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

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 = pn 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.

[1]  J. Massey,et al.  Welch’s Bound and Sequence Sets for Code-Division Multiple-Access Systems , 1993 .

[2]  Nam Yul Yu,et al.  A Construction of Codebooks Associated With Binary Sequences , 2012, IEEE Transactions on Information Theory.

[3]  Cunsheng Ding,et al.  Codebooks from almost difference sets , 2008, Des. Codes Cryptogr..

[4]  John J. Komo,et al.  Nonbinary Kasami sequences over GF(p) , 1992, IEEE Trans. Inf. Theory.

[5]  Jinsong Wu,et al.  New Constructions of Codebooks Nearly Meeting the Welch Bound With Equality , 2014, IEEE Transactions on Information Theory.

[6]  Tor Helleseth,et al.  On generalized bent functions , 2010, 2010 Information Theory and Applications Workshop (ITA).

[7]  Lloyd R. Welch,et al.  Lower bounds on the maximum cross correlation of signals (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[8]  Joseph M. Renes,et al.  Symmetric informationally complete quantum measurements , 2003, quant-ph/0310075.

[9]  Nam Yul Yu,et al.  A new class of near-optimal partial Fourier codebooks from an almost difference set , 2014, Des. Codes Cryptogr..

[10]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[11]  N. J. A. Sloane,et al.  Packing Lines, Planes, etc.: Packings in Grassmannian Spaces , 1996, Exp. Math..

[12]  Zhengchun Zhou,et al.  New nearly optimal codebooks from relative difference sets , 2011, Adv. Math. Commun..

[13]  Rudolf Lide,et al.  Finite fields , 1983 .

[14]  Keqin Feng,et al.  Two Classes of Codebooks Nearly Meeting the Welch Bound , 2012, IEEE Transactions on Information Theory.

[15]  Dustin G. Mixon,et al.  Steiner equiangular tight frames , 2010, 1009.5730.

[16]  Cunsheng Ding,et al.  Complex Codebooks From Combinatorial Designs , 2006, IEEE Transactions on Information Theory.

[17]  Jelena Kovacevic,et al.  An Introduction to Frames , 2008, Found. Trends Signal Process..

[18]  Georgios B. Giannakis,et al.  Achieving the Welch bound with difference sets , 2005, IEEE Transactions on Information Theory.

[19]  J. Seidel,et al.  SPHERICAL CODES AND DESIGNS , 1991 .

[20]  Thomas Strohmer,et al.  GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION , 2003, math/0301135.

[21]  Dilip V. Sarwate.  Meeting the Welch Bound with Equality , 1998, SETA.

[22]  Cunsheng Ding,et al.  A Generic Construction of Complex Codebooks Meeting the Welch Bound , 2007, IEEE Transactions on Information Theory.

[23]  A. Calderbank,et al.  Z4‐Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line‐Sets , 1997 .

[24]  Keqin Feng,et al.  Construction of cyclotomic codebooks nearly meeting the Welch bound , 2012, Des. Codes Cryptogr..