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A Construction of Codebooks Associated With Binary Sequences

机译：与二进制序列相关的密码本的构造

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摘要

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 $K times N$ partial matrix with $K < N$, where each code vector is equivalent to a column of the matrix. To obtain the $K times N$ matrix, $K$ rows are selected from a $J times N$ matrix $ {bm{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 $ {bm{Phi }}$-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 $ {bm{Phi }}$-transform is as small as possible. From the discovery, new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from Fourier and Hadamard matrices associated with binary sequences.
机译：$（N，K）$码本是$ K $维向量空间中的一组$ N $个单位范式代码向量。对于其应用，期望一对不同的代码向量之间的内积的最大量值应尽可能小，以严格或渐近地满足韦尔希界等式。在本文中，$（N，K）$码本是由$ K×N $的部分矩阵构造而成的，其中$ K

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