrays from m-sequences have fewer rows than covering arrays from cyclotomy for the same number of columns. In Figure 1, we give the graph of N as a function of k for the CA(N; t, k, v) that can be constructed from cyclotomy and m-sequences. The horizontal doted lines are justified since when we remove some columns from a covering array we obtain another covering array. In Table 2, we list for 2 v 25 and 2 t 10, the nonempty intervals q min (v, t) q q max (v, t), q a prime power, where a covering array from m-sequences yields fewer rows than the corresponding covering array from cyclotomy. We note that both in this work and in [8], the conditions that guarantee the existence of the covering arrays can be replaced with slightly better (weaker) but more complicated conditions found in the proofs. The compilation of Table 2 was done by taking into account these better conditions for both constructions. 18

19 v t q min q max v t q min q max v t q min q max Table 2: A sample of values of v, t, and the corresponding q min = q min (v, t) and q max = q max (v, t). For any prime power q such that q v 1 and q min q q max, covering arrays from m-sequences can be constructed with fewer rows than covering arrays from cyclotomy with the same number of columns. 19

20 Figure 1: Comparison of the size of covering arrays from m-sequences and cyclotomy, for fixed t and v Experimental results Equation (2) gives a theoretical lower bound on q to guarantee the existence of a covering array. Values of q smaller than this theoretical bound can yield covering arrays. To test this, we created the arrays described in Corollaries 2.4 and 2.5 for values of q and v, v q 1, for which experiments were computationally feasible and checked if they were covering arrays. This was often true for values of q smaller than the required condition, as demonstrated in Tables 3 and 4. As before, the theoretical guarantee is calculated using the weaker, more complicated conditions for q that follow from the proofs in Section 4, rather than the conditions in Corollaries 2.4 and 2.5. Successful q below Values of q Theoretical v theoretical guarantee tested guarantee 2 None q < 7 q 7 3 None q < 19 q 19 4 q 37, except 41 q < 61 q 61 5 q 81, except 101 q < 181 q q 103, except 109, 139, 157, 223, 277 q < 439 q q 169 q 379 q , 241, 281, 313 q 337 q q 523 q q 491 q , 397 q 463 q None q 457 q Table 3: Ranges for q such that v q 1 and q being a prime power is sufficient for the construction of a CA(v(q 2 + q + 1); 3, q 2 + q + 1, v) as per Corollary

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