A Construction of Codebooks Associated With Binary Sequences

作者：Yu Nam Yul^*

来源：IEEE Transactions on Information Theory, 2012, 58(8): 5522-5533.

DOI：10.1109/TIT.2012.2196021

• Bent functions
• codebooks
• Fourier matrices
• frames
• Golay complementary sequences
• Hadamard matrices
• Sidelnikov sequences
• Welch bound

An (N, K) codebook is a set of N unit-norm code vectors in a K-dimensional vector space. For its applications, it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible, meeting the Welch bound equality strictly or asymptotically. In this paper, an (N, K) codebook is constructed from a K x N partial matrix with K < N, where each code vector is equivalent to a column of the matrix. To obtain the K x N matrix, K rows are selected from a J x N matrix Phi, associated with a binary sequence of length J and Hamming weight K, where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence. It is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of <(Phi)over bar>-transform of the binary sequence. Thus, constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its (Phi) over bar -transform is as small as possible. From the discovery, new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from Fourier and Hadamard matrices associated with binary sequences.

• 出版日期2012-8