Codebooks from Almost Difference Sets - Semantic Scholar

Codebooks from Almost Difference Sets Cunsheng Ding∗ and Tao Feng† June 3, 2007

Abstract In direct spread CDMA systems, codebooks meeting the Welch bounds are used to distinguish among the signals of different users. Recently, constructions of codebooks with difference sets meeting Welch’s bound on the maximum cross-correlation amplitude were developed. In this paper, a generic construction of codebooks using almost difference sets is considered and several classes of codebooks nearly meeting the Welch bound are obtained. The parameters of the codebooks constructed in this paper are new. Keywords: Welch bounds, difference sets, almost difference sets, signal sets, codebooks.

1

Introduction

An (N, K) codebook C is a set {c0 , ..., cN −1 } of N unit norm 1 × K complex vectors ci , which are called codewords of the codebook. The alphabet of the codebook is the set of all different complex values that the coordinates of all the codewords of C take. The alphabet size is the number of elements in the alphabet. The root-mean-square (RMS) crosscorrelation and the maximum crosscorrelation amplitudes of such a codebook C are defined as v u X 1 2 |ci cH Irms (C) := u t N (N − 1) j | , 0≤i,j≤N −1 i6=j

Imax (C) :=

max

0≤i K, we have s N −K Irms (C) ≥ , (N − 1)K

(1)

P H with equality if and only if N i=0 ci ci = (N/K)IK , where IK denotes the K ×K identity matrix. We have also s N −K , (2) Imax (C) ≥ (N − 1)K with equality if and only if for all pairs (i, j) with i 6= j s N −K |ci cH . j |= (N − 1)K

(3)

If the equality holds in (1), the codebook CD is referred to as a Welchbound-equality (WBE) codebook. A codebook meeting the bound of (2) is called a maximum-Welch-bound-equality (MWBE) codebook. An MWBE codebook must be a WBE codebook, but a WBE codebook may not be an MWBE codebook. MWBE codebooks form a subset of WBE codebooks. In the sequel, Welch bound always refers to the one of (2), unless otherwise stated. MWBE and WBE codebooks are used in direct spread CDMA systems to distinguish among the signals of different users. A well rounded treatment of MWBE and WBE codebooks was given by Sarwate [20]. They have applications also in quantum information processing [27], packing [5], and coding theory [4, 6]. Every linear error correcting code whose dual code with Hamming distance at least 3 yields a WBE codebook [18, 20]. So it is very easy to construct WBE codebooks. There are many constructions of WBE codebooks (see Sarwate [20] for a survey). However, MWBE codebooks are extremely hard to construct, as pointed out by Sarwate in [20, p. 100]. In 2005, Xia et al. developed a generic construction of MWBE codebooks using cyclic difference sets in ZN [28]. Shortly after [28], the first author of this paper described a construction of MWBE codebooks using both cyclic and noncyclic difference sets in finite fields GF(q) [8]. Recently, we generalized these constructions in [9]. In view that Welch’s bound cannot be met by (N, K) codebooks for many parameters N and K, it is desirable to construct (N, K) codebooks that nearly meet Welch’s bound. In [8], almost difference sets in (GF(q), +) were employed to construct complex codebooks nearly meeting the Welch bound. In this paper, we extend this idea and present a generic construction of complex codebooks using almost difference sets in arbitrary finite Abelian groups. Several classes of codebooks nearly meeting the Welch bound are obtained in this paper. These codebooks may be optimal, and their parameters are new.

2

2 2.1

Constructions of codebooks from difference sets Difference sets and almost difference sets

We first introduce difference sets and almost difference sets that are the building blocks of the complex codebooks presented in this paper. Let G be an Abelian group of order v, and let D be a k-element subset of G. We define the difference function ΓD (w) := |(D + w) ∩ D|,

(4)

where D + w := {d + w : d ∈ D}. D is called a (v, k, λ)-difference set in G provided that the difference function ΓD (w) = λ for all nonzero elements w of G. In other words, for a difference set D every nonzero element of G can be expressed as d1 − d2 for exactly λ pairs (d1 , d2 ) ∈ D × D. If D is a (v, k, λ)-difference set in G, its complement C := G \ C is an (v, v − k, v − 2k + λ) difference set. D is called a (v, k, λ, t) almost difference set in G if ΓD (w) takes on λ altogether t times and λ + 1 altogether v − 1 − t times when w ranges over all the nonzero elements of G. In other words, each of t nonzero elements of G can be written as d1 − d2 for exactly λ pairs (d1 , d2 ) ∈ D × D, and each of the remaining v − 1 − t nonzero elements can be expressed as d1 − d2 for exactly λ + 1 pairs (d1 , d2 ) ∈ D × D. D is a (v, k, λ, t) almost difference set in (G, +) if and only if the complement D = G \ D is an (n, n − k, n − 2k + λ, t) almost difference set [1]. We refer the reader to [1] for a survey of almost difference sets. Difference sets were used to design binary codebooks for DS-CDMA systems [10], sequences for radar, sonar and synchronization [22], and error correcting codes. Cyclic difference sets were treated in details by Baumert [2]. Detailed information on difference sets can be found in [3].

2.2

The constructions of codebooks from difference sets

In 1999, K¨onig [14] employed a class of difference sets to construct a family of equiangular tight frames (another name for MWBE codebooks). Six years later in 2005, Xia et al. [28] presented a generic construction of complex codebooks with cyclic difference sets that contains as a special case the family of equiangular tight frames obtained by K¨onig [14]. Shortly, a construction of MWBE codebooks using difference sets in (GF(q), +) was described in [8]. It was demonstrated in [8] that none of the two constructions is a generalization of the other and that the two constructions are different. However, they coincide when and only when the underlying difference set is defined over a finite field GF(p), where p is a prime number. Recently, we generalized all the previous constructions using difference sets and our generalization is described below. Let (G, +) be an Abelian group of order N and exponent eG , and let χ0 , χ1 , · · · , χN −1 denote all the characters of G. Given any K-subset D := {d1 , d2 , ..., dK } of G, we define a codebook CD := {ci : i = 0, ..., N − 1}, 3

(5)

where for each i 1 ci := √ (χi (d1 ), χi (d2 ), ..., χi (dK )) . K

(6)

Theorem 2. [9] The set CD of (5) is an (N, K) MWBE codebook with alphabet size eG if and only if the set D is an (N, K, λ) difference set in (G, +), where K > 1. This generalization indeed yields more optimal complex codebooks that cannot be generated by the constructions in [28] and [8]. The reader is referred to [9] for detailed information.

3

Codebooks from almost difference sets

It was shown in [8] that almost difference sets in (GF(q), +) can be used to construct complex codebooks that nearly meet the Welch bound. In this section, we generalize this idea by choosing the defining set D in the construction of (5) to be an almost difference set in any Abelian group (G, +). However, in this case we do not have a general result like Theorem 2, as almost difference sets are diverse and have complicated structures. Hence, we have to deal with the obtained codebooks case by case. In the examples of codebooks presented in this section, we consider only almost difference sets in the Abelian group ZN := {0, 1, ..., N − 1}, and thus need the group characters of ZN , which are defined as ψN (n) := e2nπ

√

−1/N

.

(7)

Examples of codebooks from almost difference sets in GF(q) can be found in [8].

3.1

(N, K) complex codebooks with N = p(p + 4) ≡ 1 (mod 4), where p and p + 4 are prime

A class of cyclic almost difference sets is constructed from Whiteman’s generalized cyclotomy in [7]. To introduce the almost difference sets, we have to deal with Whiteman’s generalized cyclotomy of order 2. Let g be a fixed common primitive root of both primes p and q. Assume that 2 = gcd(p − 1, q − 1), and let f = (p − 1)(q − 1)/2. Then there exists an integer x such that x = g (mod p), x = 1 (mod q) and Z∗pq = {g s xi : s = 0, 1, ..., f − 1; i = 0, 1}, where Z∗pq denotes the set of all multiplicatively invertible elements of Zpq . Whiteman’s generalized cyclotomic classes Di of order 2 are defined as Di = {g s xi : s = 0, 1, ..., f − 1}, i = 0, 1. The generalized cyclotomic numbers of order 2 are defined by (i, j)2 = |(Di + 1) ∩ Dj |. 4

This generalized cyclotomy was used by Whiteman to find the twin-prime difference sets [26]. Lemma 3. [7] Suppose that gcd(p − 1, q − 1) = 2. Let D0 and D1 be the cyclotomic classes of order 2. Define D = D1 ∪ {p, 2p, · · · , (q − 1)p}.

(8)

If q − p = 4 and (p − 1)(q − 1)/4 is odd, then D is a (p(p + 4), (p + 3)(p + 1)/2, (p + 3)(p + 1)/4, (p − 1)(p + 5)/4) almost difference set of Zp(p+4) . The almost difference set described in Lemma 3 was stated in terms of the optimal autocorrelation values of a binary sequence in [7]. We will compute the maximum crosscorrelation amplitude of the codebook CD based on the cyclic almost difference set D of Lemma 3. For the easiness of discussion below, we define P := {p, 2p, ..., (q − 1)p}, Q := {q, 2q, ..., (p − 1)q}. In the sequel, q = p + 4 and both p and q are primes, and N := pq = p(p + 4). Let ψN be the group character defined in (7). For any integer i we define (i)

ψN (x) := ψN (ix). (i)

Hence ψN is a group character of ZN for any i. It is straightforward to prove the following lemma. Lemma 4. Let symbols and notations be  q − 1, (m) ψN (P ) = −1,  p − 1, (m) ψN (Q) = −1,

as before. We have then if gcd(m, q) = q otherwise. if gcd(m, p) = p otherwise.

Lemma 5. Let symbols and notations be as before. We have then  −(q − 1)/2, if gcd(m, N ) = q (m) ψN (D1 ) = −(p − 1)/2, if gcd(m, N ) = p. Proof. Since g is a common primitive root of both p and q and the order of g modulo N is f = (p − 1)(q − 1)/2, by the definition of x we have D1 mod p = {g s x mod p : s = 0, 1, · · · , f − 1}

= {g s+1 mod p : s = 0, 1, · · · , f − 1} = {1, 2, · · · , p − 1}.

When s ranges over {0, 1, · · · , f − 1}, g s x mod p takes on each element of {1, 2, · · · , p − 1} (q − 1)/2 times. 5

If gcd(m, N ) = q, then q divides m and m 6= 0, but gcd(m, p) = 1. Hence is a group character of ZN with order p. It then follows that q − 1 X (m/q) q−1 (m) ψN (D1 ) = . ψp (j) = − 2 2

(m) ψN

j∈Q

The second part follows by symmetry.



We shall need the following lemma in the sequel. Lemma 6. Let ΓD be the difference function defined in (4), and let D1 be the generalized cyclotomic class of order 2. Then ( 2 p −1 if w ∈ P 4 , ΓD1 (w) = p2 −9 if w ∈ Q. 4 , Proof. It follows from Lemma 5.32 in [2, p. 138].



Lemma 7. [26] If (p−1)(q−1)/4 is odd, for the generalized cyclotomic numbers of order 2, we have (0, 1)2 = (1, 0)2 = (1, 1)2 and p2 − 1 (p − 2)(q − 2) + 3 = , 4 4 (p − 2)(q − 2) − 1 p2 − 5 (0, 1)2 = = . 4 4 Lemma 8. If gcd(h, N ) = 1, then p 1 ± p(p + 4) (h) . ψN (D1 ) = 2 (0, 0)2 =

Proof. Since (p − 1)(q − 1)/4 is odd, −1 = g (p−1)(q−1)/2 ∈ D0 . It follows from (h) (h) (h) gcd(h, N ) = 1 that ψN (D1 ) = 1 − ψN (D0 ). Define δ := ψN (D1 ). By the definition of generalized cyclotomic numbers, Lemmas 4, 6, and 7, we have    X (h) X (h) ψN (ℓ)  δ2 =  ψN (−m) m∈D1

ℓ∈D1

=

X X

ℓ∈D1 m∈D1

(h)

ψN (ℓ − m) (h)

(h)

= |D1 | + (1, 1)2 ψN (D0 ) + (0, 0)2 ψN (D1 ) +

p2 − 1 (h) p2 − 9 (h) ψN (P ) + ψN (Q) 4 4

p2 − 5 (h) p2 − 1 (h) ψN (D0 ) + ψN (D1 ) + 4 4 p2 + 4p − 1 (h) = + ψN (D1 ) 4 p2 + 4p − 1 = + δ. 4 Solving the quadratic equation yields the two possible values of δ. = p+1+

6



Combining Lemmas 4, 5, and 8, we obtain the following. Lemma 9. Let symbols and notations be the same as before. Then  p+1 if gcd(h, N ) = p   − 2 , p+3 (h) , if gcd(h, N ) = q ψN (P ∪ D1 ) = 2 √  −1± p(p+4)  , if gcd(h, N ) = 1. 2

The following lemma follows from the autocorrelation values of the generalized cyclotomic sequence defined in [7, Theorem 2]. Lemma 10. Let ΓD be the difference function defined in (4), and let D be the almost difference set described before. Then ( 2 p +4p+7 , if w ∈ P ∪ D1 4 ΓD (w) = p2 +4p+3 , if w ∈ Q ∪ D0 . 4 With the preparations above, we are now read to state and prove the main result of of this subsection below. Theorem 11. Let p and q = p+4 both be prime, where p ≡ 3 (mod 4). For the (p(p + 4), (p + 3)(p + 1)/2) codebook CD of (5) defined by the almost difference set D of (8), we have q p p2 + 4p + 2 p(p + 4) + 1 Imax (CD ) = . (p + 1)(p + 3) Proof. Let 0 ≤ j < i ≤ N − 1. Note that K = (p + 1)(p + 3)/2 and ci cH j

K 1 X ψN ((i − j)dℓ ), = K ℓ=1

where the dℓ ’s are all the elements of D. It follows from Lemmas 9 and 10 that ci cH j


2

=

=

K K 1 XX ψN [(i − j)(dℓ − dm )] K2 ℓ=1 m=1   K K XX 1   ψN [(i − j)(dℓ − dm )] K + 2 K m=1 ℓ=1

=



1  K + K2

m6=ℓ

K K X X ℓ=1

(i−j)

ψN

m=1 m6=ℓ

p2 + 4p + 3 1 K − = K2 4  p2 +6p+5  2 (p+3)2 ,   (p+1) p2 +2p−3 , = (p+1)2 (p+3) √2  2   p +4p±2 p(p+4)+1 , (p+1)2 (p+3)2 

7

+



 [dℓ − dm ] (i−j) ψN (P

 ∪ D1 )

if gcd(i − j, N ) = p if gcd(i − j, N ) = q

if gcd(i − j, N ) = 1.

This completes the proof of this theorem.



Remark: Note that the Welch bound for (p(p + 4), (p + 3)(p + 1)/2) codebook CD is s s N −K p2 + 4p − 3 = . (N − 1)K (p2 + 4p − 1)(p + 3)(p + 1) Hence Imax (CD ) nearly meets the Welch bound. Examples of (p, p + 2) that yield such codebooks are

(3, 7), (7, 11), (19, 23), (43, 47), (67, 73), (97, 103).

3.2

(N, K) complex codebooks with N ≡ 2 (mod 4) and N ≡ 0 (mod 4)

Let N1 > 1 and N2 > 1 be two relatively prime integers. Let ψNi be the group character of the Abelian group ZNi defined in (7). For any (n1 , n2 ) ∈ ZN1 ×ZN2 , we define Ψ((n1 , n2 )) = ψN1 (n1 )ψN2 (n2 ).

(9)

Then Ψ is a group character of the Abelian group ZN1 × ZN2 . For any subset D := {d1 , d2 , ..., dK } of ZN1 × ZN2 , where di = (di,1 , di,2 ), we define a codebook UD := {ui : i = 0, ..., N1 N2 − 1},

(10)

where 1 u(i1 ,i2 ) = √ (Ψ((i1 d1,1 , i2 d1,2 )), Ψ((i1 d2,1 , i2 d2,2 )), ..., Ψ((i1 dK,1 , i2 dK,2 ))) (11) K for each (i1 , i2 ) ∈ ZN1 × ZN2 . Note that K X H Ψ((i1 − j1 )dℓ,1 , (i2 − j2 )dℓ,2 )). u(i1 ,i2 ) u(j1 ,j2 ) = ℓ=1

We obtain

2 K 2 u(i1 ,i2 ) uH (j1 ,j2 ) =

=

K K X X

m=1 ℓ=1 K K X X

Ψ((i1 − j1 )(dm,1 − dℓ,1 ), (i2 − j2 )(dm,2 − dℓ,2 ))

Ψ(i1 −j1 ,i2 −j2 ) ((dm,1 − dℓ,1 ), (dm,2 − dℓ,2 ))

m=1 ℓ=1 K X

= K+

m=1

K X ℓ=1 ℓ6=m

Ψ(i1 −j1 ,i2 −j2 ) ((dm,1 − dℓ,1 ), (dm,2 − dℓ,2 )),

(12)

where Ψ(i,j) ((n1 , n2 )) := Ψ((in1 , jn2 )) is a group character of ZN1 × ZN2 . Below we will describe complex codebooks UD by selecting almost difference sets D in ZN1 × ZN2 , where N1 and N2 are properly selected positive integers. 8

3.2.1

The N ≡ 2 (mod 4) case

Lemma 12. [11] Let q ≡ 5 (mod 8) be a prime. It is known that q = s2 + 4t2 for some s and t with s ≡ ±1 (mod 4). Set N = 2q. Let i, j, ℓ ∈ {0, 1, 2, 3} be three pairwise distinct integers, and define i h i h (4,q) (4,q) (4,q) (4,q) (13) D(i,j,ℓ) = {0} × (Di ∪ Dj ) ∪ {1} × (Dℓ ∪ Dj ) .
Then D(i,j,ℓ) is an N, N 2−2 , N 4−6 , 3N4−6 almost difference set in Z2 × Zq if (1) t = 1 and (i, j, ℓ) = (0, 1, 3) or (0, 2, 1); or (2) s = 1 and (i, j, ℓ) = (1, 0, 3) or (0, 1, 2) Theorem 13. Let t = 1. For the almost difference set D(0,1,3) of (13), UD(0,1,3) is a (2q, q − 1) codebook with   p2q + 2√q Imax UD(0,1,3) = . 2(q − 1) Proof. It was proved in [8] that     √ (4,q) (4,q) (2,q) ψq D0 ∪ {0} = ψq D0 ∪ D2 ∪ {0} = (1 ± q)/2. Recall the difference function ΓD of (4). It was proved in [11] that ( (4,q) (4,q) q−3 ∗ ∪ D3 )] 2 , w ∈ [{0} × Zq ] ∪ [{1} × (D1 ΓD(0,1,3) (w) = (4,q) (4,q) q−1 ∪ D2 ∪ {0}). 2 , w ∈ {1} × (D0

For any distinct pairs (i1 , i2 ) and (j1 , j2 ), it follows from (12) that 2 K 2 u(i1 ,i2 ) uH (j1 ,j2 )   q−3 (4,q) (4,q) = K− + Ψ(i1 −j1 ,i2 −j2 ) {1} × (D0 ∪ D2 ∪ {0}) 2 ( q−3 q+1 K−  2 − 2 ,  i2 = j2 = (4,q) (4,q) ψq D0 ∪ D2 ∪ {0} , i2 6= j2 ( 0, √ i2 = j2 = q± q 2 , i2 6= j2 This completes the proof.



Remark: Note that the Welch bound for (2p, p − 1) codebooks is q s 2q + 2 2q−1 q−1 N −K = . (N − 1)K 2q − 1
Hence Imax UD0,1,3 nearly meets the Welch bound. 9

Theorem 14. Let s = 1. For the almost difference set D(0,1,2) of (13), UD(0,1,2) is a (2q, q − 1) codebook with   p2q + 2√q Imax UD(0,1,2) = . 2(q − 1) Proof. It was proved in [8] that     √ (4,q) (4,q) (2,q) ψq D1 ∪ {0} = ψq D1 ∪ D3 ∪ {0} = (1 ± q)/2. Recall the difference function ΓD of (4). It was proved in [11] that ( (4,q) (4,q) q−3 , w ∈ [{0} × Z∗q ] ∪ [{1} × (D0 ∪ D2 )] 2 ΓD(0,1,2) (w) = (4,q) (4,q) q−1 ∪ D3 ∪ {0}). 2 , w ∈ {1} × (D1

For any distinct pair (i1 , i2 ) and (j1 , j2 ), it follows from (12) that 2 K 2 u(i1 ,i2 ) uH (j1 ,j2 )

  q−3 (4,q) (4,q) = K− + Ψ(i1 −j1 ,i2 −j2 ) {1} × (D1 ∪ D3 ∪ {0}) 2 ( q+1 q−3 K−  2 − 2 ,  i2 = j2 = (4,q) (4,q) ψq D1 ∪ D3 ∪ {0} , i2 6= j2 ( 0, √ i2 = j2 = q± q 2 , i2 6= j2

This completes the proof.



Lemma 15. [11] Let q ≡ 5 (mod 8) be a prime. It is known that q = s2 + 4t2 for some s and t with s ≡ ±1 (mod 4). Set N = 2q. Let i, j, ℓ ∈ {0, 1, 2, 3} be three pairwise distinct integers, and define h  i h  i (4,q) (4,q) (4,q) (4,q) D(i,j,ℓ) = {0} × Di ∪ Dj ∪ {1} × Dℓ ∪ Dj ∪ {0, 0}.


Then D(i,j,ℓ) is an N, N2 , N 4−2 , 3N4−2 almost difference set of Z2 × Zq if (1) t = 1 and (i, j, ℓ) ∈ {(0, 1, 3), (0, 2, 3), (1, 2, 0), (1, 3, 0)}; or

(2) s = 1 and (i, j, ℓ) ∈ {(0, 1, 2), (0, 3, 2), (1, 0, 3), (1, 2, 3)}. The almost difference sets D(i,j,ℓ) in Lemma 15 yield (2q, q) codebooks   UD(i,j,ℓ) that nearly meet the Welch bound. The parameter Imax UD(i,j,ℓ ) can be computed similarly.

10

3.2.2

The N ≡ 0 (mod 4) case

Difference sets with parameters (ℓ, (ℓ − 1)/2, (ℓ − 3)/4) are sometimes called Paley-Hadamard difference sets. Their complements are also difference sets with parameters (ℓ, (ℓ + 1)/2, (ℓ + 1)/4). Known cyclic Paley-Hadamard difference sets are of the following types: 1. Skew Hadamard difference sets with parameters (p, (p − 1)/2, (p − 3)/4), where p ≡ 3 (mod 4) is prime (Paley-Hadamard and other skew Hadamard difference sets [3, 12]; 2. Difference sets with Singer parameters (2t − 1, 2t−1 − 1, 2t−2 − 1) (see [19] and [29]); 3. Twin-prime difference sets with parameters (ℓ, (ℓ − 1)/2, (ℓ − 3)/4), where ℓ = p(p + 2) and both p and p + 2 are primes [3, 19]. 4. Hall difference sets with parameters (p, (p − 1)/2, (p − 3)/4), where p is a prime of the form p = 4s2 + 27 [3, 19]. Let N = 4ℓ, where ℓ is an odd positive integer. Then the ring ZN is isomorphic
to Z4 × Zℓ under the morphism φ(x) = (x mod 4, x mod ℓ). Cyclic ℓ−3 ℓ, ℓ−1 difference sets in Zℓ were used to construct almost difference sets 2 , 4 in Z4 × Zℓ , which are stated as follows.
ℓ−3 Lemma 16. [1] Let E be an ℓ, ℓ−1 difference set in Zℓ . Then 2 , 4
(14) D = ({0} × E) ∪ {1, 2, 3} × E

is a (4ℓ, 2ℓ + 1, ℓ, ℓ − 1) almost difference set in Z4 × Zℓ , where E is the complement of E.

We now consider the codebook of (10) defined by the almost difference sets of (14). Theorem 17. For the (4ℓ, 2ℓ + 1) codebook UD of (10) defined by the almost difference sets D of (14), we have √ ℓ+1 . Imax (UD ) = 2ℓ + 1 Proof. To prove this theorem, we need the notion of multiset. A multiset S is a set where an element of S may appear more than one time in S. If T is a set, 2 · T denotes the multiset which contains two copies of each element of T . If T1 and T2 are subsets of an Abelian group (A, +), we define T1 ⊖ T2 to be the multiset {t1 − t2 : t1 ∈ T1 , t2 ∈ T2 }.
ℓ−3 Since the set E employed to define the almost difference set is an ℓ, ℓ−1 2 , 4
ℓ+1 difference set, we have difference set, and its complement D is an ℓ, ℓ+1 2 , 4  ℓ−3 E ⊖ E = ℓ+1  4 · {0} ∪ 4 · Zℓ ,   ℓ+1 E ⊖ E = ℓ+1 4 · {0} ∪ 4 · Zℓ , (15) ℓ+1 ∗  E ⊖ E = 4 · Zℓ ,   ∗ E ⊖ E = ℓ+1 4 · Zℓ . 11

It then follows from (15) that D ⊖ D = [(ℓ + 1) · (Z4 × Zℓ ) ∪ (ℓ + 1) · {(0, 0)}] \ {0} × Zℓ .

(16)

Let (i1 , i2 ) and (j1 , j2 ) be two distinct elements of Z4 × Zℓ . Then the group character Ψ((i1 −j1 ,i2 −j2 )) on Z4 × Zℓ is nontrivial. Then we have Ψ((i1 −j1 ,i2 −j2 )) (Z4 × Zℓ ) = 0,

(17)

and Ψ((i1 −j1 ,i2 −j2 )) ({0} × Zℓ ) =



ℓ, 0,

if i2 = j2 if i2 = 6 j2 .

(18)

The conclusion on Imax (UD ) then follows from (12), (16), (17) and (18).  Remark: Note that the Welch bound for (4ℓ, 2ℓ + 1) codebooks is s s N −K 2ℓ − 1 = . (N − 1)K (4ℓ − 1)(2ℓ + 1) Hence Imax (UD ) nearly meets the Welch bound. With the different types of cyclic difference sets in (A), (B), (C), (D) and (E) above, Theorem (17) yields five classes of codebooks nearly meeting the Welch bound.

3.3

Other families of codebooks from almost difference sets

The following cyclic almost difference sets are stated in terms of optimal autocorrelation values of a class of binary sequences [16, 21]. (2,q)

Lemma 18. [16] Let q be odd. Define D = logα (D1

− 1). Then the set D is

1. a (q − 1, (q − 1)/2, (q − 3)/4, (3q − 5)/4) almost difference set in Zq−1 if q ≡ 3 (mod 4), and 2. a (q − 1, (q − 1)/2, (q − 5)/4, (q − 1)/4) almost difference set in Zq−1 if q ≡ 1 (mod 4). These two families of cyclic almost difference sets certainly yield codebooks. But it may not be easy to compute the maximum cross-correlation amplitude. In general, we do not know if every almost difference set yields a codebook nearly meeting the Welch bound. We have to compute the maximum cross-correlation amplitude, before drawing a conclusion. Nevertheless, we conjecture that all the examples of codebooks presented in this paper are optimal, although they do not meet the Welch bound. For those parameters (N, K), the Welch bound may not be achievable. We invite the reader to attack this conjecture.

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4

Summary and concluding remarks

In Table 1, we list some (N, K) codebooks of small sizes from almost difference sets, where N mod 4 takes on 0, 1 and 2. The (N, K) codebooks from the complements of the almost difference sets are not listed in this table. In the case N mod 4 = 3, (N, K) MWBE codebooks are produced by difference sets (see [28, 8, 9]). In the table, we also compare Imax and the Welch bound. Table 1: (N, K) N K 9 4 10 4 12 7 13 3 21 12 26 12 28 15 29 7 37 10 44 23 53 14 58 28 60 31 61 15 74 36 77 40 81 40

codebooks Imax 0.5 0.475528 0.285714 0.691437 0.232607 0.240121 0.188561 0.408895 0.324674 0.150613 0.268594 0.148085 0.129032 0.261662 0.128924 0.122187 0.125

of small sizes from almost difference sets Welch bound Construction 0.395284 Theorem 8 in [8] 0.408248 Theorem 13 here 0.254823 Theorem 17 here 0.527046 Theorem 10 in [8] 0.193649 Theorem 11 here 0.216024 Theorem 13 here 0.179161 Theorem 17 here 0.335029 Theorem 10 in [8] 0.273861 Theorem 12 in [8] 0.145717 Theorem 17 here 0.231455 Theorem 12 in [8] 0.137102 Theorem 13 here 0.125919 Theorem 17 here 0.226077 Theorem 10 in [8] 0.120248 Theorem 13 here 0.099339 Theorem 11 here 0.113192 Theorem 8 in [8]

Welch’s bound of (1) is not tight for binary codebooks in several cases. Karystinos and Pados [13] have developed new bounds that improve Welch’s bound of (1) in these cases. Binary codebooks meeting the Karystinos-Pados bounds are discussed in [13, 10, 15]. As pointed out in [17], constructing optimal codebooks with minimal Imax is very difficult in general. This problem is equivalent to line packing in Grassmannian spaces [5]. In frame theory, such a codebook with Imax minimized is referred to as a Grassmannian frame [23]. The codebooks presented in this paper should have applications in these areas.

Acknowledgments The authors would thank the referee for pointing out an error in an earlier version and his constructive comments and suggestions that much improved both the quality and readability of this paper. The research of Cunsheng Ding is supported by the Research Grants Council of the Hong Kong Special Administration Region, China (Proj. No. 612405). 13

References [1] K.T. Arasu, C. Ding, T. Helleseth, P.V. Kumar, and H. Martinsen, Almost difference sets and their sequences with optimal autocorrelation, IEEE Trans. Inform. Theory 47(7) (2001), 2834–2843. [2] L.D. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics, vol. 182, Berlin: Springer, 1971. [3] T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1999. [4] A.R. Calderbank, P.J. Cameron, W.M. Kantor, and J.J. Seidel, Z4 Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets, Proc. London Math. Soc. 75(3) (1997), 436–480. [5] J.H. Conway, R.H. Harding and N.J.A. Sloane, Packing lines, planes, etc.: packings in Grassmannian spaces, Exper. Math. 5(2) (1996), 139–159. [6] P. Delsarte, J.M. Goethals, J.J. Seidel, Spherical codes and designs, Geometriae Dedicate 67(3) (1977), 363–388. [7] C. Ding, Autocorrelation values of the generalized cyclotomic sequences of order 2, IEEE Trans. Inform. Theory 44 (1998), 1698–1702. [8] C. Ding, Complex codebooks from combinatorial designs, IEEE Trans. Inform. Theory 52(9) (2006), 4229–4235. [9] C. Ding and T. Feng, A generic construction of complex codebooks meeting the Welch bound, preprint, 2006. [10] C. Ding, M. Golin and T. Kløve, Meeting the Welch and Karystinos-Pados bounds on DS-CDMA binary signature sets, Designs, Codes and Cryptography 30 (2003), 73–84. [11] C. Ding, T. Helleseth and H.M. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inform. Theory 47 (2001), 428–433. [12] C. Ding and J. Yuan, A family of skew Paley-Hadamard difference sets, J. of Combinatorial Theory A 113(7) (2006), 1526–1535. [13] G.N. Karystinos and D.A. Pados, New bounds on the total-squaredcorrelation and perfect design of DS-CDMA binary signature sets, IEEE Trans. Communications 3 (2003), 260–265. [14] H. K¨onig, Cubature formulas on spheres, in: Advances in Multivarate Approximation. New York: Wiley, 1979, pp. 201–211. [15] V.P. Ipatov, On the Karystinos-Pados bounds and optimal binary DSCDMA signature ensambles, IEEE Comm. Letters 8(2) (2004), 81–83.

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[16] A. Lempel, M. Cohn, and W.L. Eastman, A class of binary sequences with optimal autocorrelation properties, IEEE Trans. Inform. Theory 23 (1997), 38–42. [17] D.J. Love, R.W. Heath and T. Strohmer, Grassmannian beamingforming for multiple-input multiple-output wireless systems, IEEE Trans. Inform. Theory 49(10) (2003), 2735–2747. [18] J.L. Massey and T. Mittelholzer, Welch’s bound and sequence sets for code-division multiple-access systems, in Sequences II: Methods in Communication, Security and Computer Science, Heidelberg and New York: Springer, 1993, pp. 63–78. [19] A. Pott, Finite Geometry and Character Theory, Lecture Notes in Mathematics, vol. 1601. Berlin: Springer, 1995. [20] D. Sarwate, Meeting the Welch bound with equality, in: Sequences and their Applications, Proc. of SETA’ 98, London: Springer, 1999, pp. 79– 102. [21] V.M. Sidelnikov, Some k-valued pseudo-random sequences and nearly equidistant codes, Probl. Inform. Trans. 5(1), 12–16. [22] S.W. Golomb, Cyclic Hadamard difference sets – constructions and applications, in: Sequences and their Applications, Proc. of SETA’ 98, London: Springer, 1999, pp. 39–48. [23] T. Strohmer and R.W. Heath Jr., Grassmannian frames with applications to coding and communication, Applied Computational Harmonic Analysis 14(3) (2003), 257–275. [24] M.A. Sustik, J.A. Tropp, I.S. Dhillon, R.W. Heath, Jr., On the existence of equiangular tight frames, preprint 2005. [25] L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory 20(3) (1974), 397–399. [26] A.L. Whiteman, A family of difference sets, Illinois J. Math. 6 (1962), 107–121. [27] W. Wootters and B. Fields, optimal state-determination by mutually unbiased measurements, Ann. Physics 191(2) (2005), 363–381. [28] P. Xia, S. Zhou and G.B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inform. Theory 51(5) (2005), 1900–1907. [29] Q. Xiang, Recent results on difference sets with classical parameters, in: Difference Sets, Sequences and their Correlation Properties, eds., A. Pott, P.V. Kumar, T. Helleseth and D. Jungnickel, pp. 419–434. Amsterdam: Kluwer, 1999.

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