Near-Optimal Partial Hadamard Codebook Construction Using Binary Sequences Obtained From Quadratic Residue Mapping

Abstract

In this paper, a new class of (N, K) near-optimal partial Hadamard codebooks is proposed. The construction of the proposed codebooks from Hadamard matrices is based on binary row selection sequences, which are generated by quadratic have parameters N = p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and K = (p - 1/2 p)(N + √N) + 1 for an odd prime p and an even positive integer n. We prove that the maximum magnitude of inner products between the code vectors of the proposed codebooks asymptotically achieves the Welch bound equality for sufficiently large p and derive their inner product distribution.