# New Constructions of Asymptotically Optimal Codebooks via Cyclotomic Classes of Order 8

Authors: Wanfeng Qi; Yueying Song; Rui Ma; Lingli Tang; Qian Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Asymptotically optimal codebooks are a family of codebooks that can approach an optimal codebook meeting the Welch bound when the lengths of codewords are large enough. They can be constructed easily and are a good alternative for optimal codebooks in many applications. In this paper, we construct a new class of asymptotically optimal codebooks by using the product of some special finite fields and almost difference sets, which are composed of cyclotomic classes of order eight.


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## Body

## 1. Introduction

An(N,K) codebook C is a set {c0,c1,…,cN-1}, where each codeword cl, 0≤l≤N-1, is a unit norm complex vector in CK. The value of maximum cross-correlation amplitude of an (N,K) codebook C is defined by (1)ImaxC=max0≤i<j≤N-1cicjH,where cH denotes the conjugate transpose of a complex vector c. Codebooks with minimal possible Imax(C) are desirable in many applications. From a geometric point of view, the minimal possible Imax(C) means that the smallest angle between two different lines where the codewords are located is as large as possible. Welch [1] presented the following well-known lower bound for Imax(C).Lemma 1 (see [1]).
For any(N,K) codebook C with N≥K, (2)ImaxC≥IWelch=N-KN-1K.Moreover, the equality holds if and only if, for all pairs of (i,j) with i≠j, it holds that (3)cicjH=N-KN-1K.A codebookC meeting the Welch bound is called a maximum-Welch-bound-equality (MWBE) codebook. The interests in MWBE codebooks have significantly increased because of their special properties. However, it is usually difficult to construct a family of MWBE codebooks [2, 3].Most existing methods for constructing infinite families MWBE codebooks fall into three categories: strongly regular graph methods [4], difference sets methods [5], and Steiner systems methods [6].It is known that the Welch bound is not tight whenN>K2. In fact, N<K2 is only the necessary condition of the existence of an (N,K) MWBE codebook [2]. Even when N<K2 holds, the corresponding (N,K) MWBE codebook may not exist [7]. In recent years, asymptotically optimal codebooks have attracted widespread attention, since they can be constructed easily and Imax(C) can approach the Welch bound for large enough N. Thus, they are good alternatives in many applications.Many methods can be used to construct asymptotically optimal codebooks, such as almost difference sets [8–10], binary sequences [11], and character sums in finite fields [12, 13]. Hu and Wu [14] constructed some asymptotically optimal codebooks by using difference sets and the product of Abelian groups. In this paper, we generalize this method to the case of cyclotomic classes of order 8 in finite fields and propose new classes of codebooks asymptotically meeting the Welch bound.This paper is organized as follows. In Section2, we recall some basic theory and definitions that will be used in our discussion. In Section 3, we present our main results. In Section 4, we give conclusions and future work.


## 2. Preliminaries

Letp be a prime number; q=pn,n≥1. For a finite field Fq, let θ be its one fixed primitive element, and let Fq∗ be its all nonzero elements.Letq=ef+1,e≥2,f≥2. The set Cj(e,q)≔θek+j,k∈Z=θek+j,k=0,1,…,f-1,j=0,1,…,e-1 is called a cyclotomic class of order e. Clearly, Fq∗=⋃j=0e-1Cj(e,q), and |Cj(e,q)|=f,j=0,1,…,e-1.The cyclotomic numbers of ordere with respect to Fq are given by (4)l,me=Cle,q+1∩Cme,q,where S+a≔s+a:s∈S. Cyclotomic number (l,m)e is the number of solutions to the equation 1=y-x,x∈Cl(e,q),y∈Cm(e,q).Cyclotomic class is now widely used to construct almost difference sets. An almost difference setD of a finite field Fq is a subset with k elements satisfying special properties. Define (5)△x=d,d′∈D2:d-d′=x.D is called a (q,k,λ,t) almost difference set of Fq if △x=λ for t nonzero elements of Fq, and △x=λ+1 for the remaining q-1-t nonzero elements of Fq. When q≡1(mod4), C0(2,q) is a (q,q-1/2,q-5/4,q-1/2) almost difference set, which is also called the Paley partial difference set [15].Letq=8f+1; Lehmer [16] presented the explicit form of the 64 cyclotomic numbers of order e=8 when q≡9(mod16), which are functions whose parameters are determined by the unique representation(6)q=x2+4y2=a2+2b2,x,y,a,b∈Z,x≡a≡1mod4.The 64 cyclotomic numbers have two forms determined by whether 2 is a quartic residue inFq or not. Let D=C0(8,q)∪C1(8,q)∪C2(8,q)∪C5(8,q),△j=|(D+θj)∩D|,0≤j≤7. When 2 is a quartic residue in Fq, Ding et al. [17] calculated that(7)△0=△4=16q-48+8x-8a-48y64,△1=△5=16q-80-16x+16a-32y64,△2=△6=16q-48+8x-8a-16y64,△3=△7=16q-1664.When 2 is not a quartic residue in Fq, Zhang and Feng [10] calculated that(8)△0=△4=16q-48+8x-8a+16y64,△1=△5=16q-8064,△2=△6=16q-48+8x-8a-16y64,△3=△7=16q-16-16x+16a64.Ifl is a power of a prime and l=t2+2≡3(mod8), then, for q=l2, Ding [17] proved that q≡9(mod16), x=-l, y=0, a=-l+4, b=±2t, and 2 is a quartic residue in Fq∗, and D=C0(8,q)∪C1(8,q)∪C2(8,q)∪C5(8,q) is a (q,q-1/2,q-5/4,q-1/2) almost difference set of Fq. Thus, Ck(8,q)∪C1+k(8,q)∪C2+k(8,q)∪C5+k(8,q),k=0,1,…,7 are also (q,q-1/2,q-5/4,q-1/2) almost difference sets of Fq.An additive character ofFq with characteristic p is a homomorphism from Fq to the unit circle in the complex plane, which in fact has the following form [18]: (9)χcg=exp⁡2π-1Trcgp,∀g∈Fq,c∈Fq,where Tr is the trace function from Fq to Fp. Multiplicative characters of Fq are characters of the multiplicative group Fq∗. Let θ be a fixed primitive element of Fq. All multiplicative characters ψj of Fq have the following form [18]: (10)ψjθk=exp⁡2π-1jkq-1,fork=0,1,…,q-2,j=0,1,…,q-2.The set of all additive characters ofFq is denoted by F^q, and F^q=q. When c=0, χ0 is trivial, which satisfies χ0(g)=1,∀g∈Fq. For a subset D and an additive character χ of Fq, χ(D) is defined to be χ(D)=∑g∈Dχ(g). Clearly, χ(D¯)=-χ(D).Form additive characters χ1,χ2,…,χm of Fq, we can define their product as (11)χ1χ2…χmg=χ1gχ2g…χmgfor any g∈Fq.The direct product ofn finite fields Fq1,Fq2,…,Fqn can be defined by (12)Fq1×Fq2×⋯×Fqn=g1,g2,…,gn,gi∈Fqi,i=1,…,n.Correspondingly, an additive character χ1⊗χ2⊗⋯⊗χn on the direct product is defined by (13)χ1⊗χ2⊗⋯⊗χng1,g2,…,gn=χ1g1χ2g2⋯χngn,where (g1,g2,…,gn)∈Fq1×Fq2×⋯×Fqn and χ1∈F^q1,χ2∈F^q2,…,χn∈F^qn.


## 3. Main Result

In this section, we will follow Hu’s method [14] to construct asymptotically optimal codebooks. Hu’s method [14] used the product of difference sets of finite Abelian groups of order Ni, where Ni≡3(mod4). In our paper, we use two classes of (q,(q-1)/2,(q-5)/4,(q-1)/2) almost difference sets in finite fields: one class is Ck(8,q)∪C1+k(8,q)∪C2+k(8,q)∪C5+k(8,q),k=0,1,…,7, q≡9(mod16), and the other is C0(2,q) and C1(2,q), q≡1(mod4).For the case ofD=C0(8,q)∪C1(8,q)∪C2(8,q)∪C5(8,q), we first estimate χ(D) for any nontrivial additive characters of a finite field. In fact, Zhang and Feng had given an upper bound of χ(D) in Theorem 1 of [10]. We rephrase it in our setting.Lemma 2.
LetFq be a finite field, q≡9(mod16), and x,y,a,b are determined by (6). D=C0(8,q)∪C1(8,q)∪C2(8,q)∪C5(8,q). χ(·)=χc(·)=exp⁡2π-1Trc·/p is a nontrivial additive character. If limq->∞|x-a|/q=0 and limq->∞|y|/q=0, then there exists a function f(q) satisfying limq->∞f(q)/q=0 such that (14)χD≤q+fq2,χD¯≤q+fq2.Proof.
Letηj=∑g∈Cj8,qexp⁡2π-1Tr(g)/p. ηj is a Gauss period of order 8. Since c≠0, let c∈Cλ(8,q). Then (15)χD2=∑x∈D∑y∈Dexp⁡2π-1Trcx-yp=q-12+∑x∈D∑y≠x,y∈Dexp⁡2π-1Trcx-yp=q-12+∑j=07∑x∈D∑y∈D,x-y∈Cj8,qexp⁡2π-1Trcx-yp=q-12+∑j=07ηj+λ△j=q-12+∑j=07ηj+λ△j-q-14+q-14=q-14+∑j=07ηj+λ△j-q-14.Note that, whether 2 is a quartic residue in Fq or not, △i=△i+2,i=0,1,2,3. Thus (16)χD2=q-14+∑j=03ηj+λ+ηj+4+λ△j-q-14.
Letψ be the multiplicative character satisfying ψ(θ)=ψq-1/4(θ1)=exp⁡2π-1/4. ψ(gθ-j)=ψgexp⁡-2π-1j/4. So if g∈Cj(8,q)∪Cj+4(8,q), we have ∑i=03ψigθ-j=4, or else ∑i=03ψigθ-j=0. Thus, (17)ηj+ηj+4=∑g∈Cj8,q∪Cj+48,qexp⁡2π-1Trgp=14∑g∈Fq∗exp⁡2π-1Trgp∑i=03ψigθ-j=14∑i=03Gψiexp⁡-2π-1ij4,where G(ψ)=∑g∈Fq∗ψgexp⁡2π-1Trg/p is the Gaussian sum over Fq and G(ψ0)=-1,|G(ψi)|=q,i=1,2,3 [18]. Thus, |ηj+ηj+4|≤(1/4)(1+3q).
Then(18)χD2=q-14+∑j=03ηj+λ+ηj+4+λ△j-q-14≤q-14+141+3q∑j=03△j-q-14.When 2 is a quartic residue in Fq, substituting (7) into (18) yields (19)χD2≤q+14+3q2+1+3q8x-a+3y.When 2 is a quartic nonresidue in Fq, substituting (8) into (18) yields (20)χD2≤q+14+3q2+1+3q8x-a+y.Denote f(q)=1/4+3q/2+1+3q/8(|x-a|+3|y|). If limq->∞|x-a|/q=0 and limq->∞|y|/q=0, then limq->∞f(q)/q=0. Thus (21)χD≤q+fq2.Since χ(D¯)=-χ(D), (22)χD¯=χD≤q+fq2.Remark 3.
There exist infinitely manyx,y,a,b satisfying the assumptions in Lemma 2. Ding et al. [17] proved that if q=l2≡9(mod16),l=t2+2≡3(mod8), then x=-l,y=0,a=-l+4,b=±2t, where t is an integer. In this case, we have |x-a|=4,|y|=0, D becomes a (q,(q-1)/2,(q-5)/4,(q-1)/2) almost difference set, and |χ(D)|≤q+12q+3/2.Remark 4.
Lemma2 still holds if we use D~=Cl(8,q)∪C1+l(8,q)∪C2+l(8,q)∪C5+l(8,q),l=1,…,7 instead of D=C0(8,q)∪C1(8,q)∪C2(8,q)∪C5(8,q). The reason is that new △~j=|(D~+θj)∩D~| is equal to △j-l. Thus, it has no influence on estimating |χ(D)|. However, if we use other combinations of four cyclotomic classes which do not correspond to a family of almost difference sets, there may not exist infinitely many x,y,a,b satisfying the assumptions in Lemma 2. The reason is that the eight △~j,j=0,…,7 will have more different values, which inevitably lead to more restrictions on x,y,a,b. Thus, it is more difficult to find infinitely many q such that |χ(D)| is not too large.Remark 5.
SinceD=C0(8,q)∪C2(8,q)∪C4(8,q)∪C6(8,q) can be seen as C0(2,q), we consider C0(2,q) and C1(2,q), where q≡1(mod4). Ding [19] proved that χ(C0(2,q))=-1±q/2 and χ(C1(2,q))=-1∓q/2. Thus, for D=C0(2,q) or D=C1(2,q), we have maxχD=q+1/2.Suppose thatFq1,Fq2,…,Fqn are n finite fields. Let F=Fq1×Fq2×…×Fqn. Then F^=χ1,j1⊗χ2,j2⊗⋯⊗χn,jn∣χi,ji∈F^qi,1≤i≤n.LetDi=C0(8,qi)∪C1(8,qi)∪C2(8,qi)∪C5(8,qi) be the union of the four cyclotomic classes of order 8 in Fqi,qi≡9(mod16),i=1,…,n. Let Di¯ be the complement of Di with respect to Fqi. Note that |Di|=qi-1/2.LetE1=D1. For any 1≤i≤n-1, put (23)Ei+1=Ei×D¯i+1∪E¯i×Di+1.Finally, let D=En, and K=|D|. Let N=|F|; then N=q1q2⋯qn.We can define similar sets in the case ofFqi,qi≡1(mod4),i=1,…,n. More specifically, let Di=C0(2,qi)⊂Fqi,qi≡1(mod4),i=1,…,n. Then we can define D¯i, Ei,i=1,…,n in a similar way.For both of the two cases, we have the following lemma.Lemma 6.
| E i | = ( q 1 q 2 ⋯ q i - 1 ) / 2 for any 1≤i≤n. In particular, K=(N-1)/2.Proof.
Fori=1, we have |E1|=(q1-1)/2. For i>1, suppose that |Ei|=(q1q2…qi-1)/2. Then, it follows that (24)Ei+1=q1q2⋯qi-12·qi+1+12+q1q2⋯qi+12·qi+1-12=q1q2…qi+1-12.Therefore, |En|=(q1q2⋯qn-1)/2=(N-1)/2.We define an(N,K) codebook by(25)CD=Cj1,j2,…,jn∣ji=0,1,…,qi-1,i=1,2,…,n.The vector Cj1,j2,…,jn∈CK is defined by (26)Cj1,j2,…,jn=1Kχ1,j1⊗χ2,j2⊗⋯⊗χn,jn)gi,1,gi,2,…,gi,ni=1K=1Kχ1,j1gi,1χ2,j2gi,2…χn,jngi,ni=1K,where χi,ji∈F^qi,(gi,1,gi,2,…,gi,n)∈D and χi,0 is the corresponding trivial character. It is obvious that |CD|=N, and the length of the vector Cj1,j2,…,jn is |D|=K.Next we will investigateImax(CD). For the case of Di=C0(8,qi)∪C1(8,qi)∪C2(8,qi)∪C5(8,qi)⊂Fqi,qi≡9(mod16), we need the following lemma.Lemma 7.
With the above notations, for any characterχ1,j1⊗χ2,j2⊗⋯⊗χn,jn of Fq, ji=0,1,…,qi-1,qi≡9(mod16),i=1,2,…,n. If (j1,j2,…,jn)=(0,0,…,0), we have (27)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=K.If (j1,j2,…,jn)≠(0,0,…,0), (28)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD≤q1+fq1⋯qn+fqn2.Proof.
Ifn=1, when j1=0, χ1,j1 is the trivial character; thus, χ1,j1(D)=|D|=q1-1/2=K. When j1≠0, χ1,j1 is a nontrivial character. Thus, by Lemma 2, |χ1,j1(D)|≤q1+fq1/2.
Ifn=2, we need to consider three cases:
(1) j 1 = j 2 = 0 (29) χ 1 , j 1 ⊗ χ 2 , j 2 D = D = E 2 = q 1 q 2 - 1 2 = K .
(2) j 1 = 0 , j 2 ≠ 0 (30) χ 1 , j 1 ⊗ χ 2 , j 2 D = χ 1 , j 1 D 1 χ 2 , j 2 D ¯ 2 + χ 1 , j 1 D ¯ 1 χ 2 , j 2 D 2 = D 1 χ 2 , j 2 D ¯ 2 + D ¯ 1 χ 2 , j 2 D 2 = D 1 - D ¯ 1 χ 2 , j 2 D ¯ 2 + D ¯ 1 χ 2 , j 2 F q 2 .Thus, (31)χ1,j1⊗χ2,j2D=D1-D¯1χ2,j2D¯2≤D1-D¯1q2+fq22=q2+fq22<q1+fq1q2+fq22.
In a similar way, ifj1≠0,j2=0, (32)χ1,j1⊗χ2,j2D≤q1+fq12<q1+fq1q2+fq22.
(3) j 1 ≠ 0 ,   j 2 ≠ 0 (33) χ 1 , j 1 ⊗ χ 2 , j 2 D = χ 1 , j 1 D 1 χ 2 , j 2 D ¯ 2 + χ 1 , j 1 D ¯ 1 χ 2 , j 2 D 2 = 2 χ 1 , j 1 D 1 χ 2 , j 2 D 2 ≤ q 1 + f q 1 · q 2 + f q 2 2 .
Now we assume that the formula holds forn-1; we will prove it for n.
If(j1,j2,…,jn)=(0,0,…,0), then the character χ1,j1⊗χ2,j2⊗⋯⊗χn,jn is trivial. Hence, the result is correct. In the following, we consider the case of n>2 and (j1,j2,…,jn)≠(0,0,…,0). The proof can also be divided into three cases:
Case (1) ((j1,j2,…,jn-1)=(0,0,…,0),jn≠0). We compute (34)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=χ1,j1⊗χ2,j2⊗⋯⊗χn,jnEn-1×D¯n∪E¯n-1×Dn=En-1χn,jnD¯n+E¯n-1χn.jnDn=-En-1χn.jnDn+E¯n-1χn.jnDn.By Lemma 6, |E¯n-1|-|En-1|=1. So we have (35)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=χn,jnDn≤qn+fqn2<q1+fq1⋯qn+fqn2.
Case (2) ((j1,j2,…,jn-1)≠(0,0,…,0),jn=0). In this case, we get(36)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=χ1,j1⊗χ2,j2⊗⋯⊗χn,jnEn-1×D¯n∪E¯n-1×Dn=χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1En-1D¯n+χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1E¯n-1Dn=χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1En-1D¯n-χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1En-1Dn=χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1En-1.Hence, by the assumption,(37)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1En-1≤q1+fq1⋯qn-1+fqn-12<q1+fq1⋯qn+fqn2.
Case (3) ((j1,j2,…,jn-1)≠(0,0,…,0),jn≠0). In this case, we get(38)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=χ1,j1⊗χ2,j2⊗⋯⊗χn,jnEn-1×D¯n∪E¯n-1×Dn=χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1En-1·χn,jnD¯n+χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1E¯n-1·χn,jnDn=-2χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1En-1·χn,jnDn.Hence, by the assumption and Lemma 2,(39)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=-2χ1,j1⊗χ2,j2⊗⋯⊗χn-1,jn-1En-1·χn,jnDn≤q1+fq1⋯qn+fqn2.
The proof is finished.Theorem 8.
LetCD be a codebook defined by (25) in the case of Di=C0(8,qi)∪C1(8,qi)∪C2(8,qi)∪C5(8,qi)⊂Fqi,qi≡9(mod16), and qi satisfies the assumptions in Lemma 2. Then CD is an (N,K)=(q1q2⋯qn,(q1q2⋯qn-1)/2) codebook with (40)ImaxCD≤q1+fq1⋯qn+fqnq1⋯qn-1,and Imax(CD) can reach the Welch bound if qi→∞ for any 1≤i≤n.Proof.
LetCj1,j2,…,jn and Cj1′,j2′,…,jn′ be two different codewords in CD. Then we have(41)Cj1,j2,…,jnCj1′,j2′,…,jn′H=1K∑i=1Kχ1,j1gi,1χ2,j2gi,2…χn,jngi,n·χ¯1,j1′gi,1χ¯2,j2′gi,2…χ¯n,jn′gi,n=1K∑i=1Kχ1,j1χ¯1,j1′gi,1χ2,j2χ¯2,j2′gi,2⋯χn,jnχ¯n,jn′gi,n=1Kχ1,j1χ¯1,j1′⊗χ2,j2χ¯2,j2′⊗⋯⊗χn,jnχ¯n,jn′D.Since Cj1,j2,…,jn and Cj1′,j2′,…,jn′ are different, (χ1,j1χ¯1,j1′)⊗(χ2,j2χ¯2,j2′)⊗⋯⊗(χn,jnχ¯n,jn′) is a nontrivial character of Fq. So Lemma 7 yields (42)ImaxCD≤1K·q1+fq1⋯qn+fqn2=q1+fq1⋯qn+fqnq1⋯qn-1.In this case, the Welch bound IWelch for the codebook CD is (43)IWelch=N-KN-1K=q1q2⋯qn+1q1q2⋯qn-1.Hence, (44)1≤ImaxCDIWelch≤q1+fq1⋯qn+fqnq1q2⋯qn+1→1if qi→+∞ for any 1≤i≤n.Corollary 9.
With the above notations, we assume that allFqi are the same; that is, qi=q,i=1,2,…,n. In this case, the codebook CD is a (qn,(qn-1)/2) codebook satisfying (45)ImaxCD≤q+fqnqn-1.In this case, IWelch=qn+1/(qn-1). So (46)1≤ImaxCDIWelch≤q+fqnqn+1→1if q→+∞.For the caseDi=C0(2,qi)⊂Fqi,qi≡1(mod4), we have the following lemma.Lemma 10.
For any characterχ1,j1⊗χ2,j2⊗⋯⊗χn,jn of Fq, ji=0,1,…,qi-1,qi≡1(mod4),i=1,2,…,n, if (j1,j2,…,jn)=(0,0,…,0), we have (47)χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=K.If (j1,j2,…,jn)≠(0,0,…,0), (48)max⁡χ1,j1⊗χ2,j2⊗⋯⊗χn,jnD=q1+1⋯qn+12.Proof.
The proof is very similar to the proof of Lemma7. We omit it here.Theorem 11.
LetCD be a codebook defined by (25) in the case of Di=C0(2,qi)⊂Fqi,qi≡1(mod4). Then CD is an (N,K)=(q1q2…qn,(q1q2…qn-1)/2) codebook with (49)ImaxCD=q1+1⋯qn+1q1⋯qn-1,and Imax(CD) can reach the Welch bound if qi→∞ for any 1≤i≤n.Proof.
The proof is very similar to the proof of Theorem8. We omit it here.Remark 12.
The result of Theorem11 is similar to that of [14]. The difference between our result and that of [14] is that Hu’s method is based on the product of difference sets in finite Abelian group Gi of order Ni, where Ni≡3(mod4), whereas our method is based on the product of almost difference sets in finite field Fqi, where qi≡1(mod4).Corollary 13.
With the above notations, we assume that allFqi are the same; that is, qi=q≡1(mod4),i=1,2,…,n. In this case, the codebook CD is a (qn,(qn-1)/2) codebook satisfying (50)ImaxCD=q+1nqn-1.In this case, IWelch=qn+1/(qn-1). So (51)1≤ImaxCDIWelch=q+1nqn+1→1if q→+∞.Remark 14.
Ifn=1, our codebooks (25) in the case of Di=C0(8,qi)∪C1(8,qi)∪C2(8,qi)∪C5(8,qi)⊂Fqi, qi≡9(mod16), turn into the asymptotically optimal codebooks in [10], and our condition is milder, since [10] requires that |x-a| and |y| are bounded, whereas we require limitq->∞|x-a|/q=0 and limitq->∞|y|/q=0. If n=1, in the case of Di=C0(2,qi)⊂Fqi,qi≡1(mod4), codebooks (25) turn into the asymptotically optimal codebooks in [19].Remark 15.
By Remark4 and Lemma 7, Theorem 8 still holds if we use Di=Cl(8,qi)∪C1+l(8,qi)∪C2+l(8,qi)∪C5+l(8,qi) instead of Di=C0(8,qi)∪C1(8,qi)∪C2(8,qi)∪C5(8,qi), l=1,…,7. Similarly, by Remark 5 and Lemma 10, Theorem 11 still holds if we use Di=C1(2,qi) instead of Di=C0(2,qi).


## 4. Conclusions

In this paper, we construct a new class of asymptotically optimal codebooks based on the product of two classes of almost difference sets in finite fields. Future research is required to investigate whether this strategy works for any almost difference set of(q,(q-1)/2,λ,t) type. It is also interesting to modify the strategy such that we can start with MWBE codebooks and then derive new MWBE codebooks via a similar direct product. Another interesting problem is whether any complement of asymptotically codebooks similar to the Naimark complement of MWBE codebooks exists.



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*Source: 8708161-2019-03-10.xml*