A New Construction of Codebooks Meeting the Levenshtein Bound

Codebooks with low coherence have extensive applications in many fileds such as code division multiple access (CDMA) communication systems, MIMO communications, compressed sensing and so on. In this paper, based on additive characters over finite fields, we propose a construction of optimal codebook with respect to the Levenshtein bound and verify that it is a new construction. By shortening the length of the optimal codebooks, we present a construction of codebooks asymptotically meeting the Levenshtein bound. To the best of our knowledge, the parameters of the asymptotically optimal codebooks are new.


I. INTRODUCTION
Let D = {d 0 , d 2 , · · · , d N −1 } be a set consisted of N unit norm 1 × K complex vectors over an alphabet A. The number of elements in the alphabet is said to be the alphabet size. The maximum inner-product correlation of D is given by where d H j is the conjugate transpose of the complex vector d j . Then the set D is termed a codebook (also called signal set) with the paramters (N , K ) and maximum inner-product correlation I max (D). In a code division multiple access system, the distinct vectors of a codebook are assigned to different users and the inner-product correlation is employed to distinguish among the signals of different users. As a result, the maximum inner-product correlation is an important indicator to judge the performance of a codebook. People would like to design a codebook D such that the parameter N is as great as possible and I max (D) is as small as possible for a fixed The associate editor coordinating the review of this manuscript and approving it for publication was Filbert Juwono .
K . However, there exists a tradeoff between the parameters N , K and I max (D), which is introduced by Welch [34].
Lemma 1 [34]: Let D be an (N , K ) codebook with I max (D). Then we have
Strohmer and Heath [31] has pointed out that there are no (N , K ) real codebook meeting the Welch bound with equality if N > K (K +1) 2 and (N , K ) codebook achieving the Welch bound with equality if N > K 2 . In other words, the Welch bound is not tight in these cases. When N is much bigger than K , Levenshtein [21] deduced a new bound called Levenshtein bound which is turned out to be tighter than the Welch bound. Thus, for a codebook such that the number of vectors is much greater than the length of vectors, the Levenstein bound is a preferable benchmark for the maximum inner-product correlation.
Lemma 2 [21]: Let D be an (N , K ) codebook with I max (D). If D is a real valued codebook and N > K (K +1) 2 , then If D is a complex valued codebook and N > K 2 , then In fact, it is very hard to construct codebooks achieving the Levenshtein bound. Constructing codebooks (asymptotically) achieving the Levenshtein bound is harder than the design of (asymptotically) optimal codebooks with respect to the Welch bound. Limited work has been done in the construction of codebooks (asymptotically) achieving the Levenshtein bound. Until now, there are only four constructions of optimal codebooks with respect to the Levenshtein bound. One of the optimal construction was derived from Kerdock codes [1], [36] and the other constructions were constructed by planar functions [7], bent functions over the integer rings Z 4 , bent functions over finite fields [41], respectively. Besides optimal codebooks, there are a few constructions of asymptotically optimal codebooks. Tan et al. [32] proposed a construction of codebooks asymptotically achieving the Levenshtein bound by Gauss sums over finite fields. In [36], the authors presented a construction of codebooks asymptotically meeting the Levenshtein bound from binary codes and semi-bent functions. Based on multiplicative characters over finite fields, some asymptotically optimal codebooks with respect to the Levenshtein bound were obtained in [18], [20].
Codebooks (asymptotically) achieving the Levenshtein bound have several practical applications such as the construction of mutually unbiased bases [7] which are used in quantum physics, the construction of deterministic sensing matrices with low coherence [22], the design of spreading sequences for CDMA systems [26]. Hence, it is significant to construction (asymptotically) optimal codebooks with respect to the Levenshtein bound.
We are concerned in this paper with the following two objectives. The first objective is to provide a new construction of codebooks achieving the Levenshtein bound by additive characters over finite fields. Although the optimal codebooks have the same parameters as those in [7, Theorem 4], we prove that it is indeed a different construction. The other objective is to propose a new construction of asymptotically optimal codebooks with respect to the Levenshtein bound by considering the scalability issue regarding the length of the optimal codebooks. Notably, the parameters of the asymptotically optimal codebooks are flexible and not covered by the previous literatures.
This paper is built up as follows. Section 2 is devoted to the definitions and results for the characters and character sums over finite fields. The constructions of (asymptotically) optimal codebooks are summarized in Sections 3 and 4. In Section 5, we make a conclusion.

II. CHARACTER AND CHARACTER SUMS OVER FINITE FIELDS
In this section, we recall the definitions of characters over finite fields and some results about character sums over finite fields.
Let F q be a finite field with q = p n elements, where n is a positive integer and p is a prime. The trace function from F q to F p is defined as Let m > 1 be an integer and ξ m = e 2π √ −1/m . An additive character of F q is defined to be the function χ a (x) = ξ where a, x ∈ F q . When a runs around all the elements of F q , one can obtain all the additive characters of F q . Especially, the additive character χ a of F q is called the canonical additive character if a = 1. We write F * q = F q \ {0}. Under the multiplication operation, F * q is a cyclic group of order q − 1 and a generator of it is said to be a primitive element of F q . Let ω be a primitive element of F q . For any integer j with is termed the quadratic character of F q and denoted by η for simplicity.
Finite fields are special structures since they have both additive characters and multiplicative characters. Combining the canonical additive character χ 1 and multiplicative character ϕ of F q , the Gauss sum over F q is defined as In most cases, it is very hard to determine the values of Gauss sums explicitly. Only in a few cases can Gauss sums be computed. Below, the explicit value of the Gauss sum over VOLUME 8, 2020 F q is determined when the multiplicative character ϕ is the quadratic character of F q .
Lemma 3 [27,Theorem 5.15]: Let p be a prime and n a positive integer. Write q = p n . Suppose that F q is a finite field and η is the quadratic character of F q . Then In the sequel, we need the following two lemmas. Lemma 4 [27,Theorem 5.33]: Assume that p is an odd prime. Let χ 1 be the canonical additive character of F q .
Lemma 5 [27,Theorem 5.38]: Let χ 1 be the canonical additive character of F q . Assume that f (x) ∈ F q [x] is of degree n > 0 with gcd(n, q) = 1. Then x∈F q

III. OPTIMAL CODEBOOKS
In this section, we present a new construction of optimal codebooks with respect to the Levenshtein bound and state the difference between our construction and the construction in [7].
Let χ 1 be the canonical additive character of F q . For any a, b ∈ F q , we can define a unit norm complex vector by After a, b walk along the finite field F q , one can obtain a set of q 2 unit norm complex vectors as follows: Theorem 6: Let the symbols be the same as above. Then the set G = D ∪ E q is a (q 2 + q, q) codebook with I max (G) = q −1/2 which meets the Levenshtein bound.
Proof: We divide our proof in three steps. First, we evaluate the inner-product correlation of G.
Case 1: If g 1 = g 2 ∈ E q , it is easy to verify that g 1 g H 2 = 0. Case 2: If g 1 ∈ E q and g 2 ∈ D, it can be easily seen that g 1 g H 2 = 1 √ q . Case 3: If g 1 = g 2 ∈ D, then write g 1 = d a,b and g 2 = d u,v , where (a − u, b − v) = (0, 0). It follows from Eq. (1) that If a = u, then, by the fact b = v, we derive that If a = u, then it follows from Lemma 4 that where e = 3a 2 − 3u 2 + b − v and d = a 3 − u 3 . According to the definition of additive and multiplicative characters over finite fields and Lemma 3, we obtain that g 1 g H 2 = 1 √ q . From the definition of the set G, it is easy to check that G is a (q 2 +q, q) codebook with I max (G) = q −1/2 . Clearly, I max (G) coincides with the Levenshtein bound given in Lemma 2.  (1, 2, 3, 0, 4), 1, 1, 1, 4), . This is consistent with the conclusion of Theorem 6.

B. COMPARISON
In [7], the authors presented an optimal codebook C with parameters (q 2 + q, q) and I max (C) = q −1/2 as follows: where f (x) is a planar function from F q to F q . Taking u = v = 0 in Eq. (2), one can obtain a vector e = 1 √ q (1, 1, · · · , 1) of C. If the vector e is contained in the codebook G, then there exists a vector d in G such that de H = 1. In fact, for any vector d of G, we have Thanks to Lemma 5, we have x∈F q Thus, the vector e = 1 √ q (1, 1, · · · , 1) is not contained in the codebook G which says that our construction of codebooks in Theorem 6 is not covered by the construction in [7,Theorem 4]. In a word, although the optimal codebooks generated by Theorem 6 have the same parameters as those in [7,Theorem 4], it is indeed a different construction.

IV. ASYMPTOTICALLY OPTIMAL CODEBOOKS
In this section, we consider the scalability issue regarding the length of the optimal codebooks obtained by Theorem 6. As a result, we present a new construction of asymptotically optimal codebooks with respect to the Levenshtein bound.
Suppose that p > 3 is a prime and m is a positive integer. Put q 1 = p m and q = q kp 1 , where k is a positive integer. Clearly, the finite field F q 1 is a subfield of F q . Assume that H is a subset of F q 1 with h = #H > 0, where #H denotes the cardinality of the set H . Denote by E q−h to be the standard basis of the (q − h)-dimensional Hilbert space as follows: (1, 0, 0, · · · , 0, 0), (0, 1, 0, · · · , 0, 0), . . .
Let χ 1 be the canonical additive character of F q . For any a, b ∈ F q , define a unit norm complex vector by and then define a set Thus one can obtain a codebook Theorem 7: Suppose that p > 3 is a prime and m, k are positive integers such that m is even. Let q 1 = p m and q = q kp 1 . Assume that h is an integer with 0 < h ≤ q 1 . Then the set Q(H ) defined by Eq.
Proof: Consider the codebook G constructed by Theorem 6. Clearly, G can be viewed as a q 2 + q × q matrix. By deleting h columns of the codebook G, one can obtain a new codebook Q(H ) with parameters (q 2 + q, q − h). Due to It is easy to verify that lim q→+∞ Proof: The proof is analogous to that in Theorem 7 and Remark 1. Thus, we omit the proof of the corollary. In Table 1, we list some explicit values of parameters of the codebooks in Corollary 8. Also, we compare I max (Q(H )) with the Levenshtein bound I L in Table 1. It can be seen that I max (Q(H )) is close to I L as p and m increase. This means that the codebooks defined in Corollary 8 are indeed asymptotically optimal with respect to the Levenshtein bound.

V. CONCLUDING REMARKS
In this paper, we proposed a construction of optimal codebook regarding to the Levenshtein bound and proved that the optimal codebook is different from that in [7,Theorem 4]. Considering the scalability issue regarding the length of the optimal codebooks, we obtained two families of codebooks asymptotically achieving the Levenshtein bound. In addition, the parameters of the asymptotically optimal codebooks are new. For reference, the parameters (N , K ) of some known asymptotically optimal codebooks with respect to the Levenshtein bound and the new ones are summarized in Table 2.
Li and Ge [22] have pointed out that an (N , K ) codebook with low correlation can be directly utilized to construct a deterministic sensing K ×N matrix with small coherence. The simulation results in [22] illustrated that the sensing matrices constructed by codebooks (asymptotically) meeting the Levenstein bound have a good performance in compressed sensing. Consequently, all the constructions of codebooks in this paper can be used to design deterministic sensing matrices which have a good performance.