Two constructions of asymptotically optimal codebooks via the trace functions
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Two constructions of asymptotically optimal codebooks via the trace functions Xia Wu1 · Wei Lu1 · Xiwang Cao2 Received: 27 August 2019 / Accepted: 28 June 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we present two new constructions of complex codebooks with multiplicative characters, additive characters and trace functions over finite fields, and determin the maximal cross-correlation amplitude of these codebooks. We prove that the codebooks we constructed are asymptotically optimal with respect to the Welch bound. Moreover, in the first construction, we generalize the result in Zhang and Feng (IEEE Trans. Inform. Theory 58(4), 2507–2511, 2012). In the second construction, we generalize the results in Hong et al. (IEEE Trans. Inf. Theory 60(6), 3698–3705, 2014), we can asymptotically achieve Welch bound for any odd prime p, we also derive the whole distribution of their inner products. The parameters of these codebooks are new. Keywords Codebook · Asymptotic optimality · Welch bound · Trace function Mathematics Subject Classification (2010) 94A05 · 11T24
1 Introduction An (N, K) codebook C = {c0 , c1 , ..., cN−1 } is a set of N unit-norm complex vectors ci ∈ CK over an alphabet A, where i = 0, 1, . . . , N − 1. The size of A is called the alphabet size of C . As a performance measure of a codebook in practical applications, the maximum magnitude of inner products between a pair of distinct vectors in C is defined by Imax (C ) =
max
0≤i=j ≤N−1
|ci cH j |,
Wei Lu
[email protected] Xia Wu [email protected] Xiwang Cao [email protected] 1
School of Mathematics, Southeast University, Nanjing, 210096, China
2
Department of Math, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China
Cryptography and Communications
where cH j denotes the conjugate transpose of the complex vector cj . To evaluate an (N, K) codebook C , it is important to find the minimun achievable Imax (C ) or its lower bound. The Welch bound [27] provides a well-known lower bound on Imax (C ), N −K . Imax (C ) ≥ IW = (N − 1)K The equality holds if and only if for all pairs of (i, j ) with i = j N −K H . |ci cj | = (N − 1)K A codebook C achieving the Welch bound equality is called a maximum-Welch-boundequality (MWBE) codebook [25] or an equiangular tight frame [15]. MWBE codebooks are employed in various applications including code-division multiple-access(CDMA) communication systems [22], communications [25], combinatorial designs [3, 4, 28], packing [2], compressed sensing [1], coding theory [5] and quantum computing [23]. To our knowledge, only the following MWBE codebooks are presented as follows:
• • • • • •
(N, N ) orthogonal MWBE codebooks for any N > 1 [25, 28]; (N, N − 1) MWBE codebooks for N > 1 based on discrete Fourier transformation matrices [25, 28] or m-sequences [25]; (N, K) MWBE codebooks from conference matrices [2, 26], where N = 2K = 2d+1 for a positive integer d or N = 2K = p d + 1 for a prime p and a positive integer d; (N, K) MWBE c
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