1 Introduction
An codebook
is defined to be a set of unitnorm complex vectors
in over an alphabet A. Letwhere denotes the Hermite transpose of vector . The maximum innerproduct correction is a performance measure of a codebook in practical applications. In code division multiple access (CDMA) systems, one important problem is to minimize the codebook’s maximal crosscorrelation amplitude .
For a given , it is usually desirable that is as large as possible and is as small as possible simultaneously. However, the parameters , and of a codebook have to satisfy the Welch bound [25]. That is, there is a tradeoff between the codeword length , the set size and . A codebook meeting the theoretical bound with equality is said to be optimal. Searching optimal codebooks has been an interesting research topic in recent years. Many classes of optimal codebooks has been constructed [26, 19, 1, 23, 3, 4, 5, 6, 7, 18].
It is worthwhile to point out that the constructed codebooks so far have restrictive parameters and . Hence, many researchers attempt to research asymptotically optimal codebooks, i.e., asymptotically meets the theoretical bound for sufficiently large . One important method to construct asymptotically optimal codebooks is from difference sets of finite abelian gruops which is developed by Ding and Feng [3, 4]. In [3, 4], several series of asymptotically codebooks were constructed by using almost difference sets. In [8, 27, 2], asymptotically optimal codebooks were presented by binary row selection sequences. Character sums over finite fields are considered to be useful tools for the design of asymptotically codebooks. Recently, in [9, 10], Heng et al. obtained two new constructions of infinitely many codebooks by Jacobi sums and their generalizations. In [15, 16], Luo and Cao defined a new character sum called hyper Eisenstein sum and presented two constructions of infinitely many new codebooks achieving the Wech bound. In [24], Tian presented two constructions of codebooks with additive characters over finite fields.
Bent functions are a class of Boolean functions and have important applications in cryptography, code theory and sequences for communications. In cryptography, bent vectorial functions can be used as substitution boxes in block ciphers (ensuring confusion, as explained by Shannon [20]. In code theory, they are useful for constructing error correcting codes (Kerdock codes) [17]. In sequences, they permit to construct sequences with low correlation [11].
The objective of this paper is to present two constructions of codebooks using generalised bent functions over a ring of integers modulo . The presented two kinds of codebooks have properties: (1) they are asymptotically optimal with respect to the Welch bound; (2) the parameters of these codebooks are new and flexible. As a comparison with the known ones, our codebooks are listed in Table 1.
References  Parameters  Constraints  

[8] 
is an odd prime 

[28]  is an odd prime power  
[13]  is a power of a prime  
[13]  is a prime power  
[27]  is an odd prime  
[29] 


[9] 


[9] 


[14] 


[14] 


[14] 


[24]  is a prime power  
[24]  is a prime power  
Theorem 3.1 


Theorem 4.1 

This paper is organized as follows. In Section 2, we briefly recall some definitions and notations which will be needed in our discussion. In Section 3 and Section 4, we present our constructions of codebooks and prove they are asymptotically optimal with respect to the Welch bound. In Section 5, concluding remarks of this paper is given.
2 Preliminaries
In this section, we present some notations and preliminaries which are needed for the proof of the main results. Firstly, a useful lemma is given in the following.
Lemma 2.1 (Linear Congruence Theorem, [22]).
Let , , and be integers with , and let .

If , then the congruence has no solutions.

If , then the congruence has exactly incongruent solutions.
The following is the wellknown Welch bound on , and of a codebook .
Lemma 2.2.
[25] For any codebook with ,
Moreover, the equality holds if and only if for all pairs of with , it holds that
Next, we introduce the definition of generalised bent functions. Let be a positive integer and the ring of integers modulo . Assume that is a primitive th root of unity. Denote by the dimensional vector space over . A function mapping from to is termed a generalised Boolean function on variables. For a generalised Boolean function , if the complex Fourier coefficients
preserve unit magnitude for any , then is a generalised bent function.
Bent functions are a hot research topic due to their wide applications in cryptography, information theory and coding theory. Kumar et al. [12] introduced the definition of generalised bent functions from to and gave a class of generalised bent functions in the following lemma.
Lemma 2.3 ([12]).
Assume that is a positive integer. Let be an arbitrary permutation and an arbitrary function on . Then the function
is generalised bent, where .
3 The first construction of asymptotically optimal codebooks
In this section, we present a construction of codebooks and show that the maximum innerproduct correction of these codebooks asymptotically achieves the Welch bound. Before proposing our construction, we need to do some preparations.
Suppose that is an integer and is the smallest prime factor of . Denote the standard basis of the dimensional Hilbert space by which is formed by vectors of length as follow:
In the following theorem, we give a new construction of infinite many codebooks and evaluate their maximum innerproduct correction.
Theorem 3.1.
Let symbols be the same as above. For any , , define a unitnorm complex vector of length by
where and are permutations on . Let
(1) 
Then the set is a codebook with .
Proof.
According to the definition of codebooks, we deduce that is consisted of codewords with length . In other words, is a codebook with parameters . Now we turn to the computation of . Let , be two distinct codewords. We distinguish among the following three cases to calculate the correlation of and .
(1) If , , it is easy to verify that .
(2) If and , it is obvious that .
(3) If , , write and , where . Then we deduce that
If , we obtain
where the second identity follows from the fact that .
If , then . By Lemma 2.1, the congruence has only one integer solution in . In this case, we have
where the last equality is derived from the fact that for any . Therefore, we obtain .
The analysis above shows that . This completes the proof of this theorem. ∎
The next theorem deals with the asymptotical optimality of the codebooks defined in Theorem 3.1.
Theorem 3.2.
Let symbols be the same as before. Then the maximum innerproduct correction of the codebook defined in (1) asymptotically meets the Welch bound.
Proof.
From Theorem 3.1, we know is a codebook. Obviously, Note that the corresponding Welch bound of is
Then we obtain
Observe that
which implies that the codebook asymptotically meets the Welch bound. ∎
In Table 2, we list some examples of codebooks generated by Theorem 3.1. The numerical results indicate that the codebooks defined in Theorem 3.1 asymptotically achieve the Welch bound as increases, as predicted in Theorem 3.2.
5  1225  0.02857  0.02608  0.91293  
13  48841  0.96362  
17  243049  0.97183  
31  3575881  0.98425  
43  9320809  0.98857  
61  1229410598  19829209  0.99190  
73  3123614666  42211009  0.99322  
83  4583692596  54567796  0.99403  
97  11774065058  120143521  0.99488 
4 The second construction of asymptotically optimal codebooks
In this section, we propose a construction of codebooks by slightly modifying the construction of codebooks in Section 3. In addition, we show that these codebooks are asymptotically optimal with respect to the Welch bound.
Let be an integer and the smallest prime factor of . Assume that
is a set consisted of all rows of the identity matrix
. That is to say, is the standard basis of the Hilbert space with dimension . Let and be permutations on . For any , we define a set by(2) 
where
Let
(3) 
Then is a codebook with parameters and the the maximum innerproduct correction of can be obtained in the following theorem.
Theorem 4.1.
Assume that is an integer and is the smallest prime factor of . Then defined by (3) is a codebook with .
Proof.
Let , be two distinct codewords in . Now we calculate the correlation of and by distinguishing among the following cases.
(1) If , , it is obvious that .
(2) If and , we have .
(3) If , , we assume that and , where . Then we have
If and , ‘we obtain
If and , it follows from the fact that that
If , then . According to Lemma 2.1, the congruence has only one integer solution in .
When , we can deduce that
When , we have
Hence, we get that
Summarizing the conclusions in the three cases above, we obtain
This completes the proof. ∎
Theorem 4.2.
Let symbols be the same as before. If , then the codebook is asymptotically optimal with respect to the Welch bound.
Proof.
Table 3 presents some parameters of codebooks derived from Theorem 4.1. From Table 3, we can see is very close to as increases. This means that the codebooks defined in Theorem 4.1 are asymptotically optimal with respect to the Welch bound for large , as predicted in Theorem 4.2.
7  5852  0.013072  0.01224  0.93618  
19  190532  0.97474  
29  1150256  0.98321  
41  4719756  0.98803  
59  12949202  0.99163  
67  1538770575  22624292  0.99262  
79  3439533363  42987692  0.99373  
89  6707573377  74520056  0.99443  
101  12362193253  121187072  0.99509 
5 Concluding remarks
Employing generalised bent functions, we presented two classes of codebooks and proved that these constructed codebooks nearly meet the Welch bound. As a comparison, the parameters of some known classes of asymptotically optimal codebooks with respect to the Welch bound and the new ones are listed in Table 1. Obviously, the parameters of our classes of codebooks have not been covered in the literature.
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