Optimal Frequency Hopping Sequences: A ... - Semantic Scholar

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 0, NO. 00, JUNE 2004

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Optimal Frequency Hopping Sequences: A Combinatorial Approach Ryoh Fuji-Hara, Ying Miao and Miwako Mishima

Abstract— Frequency hopping multiple access (FHMA) spread spectrum communication systems employing multiple frequency shift keying (MFSK) as data modulation technique are investigated from a combinatorial approach. A correspondence between optimal frequency hopping (FH) sequences and partition type difference packings is first established. By virtue of this correspondence, FHMA systems with a single optimal FH sequence each are constructed from various combinatorial structures such as affine geometries, cyclic designs and difference families. Combinatorial recursive constructions are also presented. Many new infinite series of optimal FH sequences are thus obtained. These new FH sequences are also useful in ultra wide band (UWB) communication systems.

collisions of frequencies or hits of frequencies) occur, then a part of the messages may be corrupted. Therefore we want to use FH sequences which give rise to as few hits of frequencies as possible. Let X (v; F ) be the set of all sequences of length v over a frequency library F with |F | = m. In order to measure hits of frequencies, we define the Hamming correlation HXY between two sequences X, Y ∈ X (v; F ) to be

Index Terms— Affine geometries, cyclic designs, difference families, difference matrices, difference packings, frequency hopping (FH) sequences, partition type, resolvable designs, 1rotational designs, ultra wide band (UWB).

where

I. I NTRODUCTION

I

N this paper, we are concerned with frequency hopping multiple access (FHMA) spread spectrum communication systems, employing multiple frequency shift keying (MFSK) as data modulation technique. Detailed description for such a system can be found in, for example, [19] and [33]. Let F = {f0 , . . . , fm−1 } be a set of frequencies called a frequency library and X = (x0 , . . . , xv−1 ), where xi ∈ F , be a sequence of frequencies called a frequency hopping (FH) sequence of length v over F . In an FHMA system, each sender transmits a message along with switching frequencies in every time slot according to an FH sequence X provided to him. In practice, the switching occurs very frequently, say, 1600 times per second in ‘Bluetooth’ wireless technology (for the detailed technical information about Bluetooth, see [6]). FH sequences are used iteratively, i.e., they appear as . . . , xv−2 , xv−1 , x0 , x1 , . . . . The corresponding receiver then dehops the received signals using the same hopping pattern. Suppose that a second sender wants to transmit messages over the same frequency library using another FH sequence Y = (y0 , . . . , yv−1 ) starting at some slot t. Then it may happen that the two senders transmit messages on the same frequency in several slots. If such signal interferences (called This work was supported in part by JSPS under Grant-in-Aid for Scientific Research (B)13558046 and (C)14540100 for the first and the second authors, and by Ministry of Education, Culture, Sports, Science and Technology of Japan under Grant-in-Aid for Young Scientists (B)1570023 for the third author, respectively. R. Fuji-Hara and Y. Miao are with the Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan. E-mail: [email protected]; [email protected]. M. Mishima is with the Information and Multimedia Center, Gifu University, 1-1 Yanagido, Gifu 501-1193, Japan. E-mail: [email protected].

HXY (t) =

v−1 X

h(xi , yi+t ),

0 ≤ t < v,

i=0

½ h(x, y) =

1, 0,

if x = y, otherwise

and all operations among the position-indices are performed modulo v. When X = Y , the Hamming correlation HXY coincides with the usual concept of auto-correlation HXX . Suppose that the FHMA system under consideration has only one FH sequence X ∈ X (v; F ), which is used by two senders for transmission. Define H(X) = max {HXX (t)}. 1≤t m, if H(X) = k, then the sequence X is optimal. Since the case v ≤ m is trivial, we assume that v > m throughout this paper. Our objective is to construct as many FH sequences attaining the above bound as possible. Let FHS(v, m, λ) with λ = H(X) denote an FH sequence X of length v over a frequency library of size m. Most of the known FH sequences are derived from sequence techniques like m-sequences over finite fields or rings (see, for example, [20], [26], [28], [29], [35] and [36]). Table 1.1 is a list of known series of optimal FHS(v, m, λ) (these series are also listed in [36]), where p is a prime number and t, n, and c are integers. In this paper, we investigate FHMA systems with a single FH sequence each from the standpoint of combinatorial designs. In Section II, a correspondence between FH sequences and combinatorial structures called partition type difference packings is established. This correspondence reveals that in order to construct optimal FH sequences, we need only construct their corresponding difference packings. In Section III, it is shown that a geometrical construction can produce optimal FH sequences with the same parameters as most of those listed in Table 1.1. Sections IV and V are devoted to direct algebraic constructions, by which many new optimal FH sequences are produced. These new optimal FH sequences can be used as ingredients in our combinatorial recursive constructions presented in Section VI. Many new infinite series of optimal FH sequences are thus obtained due to this completely new approach. These new series of optimal FH sequences are listed in Table 1.2, where p, p1 and p2 are all prime and g(p2 ) is an integer defined by (4.3). What is remarkable is that these series, except the series No.13, do not require v − 1 to be a prime power. This is totally different from the previously known series (listed in Table 1.1). II. C OMBINATORIAL C HARACTERIZATIONS A convenient way of viewing FH sequences is from a settheoretic perspective. For simplicity, we hereafter write just F = {0, . . . , m−1} for a frequency library. An FHS(v, m, λ), X = (x0 , . . . , xv−1 ), over a frequency library F can be interpreted as a family of m sets B0 , . . . , Bm−1 such that each set Bi corresponds to frequency i ∈ F and the elements in each set Bi specify the position-indices in the FH sequence X at which frequency i appears. For example, X = (x0 , x1 , x2 , x3 , x4 , x5 , x6 ) = (0, 0, 0, 1, 0, 1, 1) is an

Lemma 2.1: There exists an FHS(v, m, λ), X, if and only if there exists a partition P = {B0 , . . . , Bm−1 } of Zv such that m−1 X H(X) = max { |Bi ∩ (Bi + t)|} = λ 1≤t 0, the inequality λ ≥ k must hold, which implies that a partition type m-DP(v, K, λ) with λ = k corresponds to an optimal FHS(v, m, k). Lemma 2.4: Let v = km + m − 1 with k ≥ 1. Then there exists an optimal FHS(v, m, k) if and only if there exists a partition type (v, {k, k+1}, k)-difference family in which m− 1 blocks are of size k + 1 and the remaining one is of size k. Proof: The sufficiency follows directly from Theorem 2.3 with ² = m − 1. Now we consider its necessity. Let ni be the number of blocks of size i in the corresponding mDP(v, K, k) of partition type, where i is an integer such that 1 ≤ i ≤ v. Then we know that v X

ni = m,

i=1

v X

ni i = v,

i=1

and for the number of differences arising from the blocks of the partition type m-DP(v, K, k), v X

Pv

ni i(i − 1) ≤ k(v − 1).

i=1

Meanwhile, i=1 ni i(i − 1) ≥ k(v − 1) P follows from (2.1) v with ² = m − 1. Therefore the values of i=1 ni i(i − 1) is

If we take m = 2 in Lemma 2.4, then we know that there exists an optimal FHS(2k +1, 2, k) if and only if there exists a partition type (2k+1, {k, k+1}, k)-difference family in which one block is of size k +1 and the other is of size k. For odd k, such a partition type (2k + 1, {k, k + 1}, k)-difference family can be constructed from a (2k + 1, k, (k − 1)/2)-difference family and its complement (2k+1, k+1, (k+1)/2)-difference family over Z2k+1 . These two difference families are closely related to Hadamard matrices of order 2k+2 (see, for example, [16]). Considering the set-theoretic interpretation of FH sequences in Lemmas 2.1, 2.4 and Theorem 2.3, many of the constructions for FH sequences are expected to be design-theoretic in nature. Instead of constructing optimal FH sequences directly, we try to make use of partition type difference packings to obtain optimal FH sequences. In the remainder of this paper, it is shown that this approach is quite effective. III. G EOMETRICAL C ONSTRUCTION We first explain several terms related to combinatorial designs needed in this section. For a set V of v elements (points) and a collection B of k-subsets (blocks) of V , a pair (V, B) is called a balanced incomplete block design (BIB design for short) or 2-design, and is denoted by B(v, k, λ), if every pair of distinct points of V is contained in precisely λ blocks of B. A BIB design with λ = 1 is called a Steiner 2-design in particular. These combinatorial designs are originally introduced in 19th century. They have blossomed into an area of mathematics, called combinatorial design theory, which is now a useful mathematical platform for coding theory, experimental designs, cryptography and many other areas. The books [37] by van Lint and Wilson and [4] by Beth, Jungnickel and Lenz give good historical and theoretical backgrounds. For a BIB design (V, B), let τ be a permutation on V . For any block B = {b1 , . . . , bk } ∈ B, define τ (B) = {τ (b1 ), . . . , τ (bk )}. If τ (B) = {τ (B) : B ∈ B} = B, then τ is called an automorphism of the BIB design (V, B). If there exists an automorphism τ consisting of a single fixed point and one cycle of length v − 1, then the design is said to be 1-rotational. In this case the point-set V can be identified with Zv−1 ∪ {∞} in such a way that the 1rotational BIB design (V, B) has an automorphism τ : i 7−→ i + 1 (mod v − 1) and ∞ 7−→ ∞. The orbit containing a block B ∈ B is defined to be the set of the following distinct blocks τ i (B) = B+i = {b1 +i (mod v − 1), . . . , bk +i (mod v − 1)} for i ∈ Zv−1 , where ∞+i = i+∞ = ∞. The orbit containing B is also called the development of B. The length of an orbit is its cardinality. If the length of an orbit is v − 1, then the

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orbit is said to be full, otherwise short. An orbit that contains a block either of form ¾ ½ v−1 v−1 ∞, 0, , . . . , (k − 2) or k−1 k−1 ½ ¾ v−1 v−1 0, , . . . , (k − 1) k k (thus the length of the orbit is (v − 1)/(k − 1) or (v − 1)/k, respectively) is called a regular short orbit if it exists. A block chosen arbitrarily from each orbit is called a base block. A BIB design (V, B) is said to be resolvable if the collection B of blocks can be partitioned into classes R1 , . . . , Rr such that every point of V is contained in exactly one block in each class. The classes Ri are called resolution classes and R = {R1 , . . . , Rr } is called a resolution. Let q be a prime power, n a positive integer and let Vn (q) denote the n-dimensional vector space over GF(q). When T is a t-dimensional linear subspace of Vn (q), a coset (called a t-flat) of T is a set of the form T + y for any vector y ∈ Vn (q). 0-Flats, 1-flats and (n − 1)-flats are called points, lines and hyperplanes, respectively. A system consisting of all the vectors (points), all the t-flats of Vn (q) and their incidence relation is called an affine geometry, denoted by AG(n, q), and it is well known (see, for example, [3], [4] and [5]) that the set Bt of all t-flats in AG(n, q) forms the set of blocks of a BIB design. For a t-flat T of AG(n, q), we define a parallel class Ct (T ) containing T as the set of all the q n−t distinct cosets (t-flats) of T , i.e., Ct (T ) = {T, T + y 1 , . . . , T + y qn−t −1 } for suitable y i ∈ Vn (q), 1 ≤ i ≤ q n−t − 1. The set of tflats is partitionable into parallel classes and the partition is a resolution. On the other hand, let T be an additive subgroup of order q t , where 1 ≤ t < n, in GF(q n ), and let Ct (T ) denote the set of all additive cosets of T in GF(q n ). In a geometrical sense, T can be considered as a t-flat containing the origin 0 in AG(n, q) and Ct (T ) is the parallel class of t-flats containing T . Since the elements of GF(q n ) can be represented by ω ∞ (= 0), ω 0 , n . . . , ω q −2 for a primitive element ω of GF(q n ), there exists a one-to-one correspondence between the point-set of AG(n, q) and Zqn −1 ∪ {∞}, and in this case, the mapping τ : i 7−→ i + 1 (mod q n − 1) and ∞ 7−→ ∞ on Zqn −1 ∪ {∞} is an automorphism of AG(n, q). So, in what follows, we identify the point-set of AG(n, q) with Zqn −1 ∪ {∞}. Let Bt be the set of all t-flats in AG(n, q) and Bt∗ the set of t-flats passing through the origin ∞. Clearly Bt can be partitioned into parallel classes each of which contains exactly one t-flat passing through the origin and q n−t − 1 t-flats not passing through the origin. It is well known (see, for example, [3], [4] and [5]) that the numbers of the elements of Bt and Bt∗ are q n−t [ nt ]q and [ nt ]q , respectively, where [ nt ]q is the Gaussian coefficient defined by  n (q − 1)(q n−1 − 1) · · · (q n−t+1 − 1)   · ¸ ,  n (q t − 1)(q t−1 − 1) · · · (q − 1) = t q  if 1 ≤ t ≤ n,   1, if t = 0. It is also known that by taking 0-flats as points of V = Zqn −1 ∪ {∞} and t-flats as blocks, (V, Bt ) is a resolvable 1-rotational

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£ ¤ B(v, k, λ) with v = q n , k = q t and λ = n−1 t−1 q . The lines of AG(n, q) consist of a single short orbit of length (q n −1)/(q − 1) and (q n−1 − 1)/(q − 1) full orbits of length q n − 1. Any parallel class of lines contains one line from the unique short orbit and q − 1 lines from each of the (q n−1 − 1)/(q − 1) full orbits. The basic facts on AG(n, q) mentioned above immediately imply the following result. Lemma 3.1: A parallel class of lines in AG(n, q) forms a partition type (q n , q, q − 1)-difference family over Zqn −1 ∪ {∞}. We can generalize Lemma 3.1 to work for a parallel class of t-flats. The development of lines in Lemma 3.1 contains all lines of AG(n, q). The following lemma shows that a parallel class of t-flats forms a difference family. However, when 2 ≤ t ≤ n − 2, the developments of the t-flats do not generate all t-flats of AG(n, q). The following result might be known [23], but unfortunately we could not find the original in the literature. So, for the aid of the reader’s understanding, we give a sketch of its proof. Lemma 3.2: A parallel class of t-flats in AG(n, q) forms a partition type (q n , q t , q t − 1)-difference family over Zqn −1 ∪ {∞} for any t, 1 ≤ t < n. Proof: We prove this lemma by induction. When t = 1, see Lemma 3.1. We consider the case t ≥ 2. Let Ct−1 be a parallel class of (t − 1)-flats in AG(n, q). We assume that it is a partition type (q n , q t−1 , q t−1 − 1)-difference family with q n−t+1 blocks. Take a parallel class Ct of t-flats, where each t-flat contains q (t − 1)-flats of Ct−1 . Let us count the number of ordered pairs of points i and j in Ct such that d ≡ i−j (mod q n − 1) for each difference d, 1 ≤ d ≤ q n −2, and d = ±∞. There are two cases need to be considered: 1) i and j are both in a (t − 1)-flat of Ct−1 ; 2) i and j are in different (t − 1)-flats of Ct−1 but in an identical t-flat of Ct . For the case 1, it is counted in Ct−1 . For the case 2, consider the lines that are contained in a t-flat of Ct and transverse q parallel (t − 1)-flats of Ct−1 . The set of such lines consists of q t−1 distinct parallel classes of lines in AG(n, q), which implies that this set of lines involves q t−1 distinct (q n , q, q−1)difference families. Then, for each difference d, the number of pairs (i, j) satisfying the required condition is λ = (q t−1 − 1) + q t−1 (q − 1) = q t − 1 in total. By deleting the point ∞ from the point-set of AG(n, q), for any t, 1 ≤ t < n, a parallel class becomes a partition type (q n − 1, {q t − 1, q t }, q t − 1)-difference family with q n−t blocks over Zqn −1 , where all blocks are of size q t except the shortened one, which is of size q t − 1. This enables us to state the following theorem. Theorem 3.3: There exists an optimal FHS(pcn − 1, pc(n−t) , pct − 1) for any prime number p, 1 ≤ t < n and 1 ≤ c.

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Note that Theorem 3.3 covers all of the parameters of the optimal FH sequences listed in Table 1.1 except the series No.2. At the same time, it should be noted that although Theorem 3.3 can yield most of the known optimal FH sequences based on m-sequences with the same parameters, we do not know whether they are essentially the same or not. We discuss some more about this in Section VII. There are also combinatorial designs similar to affine geometries, which are resolvable and 1-rotational. By deleting ∞ from the point-set of such a design, a resolution class can be regarded as a difference family of partition type. In [1], we can find a number of such designs with explicit base blocks. As examples, here we just give some of their parameters: (v, k, λ) = (20, 5, 4), (40, 5, 4), (24, 6, 5), (30, 6, 5), (42, 6, 5), (28, 7, 6), (42, 7, 6), (32, 8, 7) and (40, 8, 7). These resolvable 1-rotational B(v, k, λ) produce optimal FHS(v − 1, m, λ) with (v − 1, m, λ) = (19, 4, 4), (39, 8, 4), (23, 4, 5), (29, 5, 5), (41, 7, 5), (27, 4, 6), (41, 6, 6), (31, 4, 7) and (39, 5, 7), respectively. More such examples are available in Section 10.6 of [1].

and cyclic resolvable group divisible designs, respectively. The interested reader is referred to [30] for the details. Among the three types of CRB(v, k, 1), type (T2) is valid for constructing optimal FH sequences.

IV. D IRECT C ONSTRUCTIONS W HEN v = km

Let q be a prime power. A multiple of a block B = {b1 , . . . , bk } in GF(q) is tB = {tb1 , . . . , tbk } for some t ∈ GF(q) and a multiplier of order n of B is an integer t ∈ GF(q) such that tB + j = {tb1 + j, . . . , tbk + j} = B for some j ∈ GF(q) and tn = 1 but ti 6= 1 for 0 < i < n. In a (q, k, λ)-difference family over GF(q), if there is a block B which has a multiplier of order k or k − 1, and all the other blocks are multiples of B, then the difference family is said to be radical (see [8]). Note that for a prime number p, a radical (p, k, λ)-difference family is defined over Zp .

In this section, direct constructions for optimal FH sequences are discussed through partition type difference packings with uniform block size. Once again, let the pair (V, B) be a BIB design B(v, k, λ), and π be an automorphism of (V, B). If π is of order v = |V |, then the BIB design is said to be cyclic. For a cyclic BIB design (V, B), the point-set V can be identified with Zv . In this case, the automorphism is represented by π : i 7−→ i + 1 (mod v). The orbit containing a block B ∈ B is defined similarly to that of a 1-rotational BIB design defined in Section III, but is developed modulo v. A base block and the length of an orbit are also defined similarly. In a cyclic BIB design, if the length of an orbit is v, then the orbit is said to be full, otherwise short. Note that if the orbit containing a block B is short, then the points in B can be partitioned into cosets of a subgroup of Zv . The orbit containing the block n v v vo Zk = 0, , . . . , (k − 1) (4.1) k k k is called a regular short orbit in the case of a cyclic BIB design. Let (V, B) be a cyclic B(v, k, λ) which has a nontrivial automorphism π of order v = |V |. If the cyclic B(v, k, λ) is resolvable and its resolution R = {R1 , . . . , Rr } is preserved by π, i.e., π(R) = {π(R1 ), . . . , π(Rr )} = R, where π(Ri ) = {π(B) : B ∈ Ri }, then the design is called a cyclic resolvable BIB design and is referred to as a CRB(v, k, λ). A cyclic resolvable BIB design is also referred to as a “cyclically resolvable cyclic BIB design” (see [21], [27] and [30]). When λ = 1, the design is called a cyclic resolvable Steiner 2design. It is known (see [21]) that a CRB(v, k, 1) consists of (v − k)/{k(k − 1)} full orbits and a single regular short orbit, and that a necessary condition for its existence is v ≡ k (mod k(k − 1)). Mishima and Jimbo [30] classified CRB(v, k, 1) into three types (T1), (T2) and (T3) according to their connections with cyclic quasiframes, cyclic semiframes

Construction 4.1: If there exists a CRB(km, k, 1) of type (T2), then there exists an optimal FHS(km, m, k) derived from a partition type m-DP(km, k, k) over Zkm . Proof: According to [30], a CRB(v, k, 1) of type (T2) satisfies a property that each resolution class contains at most one block of the regular short orbit. From this property, it follows that in a CRB(v, k, 1) of type (T2) with v = km, there must exist m = v/k resolution classes consisting of (m − 1)/k full orbits each of which is arranged in an m × k array and the regular short orbit in an m × 1 array. That is, each of the m resolution classes consists of every m blocks in each of the (m − 1)/k full orbits and a single block in the regular short orbit. This means that in a CRB(km, k, 1) of type (T2), an arbitrary resolution class with a block belonging to the regular short orbit can be viewed as a partition type m-DP(km, k, k) over Zkm .

Theorem 4.2: Let p be a prime number. Then there exist optimal FHS(kp, p, k) for the following cases: (1) (k, p) = (3, 6f + 1) for any positive integer f ; (2) (k, p) = (4, 12f + 1) for any odd integer f ; (3) (k, p) = (5, 20f +1) for any positive integer f satisfying the condition in Theorem 2.1 of [7]; (4) (k, p) = (7, 42f +1) for any positive integer f satisfying the condition in Theorem 12 of [8]. Proof: Netto [31] and Buratti [7], [8] constructed a radical (p, k, 1)-difference family for any prime p ≡ 1 (mod k(k − 1)) satisfying (1), (3), (4), which was used by Genma, Mishima and Jimbo [21] to construct a CRB(kp, k, 1) of type (T2). Lam and Miao [27] constructed a CRB(4p, 4, 1) of type (T2) for any prime number p = 12f + 1 where f is an odd integer. Then apply Construction 4.1. As previously mentioned, difference families with some special properties are very useful in constructing optimal FH sequences. The next construction needs a difference family with the property that the blocks are mutually disjoint. Such a difference family is said to be free (cf. [32]). Any difference family of partition type is free. For m ≥ k, if an (m, k, λ)difference family is free, then the inequality λ ≤ k must be satisfied and equality holds if and only if m = k. Construction 4.3: Let m and k be two positive integers

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which are relatively prime. Suppose that there exist a free (m, k, 1)-difference family over Zm and a set of k distinct invertible elements (called units) u0 , . . . , uk−1 in Zm such that the differences ui − uj are also units in Zm for 0 ≤ i < j ≤ k − 1. Then there exists an optimal FHS(km, m, k) which is derived from a partition type m-DP(km, k, k) over Zm × Zk ∼ = Zkm .

The differences arising from the blocks ui ◦Bti are all pure differences. Then from the hypothesis on units in Zm , we have the following list of pure differences. XX XX X X ∆00 (ui ◦Bti ) = ui ∆00 Bti = ui ∆At

To prove Construction 4.3, the “method of pure and mixed differences” is a useful tool. Although the method can be applied over any Abelian group in general (see, for example, [3] or [4]), we apply it only to cyclic groups. Suppose that m and n are relatively prime. Then Zmn is isomorphic to Zm × Zn . Let F be a collection of k-subsets of Zm × Zn and each k-subset B ∈ F be of form B = {(b1 , `1 ), . . . , (bk , `k )},

bj ∈ Zm , `j ∈ Zn

for 1 ≤ j ≤ k. Note that since gcd(m, n) = 1, 0

π : (i, j) 7−→ (i + 1 (mod m), j + 1 (mod n)) is a cyclic automorphism on the point-set Zm × Zn , which is substantially the same as the automorphism π : i 7−→ i + 1 (mod mn) if we take Zmn as our point-set. For a block B = {(b1 , `1 ), . . . , (bk , `k )} ∈ F, the differences defined by the list ∆ij B = (bh − bl : `h − `l ≡ j − i (mod n)) are called the pure (i, i) differences arising from B if i = j, and the mixed (i, j) differences otherwise. Clearly, for i, j ∈ Zn , ∆0,j−i B = ∆1,j−i+1 B = · · · = ∆ij B = · · · = ∆i−j,0 B. 0

Under the automorphism π , F forms an (mn, k, λ)-difference family if every element of ZP m \ {0} occurs exactly λ times in the list of pure differences B∈F ∆00 B and every element of Zm occurs P exactly λ times in the list of mixed (0, d) differences B∈F ∆0d B for every d ∈ P Zn \ {0} (equivalently in the list of mixed (i, j) differences B∈F ∆ij B for any i, j ∈ Zn such that d ≡ j − i (mod n)), where the addition of lists means their concatenation. Note that all of these discussions are still valid if the roles of the first and second coordinates of points are exchanged. Proof of Construction 4.3: Let {A1 , . . . , As } be a free (m, k, 1)-difference family over Zm . A simple counting argument shows that s = (m − 1)/(k(k − 1)) and thus ks = (m − 1)/(k − 1) < m. Without loss of generality, we may assume that 0 6∈ At for any t, 1 ≤ t ≤ s. Note that Zm × Zk is isomorphic to Zkm because gcd(m, k) = 1. Now form ks blocks over the point-set Zm × Zk as follows: Bti = At × {i} = {(at1 , i), . . . , (atk , i)},

t ∈ Is , i ∈ Zk ,

{at1 , . . . , atk }

where At = is a block of the given free difference family and Is = {1, . . . , s} is a set of indices. Furthermore, replace the blocks Bti by ui ◦Bti = ui ◦{(at1 , i), . . . , (atk , i)} = {(ui at1 , i), . . . , (ui atk , i)}. (4.2)

t∈Is i∈Zk

t∈Is i∈Zk

=

X

t∈Is

i∈Zk

ui (Zm \ {0}) = k(Zm \ {0}).

i∈Zk

In order to form an m-DP(km, k, k) of partition type, we need m − ks more blocks satisfying the required condition for the mixed differences, besides km blocks of (4.2). For these m − ks blocks, take Dx = x◦{(u0 , 0), . . . , (uk−1 , k − 1)}

[

= {(xu0 , 0), . . . , (xuk−1 , k − 1)}, x ∈ Zm \

At .

t∈Is

Then it is easy to see that Dx gives mixed (0, d) differences (and thus mixed (i, j) differences for any pair (i, j) such that d ≡ j − i (mod k), i, j ∈ Zk ) X [ X At ) ∆0d Dx = (uj − ui )(Zm \ x∈Zm \∪t∈Is At

(i,j)∈Hd

= kZm \

X

(i,j)∈Hd

t∈Is

[

(uj − ui )

At

t∈Is

for each d ∈ Zk \ {0}, where Hd = {(i, j) : d ≡ j − i (mod k), i, j ∈ Zk }. Here let [ P = {ui ◦Bti : t ∈ Is , i ∈ Zk } ∪ {Dx : x ∈ Zm \ At }. t∈Is

Then we can verify that P is an m-DP(km, k, k) over Zm × Zk and a partition of Zm × Zk as well. Thus the proof is completed. If m is a prime number, then all non-zero elements of Zm will be units. Then, as an immediate consequence of Construction 4.3, we have the following. Corollary 4.4: Let p be a prime number and p > k. Then the existence of a free (p, k, 1)-difference family over Zp implies the existence of an optimal FHS(kp, p, k). From the fact that any radical (p, k, 1)-difference family with k odd is free and the available results on radical difference families (see [7], [8] and [31]), we now know that there does exist a free (p, k, 1)-difference family for k = 3, 5, 7 and p ≡ 1 (mod k(k − 1)) a prime number satisfying certain conditions. The reader may have a question on the existence of a set of units which satisfies the condition in Construction 4.3 for general positive integer m. Below we review several known results on units in Zm . Let U (Zm ) be the set of all units in Zm . It is well known that U (Zm ) forms a group under multiplication with ϕ(m) elements, where ϕ is referred to as Euler function defined by ϕ(m) = |{i : 1 ≤ i ≤ m, gcd(m, i) = 1}|.

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Qr If m has a prime power factorization as m = i=1 pei i , where pi , 1 ≤ i ≤ r, are distinct prime numbers, then it follows that ¶ r µ Y 1 ϕ(m) = m . 1− pi i=1 Let S(Zm ) be any subset of U (Zm ) in which ui − uj ∈ U (Zm ) for any ui , uj ∈ S(Zm ), i 6= j, and let d(m) = max{|S(Zm )|}. Clearly d(m) ≤ ϕ(m) for any positive integer m. In fact, a tighter upper bound for d(m) was also found in [24]. The following lower bounds for d(m) can be derived from the results due to Kageyama and Miao [24]. Lemma 4.5 ([24]): For any two positive integers x and y, d(xy) ≥ min{d(x), d(y)}. Corollary 4.6:QSuppose that m has the prime power factorr ization as m = i=1 pei i . Then d(m) ≥ min{pei i − 1 : 1 ≤ i ≤ r}. By using Corollary 4.6 together with the result due to Dinitz and Rodney [18], we can produce a series of optimal FH sequences. Theorem 4.7: There exists an optimal FHS(3m, m, 3) for any positive integer m ≡ 1 (mod 6). Proof: The case m = 1 is trivial. Dinitz and Rodney [18] proved that there exists a free (m, 3, 1)-difference family whenever m ≡ 1 (mod 6). In order to apply Construction 4.3, we have only to show that there exists a set of three Q units that satisfies the condition of the construction. Let i pei i be the prime power factorization of m. Since m ≡ 1 (mod 6), we know that pi ≥ 5 for any i. Then Corollary 4.6 guarantees that d(m) ≥ ϕ(5) = 4, which completes the proof. Let F be a collection of k-subsets (blocks) of Zv and N be an additive subgroup of Zv . If the difference list ∆F contains each element of Zv \N precisely λ times, whereas no element of N has a difference representation from F, that is, if ∆F = λ(Zv \ N ), then F is called a relative (Zv , N, k, λ)difference family (or a relative (v, |N |, k, λ)-difference family) over Zv (see [9]). Obviously, when N = {0}, a relative (v, 1, k, λ)-difference family is merely a (v, k, λ)-difference family. We should mention that for a relative difference family consisting of a single block, which is specifically called a relative difference set, the reader may find a slightly different notation in the literature ([4] for example). Next, we show a construction for an optimal FHS(v, m, k) through a relative (Zv , N, k, k − 1)-difference family which is a partition of Zv \ N , where v = km and N is an additive subgroup of order k of Zv (or alternatively, a relative (v, k, k, k − 1)-difference family with blocks partitioning Zv \ N , where the second parameter k denotes the cardinality of N and the third parameter k denotes the size of each block in F.) Construction 4.8: Let N be an additive subgroup of order k of Zv , where v = km. If there exists a relative (v, k, k, k − 1)difference family which is a partition of Zv \ N , then there exists a partition type m-DP(v, k, k) over Zv . This gives an optimal FHS(v, m, k).

8

Proof: If there exists a relative (v, k, k, k − 1)-difference family, F = {B0 , . . . , Bm−2 }, over Zv such that F is a partition of Zv \ N , then a difference packing m-DP(v, k, k) of partition type can be obtained in an obvious way, i.e. by adding a new block Bm−1 = N of size k to F. Since N is an additive subgroup of order k of Zv , any difference arising from N is still an element of N and thus ∆N = k(N \ {0}). Therefore, F ∪ {N } forms a partition type m-DP(v, k, k) over Zv . Then the existence of an optimal FHS(v, m, k) follows from Theorem 2.3. Several infinite series of relative (G, N, k, k − 1)-difference families with |N | = k, which are partitions of G \ N , were found by Abel et al. in [2] over a general Abelian group G. Some of them can be modified to generate optimal FH sequences. In the remainder of the present section, we list those slight modifications without proofs. Theorem 4.9: Let p1 ≡ 3 (mod 4) and p2 be two odd prime numbers with p1 < p2 . Then there exists an optimal FHS(p1 p2 , p2 , p1 ). For constructing an optimal FHS(p1 p2 , p2 , p1 ) with p1 ≡ 1 (mod 4) a prime number, we need to associate an integer g(p2 ) with any odd prime number p2 according to the following definition: Let p2 be an odd prime number such that p2 = ef + 1, where e is the largest power of 2 in p2 − 1. Consider the cyclotomic classes C0 , . . . , Ce−1 of e-th powers in Zp2 , and denote by n(i, j) the number of non-zero elements z of Zp2 such that z − 1 ∈ Ci and z + 1 ∈ Cj (see, for example, [17] and [22]). Let I be the set of pairs defined by: I = {(i, j) : 0 ≤ i < e/2, 0 ≤ j < e/2, i 6= j} ∪ {(i, j) : 0 ≤ i < e/2, i + e/2 ≤ j < e} ∪ {(i, j) : e/2 ≤ i ≤ e, i − e/2 ≤ j < e/2}. Further let J be the subset of I defined by J = {(i, j) : i ≡ j (mod e/2)}. Then g(p2 ) is defined to be the value ¹ º X ¹ n(i, j) º f −1 g(p2 ) = + + 4 4 (i,j)∈J

X (i,j)∈I\J

¹

º n(i, j) . 2 (4.3)

Theorem 4.10: Let p1 ≡ 1 (mod 4) and p2 be two odd prime numbers such that g(p2 ) ≥ (p1 − 1)/4. Then there exists an optimal FHS(p1 p2 , p2 , p1 ). By cyclotomy, it is known that for any prime p ≡ 3 (mod 4) but p 6≡ 3 (mod 16) we have g(p) = (p − 7)/4, while for p ≡ 3 (mod 16) we have g(p) = (p − 3)/4. Therefore, the following corollary can be obtained. Corollary 4.11: Let p1 ≡ 1 (mod 4) and p2 ≡ 3 (mod 4) be two odd prime numbers. Then there exists an optimal FHS(p1 p2 , p2 , p1 ) in each of the following cases: (1) p2 ≡ 7, 11, 15 (mod 16) and p2 > p1 + 2; (2) p2 ≡ 3 (mod 16) and p2 > p1 .

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Summarizing Theorem 4.9 and Corollary 4.11, we can establish a general result for the case p2 ≡ 3 (mod 4).

Let B(x) be the set of (v − 1)/(k − 1) blocks in B each of which contains the point x ∈ Zv . Since every element of Zv appears k times in any full orbit, B(x) includes k blocks from each full orbit. Without loss of generality we now take x = 0. When a cyclic B(v, k, 1) has no short orbit, B(0) consists of exactly k blocks from each of the full orbits, which implies that every non-zero element of Zv occurs precisely k times as the differences arising from B(0). Therefore B(0) can be viewed as a (v, k, k)-difference family with (v − 1)/(k − 1) blocks over Zv . For B(0) = {B0 , . . . , Bm−1 }, where m = (v − 1)/(k − 1), consider the following partition of Zv ,

Theorem 4.12: Let p1 and p2 ≡ 3 (mod 4) be two odd prime numbers with p1 < p2 . Then there exists an optimal FHS(p1 p2 , p2 , p1 ) with the only possible exceptions of p2 = p1 + 2 when p2 ≡ 7, 11, 15 (mod 16). The value of g(p) is much less clear for the case p ≡ 1 (mod 4), although there were several lower and upper bounds found by Abel et al. in [2]. The interested reader is referred to [2] for more details. Now we mention two special cases of p1 = 3 and 5. By Theorem 4.9, there exists an optimal FHS(3p2 , p2 , 3) for any odd prime number p2 > 3. Theorem 4.13: Let p be an odd prime number with p > 3. Then there exists an optimal FHS(3p, p, 3). Note that Theorem 4.13 differs from Theorem 4.7 on the requirement for p (or m in Theorem 4.7). For the case p1 = 5, a complete solution for the existence of a relative (5p2 , 5, 5, 4)-difference family which is a partition of Z5 ×Zp2 \Z5 ×{0} was also given in [2] for any odd prime number p2 > 5. As a consequence, we have the following. Theorem 4.14: Let p be an odd prime number with p > 5. Then there exists an optimal FHS(5p, p, 5). V. D IRECT C ONSTRUCTIONS W HEN v = km + 1 In the previous section, we have considered optimal FH sequences derived from partition type difference packings in which all blocks are of the same size. We are also interested in the case that blocks are of different sizes. In this section, we consider the simplest generalized case where all blocks are of size k except one of size k + 1. In this case, v = k(m − 1) + (k + 1) = km + 1. Let 1 ≤ i ≤ m − 1. An FHS(v, m, λ) derived from a partition type m-DP(v, {k, k+1}, λ) in which m−i blocks are of size k and the remaining i blocks are of size k+1 is optimal if λ = k. Here we describe two constructions for such optimal FHS(v, m, k) with i = 1 which can be derived from their corresponding m-DP(v, {k, k + 1}, k). The first construction makes use of cyclic Steiner 2-designs and the second one is due to a direct construction for m-DP(v, {k, k + 1}, k) of partition type. Construction 5.1: If there exists a cyclic Steiner 2-design B(v, k, 1), then there exists a partition type m-DP(v, {k − 1, k}, k − 1) over Zv , where m = (v − 1)/(k − 1). This gives an optimal FHS(v, m, k − 1). Proof: Suppose that (Zv , B) is a cyclic Steiner 2-design B(v, k, 1). It is known (see [21] and [30]) that v ≡ 1 or k (mod k(k − 1)) is a necessary condition for the existence of a cyclic B(v, k, 1), and that a cyclic B(v, k, 1) with v ≡ 1 (mod k(k − 1)) has no short orbit and a cyclic B(v, k, 1) with v ≡ k (mod k(k − 1)) has a single regular short orbit containing a block of the form (4.1).

P = {B0 , B1 \ {0}, . . . , Bm−1 \ {0}}.

(5.1)

Since (Zv , B) is a Steiner 2-design and d − 0 = 0 − (v − d) in Zv , the elements d and v − d cannot be contained in a block of B(0) simultaneously. Therefore each non-zero element d ∈ Zv occurs k − 1 times if d ∈ B0 , otherwise k − 2 times as the differences arising from the blocks in P of the form (5.1), which means that P forms a partition type m-DP(v, {k − 1, k}, k − 1) over Zv . When a cyclic B(v, k, 1) has a single regular short orbit, B(0) contains the block D = {0, v/k, . . . , (k − 1)v/k}. Since every multiple of v/k occurs precisely k times as the differences arising from the block D and does not occur from any of the other blocks of B(0), by letting D = Bi for some i, 1 ≤ i ≤ m − 1, the partition P defined by (5.1) becomes an m-DP(v, {k − 1, k}, k − 1) of partition type. Then Theorem 2.3 gives an optimal FHS(v, m, k − 1). It is well known (see [1]) that the existence of a cyclic B(v, k, λ) and that of a (v, k, λ)-difference family over Zv are equivalent. Many difference families over Zv are available in [1] and several series of (v, k, 1)-difference families over Zv (thus cyclic B(v, k, 1)) are known due to Chen and Zhu [10], [11], and Colbourn and Mathon [15]. However, the spectrum of (v, k, 1)-difference families over Zv is still unknown in general. Combining the known results on difference families over Zv with our results, we obtain the following theorems. Theorem 5.2: There exists an optimal FHS(v, (v − 1)/2, 2) for any positive integer v ≡ 1, 3 (mod 6). Theorem 5.3: Let p ≡ 1 (mod k(k − 1)) be a prime number. Then there exists an optimal FHS(p, (p−1)/(k−1), k−1) for k = 4, 5 and 6 except when k = 6 and p = 61. From now on, we describe a direct construction for an optimal FHS(v, m, k) which can be derived from a partition type m-DP(v, {k, k + 1}, k) in which m − 1 subsets are of size k and the remaining one is of size k + 1. Construction 5.4: Let G = {θ1 , . . . , θk } be a multiplicative subgroup of U (Zv ) such that the list of differences arising from G is a subset of U (Zv ), where U (Zv ) is the set of all units in Zv . If θi +θj 6≡ 0 (mod v) for any i 6= j, 1 ≤ i, j ≤ k, then there exists a partition type m-DP(v, {k, k + 1}, k) over Zv in which m − 1 blocks are of size k and the remaining one is of size k + 1. This gives an optimal FHS(v, m, k). Proof: For x, y ∈ Zv \ {0}, the binary relation ∼ defined

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by x ∼ y if and only if there exists a θi ∈ G such that xθi = y is an equivalence relation over Zv \ {0}. Then its equivalence classes are the subsets xG, x ∈ Zv \ {0}, of Zv . Consider the collection F = {sG : s ∈ S}, where S is a system of distinct representatives for the equivalence classes modulo G of Zv \ {0}. It is easy to see that the collection F is a partition of Zv \ {0} and that |sG| = |G| = k for any s ∈ S. Since the list of differences arising from G is given by

forms a partition type e-DP(p, {k, k + 1}, k) over Zp . This gives an optimal FHS(p, e, k). Proof: If p is prime, then Zp becomes a field. Therefore, we need only check whether θ + θ0 6≡ 0 (mod p) for any θ 6= θ0 , θ, θ0 ∈ G. Since −1 = ω (p−1)/2 = ω ke/2 , we know that −1 ∈ G if and only if k is even. It implies that when k is odd, θ + θ0 6≡ 0 (mod p) for any θ 6= θ0 , θ, θ0 ∈ G. Then apply Construction 5.4

∆G =

k X k X

(θi − θj ) =

j=1 i=1 i6=j

=

k X k X j=1 i=1 i6=j

k X k X

(θh − 1)θj =

h=1 j=1 θh 6=1

we have ∆F =

X

s∆G =

s∈S

=

X

(θi θj−1 − 1)θj

X

(θ − 1)G,

θ∈G\{1}

X θ∈G\{1}

(θ − 1)

X

sG

s∈S

(Zn \ {0}) = (k − 1)(Zv \ {0}).

θ∈G\{1}

Hence every non-zero element of Zv occurs exactly k − 1 times in ∆F. Add the element 0 to the block G to get G ∪ {0} and let P = (F \ {G}) ∪ {G ∪ {0}}. Since for G = {θ1 , . . . , θk } we require that θi + θj 6≡ 0 (mod v) for any i 6= j, 1 ≤ i, j ≤ k, every element in G occurs exactly k times in ∆P. Clearly, the other non-zero element of Zv occurs exactly k − 1 times in ∆P. Thus P can be viewed as a partition type mDP(v, {k, k + 1}, k) over Zv in which m − 1 = (v − k − 1)/k subsets are of size k and the remaining one is of size k + 1. Then the desired optimal FHS(v, m, k) can be derived from P. Construction 5.4 gives a way of constructing a partition type m-DP(v, {k, k + 1}, k) (and thus an optimal FHS(v, m, k)) over Zv in which all blocks are of size k except one of size k + 1. However, the problem of finding such a multiplicative subgroup G in U (Zv ) such that θ + θ0 6≡ 0 (mod v) for any θ 6= θ0 , θ, θ0 ∈ G and that any difference from G is still a unit in Zv appears to be quite difficult. Problem 5.5: Find a maximum multiplicative subgroup G of U (Zv ) such that θ + θ0 6≡ 0 (mod v) for any θ 6= θ0 , θ, θ0 ∈ G, and that any difference from G is still a unit in Zv . Here the term “maximum” means that the subgroup is of maximum cardinality among all subgroups satisfying the above conditions. When v = km + 1 is a prime number with k odd, we have the following corollary. Corollary 5.6: Let p = ke + 1 be a prime number with k odd, ω be a primitive element of Zp and G = hω e i be the multiplicative subgroup of order k of Zp generated by ω e . For 0 ≤ j ≤ e − 1, define Aj = ω j G. Then {A1 , . . . , Ae−1 } ∪ {A0 ∪ {0}}

The following is an immediate generalization of Corollary 5.6. Corollary 5.7:QIf v is decomposable into distinct prime r factors as v = i=1 pi and each of the prime factors is of the form pi = kei + 1 for an odd integer k, then there exists an optimal FHS(v, (v − 1)/k, k). Proof: Consider the Galois ring Zp1 × · · · × Zpr , which is isomorphic to Zp1 ...pr = Zv . Since pi = kei + 1 for each i, 1 ≤ i ≤ r, Zpi has a multiplicative subgroup Gi = hωiei i of order k generated by ωiei , where ωi is a primitive element of Zpi . Let θj = (ω1je1 , . . . , ωrjer ) for each j, 1 ≤ j ≤ k. Then G = {θ1 , . . . , θk } is a multiplicative subgroup of U (Zp1 × · · · × Zpr ) with each difference arising from G being still a unit in U (Zp1 × · · · × Zpr ). Since k is odd, θi + θj 6≡ (0 (mod p1 ), . . . , 0 (mod pn )) for any i 6= j, 1 ≤ i, j ≤ k. Then apply Construction 5.4. VI. R ECURSIVE C ONSTRUCTIONS In order to describe our recursive constructions, we first need to introduce the new conception of a partition type difference packing with a hole. Let H be a subset of Zv and P = {B0 , . . . , B`−1 } be a partition of Zv \ H which satisfies the property that ½ `−1 X ≤ λ for any integer t ∈ Zv \ H, |Bi ∩ (Bi + t)| = 0 for any integer t ∈ H, i=0

or equivalently, the following property stated in terms of differences: any integer d ∈ Zv \ H can be represented as the difference b − b0 , b, b0 ∈ Bi , 0 ≤ i ≤ ` − 1, in at most λ ways, whereas no integer in H can be represented in such a way, then P is called a partition type difference packing `DP(v, K, λ) with a hole H, where K = {|Bi | : 0 ≤ i ≤ `−1}. Example 6.1: Consider the case that v = 35 and H = {0, 7, 14, 21, 28} which is the additive subgroup of order 5 of Z35 . Let P be a set of the following six blocks: B0 = {1, 4, 6, 24, 30}, B1 = {2, 5, 13, 17, 18}, B2 = {3, 12, 20, 22, 23}, B3 = {8, 10, 27, 32, 33}, B4 = {9, 11, 15, 19, 31}, B5 = {16, 25, 26, 29, 34}. Then it can be verified that P = {B0 , . . . , B5 } forms a partition type 6-DP(35, 5, 4) with a hole H = {0, 7, 14, 21, 28}, i.e., H = 7Z5 , over Z35 . A case of several g1 , . . . , gr 1 ≤ i, j

of particular interest is when H is the union distinct additive subgroups G1 , . . . , Gr of orders of Zv satisfying Gi ∩ Gj = {0} for any i 6= j, ≤ r. Such a difference packing is said to be

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(g1 , . . . , gr )-regular Pror with a (g1 , . . . , gr )-regular hole. In this case, |H| = 1 + i=1 (gi − 1). When r = 1, we simply say g1 -regular. The partition type difference packing with a hole described in Example 6.1 is in fact 5-regular. Note that a partition type ((v − k)/k)-DP(v, k, λ) with a k-regular hole N is just the same as a relative (Zv , N, k, λ)difference family which partitions Zv \ N , where N is an additive subgroup of order k of Zv . The combinatorial structure exhibited in Example 6.1 is, in fact, a relative (Z35 , 7Z5 , 5, 4)difference family which partitions Z35 \ 7Z5 . If there exist a partition type difference packing P with a g-regular hole H, i.e., the additive subgroup H of order g, over Zv , and a partition type difference packing Q over Zg , then by embedding (v/g)Q = {{av/g : a ∈ A} : A ∈ Q} into the hole H of P, we obtain a partition type difference packing P ∪ (v/g)Q over Zv . Thus an FH sequence can be derived from P ∪ (v/g)Q.

For 1 ≤ i, i0 ≤ `, 1 ≤ j, j 0 ≤ n, 1 ≤ t ≤ s and 1 ≤ t0 ≤ s0 , where s, s0 ≤ k and s = s0 in the case i = i0 , if

We also need the notion of a difference matrix over Zn (see [12]). Let Σ = (σij ) be a t × λn matrix with entries from Zn . If each element of Zn occurs exactly λ times among the differences σhj − σij , j = 1, . . . , λn, for any h 6= i, where 1 ≤ h, i ≤ t, then Σ is called an (n, t; λ)-difference matrix over Zn . It is easy to see that the property of a difference matrix is preserved even if we add any element of Zn to all entries in any row or column of the difference matrix. Then, without loss of generality, we can assume that all entries in the first row are zero. Such a difference matrix is said to be normalized. The difference matrix obtained from a normalized (n, t; λ)-difference matrix by deleting the entries in its first row is said to be homogeneous. In a homogeneous difference matrix, any element of Zn appears in every row exactly λ times. The existence of a homogeneous (n, t−1; λ)-difference matrix is obviously equivalent to the existence of an (n, t; λ)difference matrix. Construction 6.2: Suppose that there exist a partition type `-DP(v, K, λ) with a (g1 , . . . , gr )-regular hole H over Zv and a homogeneous (n, k; 1)-difference matrix over Zn , where k is the maximum integer in K. Then there exists a partition type n`-DP(nv, K, λ) with a (g1 n, . . . , gr n)-regular hole H + vZn over Znv , where H + vZn = {h + zv : h ∈ H, z ∈ Zn }. Proof: Let Σ = (σij ) be a homogeneous (n, k; 1)difference matrix over Zn . Let G1 , . . . , Gr be additive subgroups of orders g1 , . . . , gr of Zv such that H = {0} ∪ (G1 \ {0}) ∪ · · · ∪ (Gr \ {0}) is the hole of a partition type `-DP(v, K, λ). For each block Bi = {bi1 , . . . , bis } of the partition type `-DP(v, K, λ) with a (g1 , . . . , gr )-regular hole H, where s ≤ k, construct the following n new blocks: (j)

Bi

= {bi1 + σ1j v, . . . , bis + σsj v}, j = 1, . . . , n.

(6.1)

Then it can be readily verified that each element of Znv \ H 0 , where H 0 is defined to be the set H + vZn = {h + zv : h ∈ H, z ∈ Zw }, can be represented as the difference x − x0 of (j) two distinct elements x, x0 ∈ Bi , 1 ≤ i ≤ `, 1 ≤ j ≤ n, in at most λ ways. On the other hand, it can also be checked that the newly (j) defined blocks Bi , 1 ≤ i ≤ `, 1 ≤ j ≤ n, partition Znv \ H 0 .

bit + σtj v ≡ bi0 t0 + σt0 j 0 v

(mod nv),

then bit ≡ bi0 t0

(mod v),

which implies that i = i0 and t = t0 , since {B1 , . . . , B` } is a partition type `-DP(v, K, λ) with a (g1 , . . . , gr )-regular hole H over Zv . In this case, we have σtj ≡ σt0 j 0

(mod n),

which implies that j = j 0 , since there is no element of Zn that can appear twice in any row of the homogeneous difference matrix. Meanwhile, from the definition (6.1) of new blocks (j) Bi and the property of a homogeneous (n, k; 1)-difference matrix, it follows that any element of H 0 cannot be contained in any newly defined blocks. The proof is then completed. In Construction 6.2, the existence of a homogeneous difference matrix is assumed. Homogeneous difference matrices have been investigated extensively, see, for example, [12] and the references therein. Here is a well-known simple construction. Lemma 6.3 ([14]): Let n and k be positive integers such that gcd(n, k!) = 1, and let σij = ij (mod n) for 1 ≤ i ≤ k and 0 ≤ j ≤ n − 1. Then Σ = (σij ) is a homogeneous (n, k; 1)-difference matrix. In particular, if n is an odd prime number, then there exists a homogeneous (n, k; 1)-difference matrix for any integer k ≤ n − 1. We now illustrate how Construction 6.2 can be used. Corollary 6.4: Suppose that p ≥ k + 1 is an odd prime number. Then there exists a partition type p-DP(p(k+1), k, k− 1) with a p-regular hole over Zp(k+1) . Proof: Clearly, over Zk+1 , {1, . . . , k} can be regarded as a partition type 1-DP(k + 1, k, k − 1) with the 1-regular hole {0}. Since Lemma 6.3 ensures that there exists a homogeneous (p, k; 1)-difference matrix for any odd prime number p ≥ k + 1, by applying Construction 6.2, we can obtain a partition type p-DP(p(k + 1), k, k − 1) with a p-regular hole (k + 1)Zp . Corollary 6.5: Suppose that p1 = ke + 1 and p2 ≥ k + 1 are two odd prime numbers. Then there exists a partition type ep2 -DP(p1 p2 , k, k − 1) with a p2 -regular hole over Zp1 p2 . Proof: Let p1 = ke + 1 be an odd prime number, ω1 a primitive element of Zp1 and G = hω1e i the multiplicative subgroup of order k of Zp1 generated by ω1e . Then through [38], we know that the collection {B0 , . . . , Be−1 }, where Bj = ω1j G, 0 ≤ j ≤ e − 1, gives a partition type eDP(p1 , k, k −1) with the 1-regular hole {0} over Zp1 . For any odd prime number p2 ≥ k +1, it follows from Lemma 6.3 that there exists a homogeneous (p2 , k; 1)-difference matrix. Then, Construction 6.2 gives a partition type ep2 -DP(p1 p2 , k, k − 1) with a p2 -regular hole p1 Zp2 over Zp1 p2 . If we can embed some suitable “small” partition type difference packing into the hole of the resulting partition type

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difference packing obtained by Corollary 6.4 or 6.5, we can produce a new partition type difference packing which can yield an optimal FH sequence. This means that, by virtue of Corollaries 6.4 and 6.5, we just need to find suitable “small” partition type difference packings in order to obtain optimal FH sequences. For example, by applying Corollaries 6.4 and 6.5 with Construction 5.1 and Theorem 5.3, we can obtain the following results.

In Section III, we raised an open problem whether the optimal FH sequences given in Theorem 3.3 and the known optimal FH sequences with the same parameters based on msequences are essentially the same. However, we should first clarify the exact meaning when we say that two FH sequences are “essentially the same.” Often we are interested in properties of FH sequences, such as auto-correlation, randomness and generating method, which remain unchanged when passing from one FH sequence to another that is essentially the same. Providing an exact definition for this concept and enumerating how many non “essentially the same” FH sequences are also interesting problems deserving of attention.

Theorem 6.6: Suppose that p = tk(k − 1) + 1 is an odd prime number with t a positive integer. Then there exists an optimal FHS(p(k +1), tk 2 +1, k −1) for k = 4, 5 and 6 except when k = 6 and p = 61. Theorem 6.7: Suppose that p1 = ke + 1 and p2 = tk(k − 1) + 1 are two odd prime numbers with t a positive integer. Then there exists an optimal FHS(p1 p2 , ep2 + tk, k − 1) for k = 4, 5 and 6 except when k = 6 and p2 = 61. We also present a recursive construction for free difference families to enrich Construction 4.3. Lemma 6.8: Suppose that m and n are relatively prime. If there exist a free (m, k, 1)-difference family over Zm , a free (n, k, 1)-difference family over Zn and a homogeneous (n, k; 1)-difference matrix over Zn , then there exists a free (mn, k, 1)-difference family over Zmn . Lemma 6.8 can be proved in a manner analogous to the proof of Construction 6.2. Note that in Lemma 6.8 if d(m) ≥ k and d(n) ≥ k, then Construction 4.3 can be applied with Corollary 4.6 in order to give an optimal FHS(kmn, mn, k), where d(m) is the number of units in Zm satisfying the hypothesis in Construction 4.3. VII. C ONCLUDING R EMARKS In this paper, we investigated frequency hopping multiple access (FHMA) systems with a single optimal frequency hopping (FH) sequence each from a combinatorial designtheoretic point of view. In Section II, we established a connection between FH sequences and partition type difference packings. This connection allowed us to obtain optimal FH sequences by constructing their corresponding difference packings of partition type. In Sections III, IV and V, various combinatorial structures such as affine geometries (Theorem 3.3), cyclic Steiner 2-designs (Construction 5.1), cyclically resolvable Steiner 2-designs (Construction 4.1), and difference packings and families (Constructions 4.3, 4.8 and 5.4) were utilized to construct optimal FH sequences. We further showed in Section VI that these new optimal FH sequences can be ingredients in our combinatorial recursive constructions. These FH sequences are also useful in ultra wide band (UWB) communication systems [34]. However, we have investigated neither generating methods like linear recurrence equations for our new optimal FH sequences, nor randomness properties of these FH sequences. We have not considered constructions for FHMA systems with plural optimal FH sequences, either. These interesting problems are definitely worthy of further research.

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Ryoh Fuji-Hara was born in Hyogo, Japan, on November 4, 1949. He received the B. Eng. degree from Tokai University, Hiratsuka, Kanagawa, Japan, the M. Eng. degree from Waseda University, Tokyo, Japan, and the Ph.D. degree in combinatorics and optimization from the University of Waterloo, Waterloo, ON, Canada, in 1981. From 1981 to 1983, he was a Postdoctoral Fellow at the University of Waterloo. He was an Assistant Professor from 1983 to 1988, an Associate Professor from 1988 to 1994, and is currently a Professor at the Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki, Japan. With M. Jimbo, he coauthored a book Mathematical Theory of Coding and Cryptography (in Japanese) (Tokyo, Japan: Kyoritsu Shuppan, 1993). Dr. Fuji-Hara is on the Editorial Board of Journal of Combinatorial Mathematics and Combinatorial Computing and a Member of the Council of the Institute of Combinatorics and its Applications.

Ying Miao was born in Jiangsu, China, in 1963. He received the B. S. degree from Wuhan University, Wuhan, Hubei, China, in 1985, the M. S. degree from Suzhou University, Suzhou, Jiangsu, China, in 1989, and the D. Sci. degree from Hiroshima University, Hiroshima, Japan, in 1997, all in mathematics. From 1989 to 1993, he was with Suzhou Institute of Silk Textile Technology, Suzhou, Jiangsu, China, as a Teaching-Research Assistant and then an Assistant Professor. From 1995 to 1997, he was a Research Fellow of the Japan Society for the Promotion of Science. During 1997 1998, he was a Postdoctoral Fellow at the Department of Computer Science, Concordia University, Montreal, QC, Canada. From 1998 to 2004, he was an Assistant Professor at the Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki, Japan. He is currently an Assistant Professor at the Graduate School of Systems and Information Engineering, University of Tsukuba. His research interests include combinatorics, coding theory, cryptography, and their interactions. With S. Furino and J. Yin, he coauthored a book Frames and Resolvable Designs: Uses, Constructions, and Existence (Boca Raton, FL: CRC, 1996). Dr. Miao is on the Editorial Board of both Graphs and Combinatorics and Journal of Combinatorial Designs. He received The 2001 Kirkman Medal from the Institute of Combinatorics and its Applications.

Miwako Mishima received the B. Eng. and the M. Eng. degrees from Gifu University, Gifu, Japan, in 1991 and 1993, respectively, and the degree of Ph.D. of Science from Keio University, Yokohama, Japan, in 1999. From 1993 to 1996, she worked for Nippon Telegraph and Telephone Corporation (NTT), Japan. From 1996 to 2003, she was a Research Associate at the Department of Information Science, Gifu University, Gifu, Japan, and is currently an Associate Professor at the Information and Multimedia Center, Gifu University. Her research interests include design theory, graph theory, and their applications.