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.
1. Introduction
An codebook is a set , where each codeword , , is a unit norm complex vector in . The value of maximum cross-correlation amplitude of an codebook is defined by where denotes the conjugate transpose of a complex vector . Codebooks with minimal possible are desirable in many applications. From a geometric point of view, the minimal possible 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 .
Lemma 1 (see [1]). For any codebook with , Moreover, the equality holds if and only if, for all pairs of with , it holds that
A codebook 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 when . In fact, is only the necessary condition of the existence of an MWBE codebook [2]. Even when holds, the corresponding MWBE codebook may not exist [7]. In recent years, asymptotically optimal codebooks have attracted widespread attention, since they can be constructed easily and can approach the Welch bound for large enough . 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 Section 2, 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
Let be a prime number; . For a finite field , let be its one fixed primitive element, and let be its all nonzero elements.
Let . The set is called a cyclotomic class of order . Clearly, , and .
The cyclotomic numbers of order with respect to are given by where . Cyclotomic number is the number of solutions to the equation .
Cyclotomic class is now widely used to construct almost difference sets. An almost difference set of a finite field is a subset with elements satisfying special properties. Define is called a almost difference set of if for nonzero elements of , and for the remaining nonzero elements of . When , is a almost difference set, which is also called the Paley partial difference set [15].
Let ; Lehmer [16] presented the explicit form of the 64 cyclotomic numbers of order when , which are functions whose parameters are determined by the unique representation
The 64 cyclotomic numbers have two forms determined by whether 2 is a quartic residue in or not. Let . When 2 is a quartic residue in , Ding et al. [17] calculated thatWhen 2 is not a quartic residue in , Zhang and Feng [10] calculated that
If is a power of a prime and , then, for , Ding [17] proved that , , , , , and 2 is a quartic residue in , and is a almost difference set of . Thus, are also almost difference sets of .
An additive character of with characteristic is a homomorphism from to the unit circle in the complex plane, which in fact has the following form [18]: where Tr is the trace function from to . Multiplicative characters of are characters of the multiplicative group . Let be a fixed primitive element of . All multiplicative characters of have the following form [18]:
The set of all additive characters of is denoted by , and . When , is trivial, which satisfies . For a subset and an additive character of , is defined to be . Clearly, .
For additive characters of , we can define their product as for any .
The direct product of finite fields can be defined by Correspondingly, an additive character on the direct product is defined by where and .
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 , where . In our paper, we use two classes of almost difference sets in finite fields: one class is , , and the other is and , .
For the case of , we first estimate for any nontrivial additive characters of a finite field. In fact, Zhang and Feng had given an upper bound of in Theorem 1 of [10]. We rephrase it in our setting.
Lemma 2. Let be a finite field, , and are determined by (6). . is a nontrivial additive character. If and , then there exists a function satisfying such that
Proof. Let . is a Gauss period of order 8. Since , let . Then Note that, whether 2 is a quartic residue in or not, . Thus Let be the multiplicative character satisfying . . So if , we have , or else . Thus, where is the Gaussian sum over and [18]. Thus, .
ThenWhen 2 is a quartic residue in , substituting (7) into (18) yields When 2 is a quartic nonresidue in , substituting (8) into (18) yields Denote . If and , then . Thus Since ,
Remark 3. There exist infinitely many satisfying the assumptions in Lemma 2. Ding et al. [17] proved that if , then , where is an integer. In this case, we have , becomes a almost difference set, and .
Remark 4. Lemma 2 still holds if we use instead of . The reason is that new is equal to . Thus, it has no influence on estimating . 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 satisfying the assumptions in Lemma 2. The reason is that the eight will have more different values, which inevitably lead to more restrictions on . Thus, it is more difficult to find infinitely many such that is not too large.
Remark 5. Since can be seen as , we consider and , where . Ding [19] proved that and . Thus, for or , we have .
Suppose that are finite fields. Let . Then .
Let be the union of the four cyclotomic classes of order 8 in . Let be the complement of with respect to . Note that .
Let . For any , put Finally, let , and . Let ; then .
We can define similar sets in the case of . More specifically, let . Then we can define , in a similar way.
For both of the two cases, we have the following lemma.
Lemma 6. for any . In particular, .
Proof. For , we have . For , suppose that . Then, it follows that Therefore, .
We define an codebook byThe vector is defined by where and is the corresponding trivial character. It is obvious that , and the length of the vector is .
Next we will investigate . For the case of , we need the following lemma.
Lemma 7. With the above notations, for any character of , . If , we have If ,
Proof. If , when , is the trivial character; thus, . When , is a nontrivial character. Thus, by Lemma 2, .
If , we need to consider three cases:
(1) (2) Thus, In a similar way, if , (3) Now we assume that the formula holds for ; we will prove it for .
If , then the character is trivial. Hence, the result is correct. In the following, we consider the case of and . The proof can also be divided into three cases:
Case (1) (). We compute By Lemma 6, . So we have Case (2) (). In this case, we getHence, by the assumption,Case (3) (). In this case, we getHence, by the assumption and Lemma 2,The proof is finished.
Theorem 8. Let be a codebook defined by (25) in the case of , and satisfies the assumptions in Lemma 2. Then is an codebook with and can reach the Welch bound if for any .
Proof. Let and be two different codewords in . Then we haveSince and are different, is a nontrivial character of . So Lemma 7 yields In this case, the Welch bound for the codebook is Hence, if for any .
Corollary 9. With the above notations, we assume that all are the same; that is, . In this case, the codebook is a codebook satisfying In this case, . So if .
For the case , we have the following lemma.
Lemma 10. For any character of , , if , we have If ,
Proof. The proof is very similar to the proof of Lemma 7. We omit it here.
Theorem 11. Let be a codebook defined by (25) in the case of . Then is an codebook with and can reach the Welch bound if for any .
Proof. The proof is very similar to the proof of Theorem 8. We omit it here.
Remark 12. The result of Theorem 11 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 of order , where , whereas our method is based on the product of almost difference sets in finite field , where .
Corollary 13. With the above notations, we assume that all are the same; that is, . In this case, the codebook is a codebook satisfying In this case, . So if .
Remark 14. If , our codebooks (25) in the case of , , turn into the asymptotically optimal codebooks in [10], and our condition is milder, since [10] requires that and are bounded, whereas we require and . If , in the case of , codebooks (25) turn into the asymptotically optimal codebooks in [19].
Remark 15. By Remark 4 and Lemma 7, Theorem 8 still holds if we use instead of , . Similarly, by Remark 5 and Lemma 10, Theorem 11 still holds if we use instead of .
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 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (nos. 61502217, 11404051, and 61702244).