Moments of crosscorrelation demerit factors of binary sequences
Abstract
Families of sequences with low mutual aperiodic crosscorrelation assist the design of systems for multi-user asynchronous communications and multiple-input multiple-output radar. The crosscorrelation demerit factor of a pair of sequences is the sum of the squared magnitudes of their crosscorrelation values at every shift when the sequences are normalized to unit Euclidean norm, and the merit factor is the reciprocal of the demerit factor. For each positive integer ℓ, we endow the 22 ℓ pairs of binary sequences of length ℓ with uniform probability measure and study the distribution of their crosscorrelation demerit factors. Sarwate showed that the mean value is always $1$ regardless of length ℓ. We develop a method for finding an exact formula for the pth central moment (for any positive integer p) as a function of ℓ. Formulae for the variance and third central moment (p=2 and $3$) are then obtained by hand calculations, while the fourth through sixth central moments are obtained by computer-assisted calculations. Our theory also shows that all the central moments must be strictly positive for p≥ 2 and ℓ ≥ 3.
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BibTeX
@misc{katz2026moments,
title = {{Moments of crosscorrelation demerit factors of binary sequences}},
author = {Daniel J. Katz and Harmony M. Vargas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05771},
eprint = {2609.05771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05771},
}
RIS
TY - UNPB
TI - Moments of crosscorrelation demerit factors of binary sequences
AU - Daniel J. Katz
AU - Harmony M. Vargas
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05771
UR - https://arxiv.org/abs/2609.05771
AB - Families of sequences with low mutual aperiodic crosscorrelation assist the design of systems for multi-user asynchronous communications and multiple-input multiple-output radar. The crosscorrelation demerit factor of a pair of sequences is the sum of the squared magnitudes of their crosscorrelation values at every shift when the sequences are normalized to unit Euclidean norm, and the merit factor is the reciprocal of the demerit factor. For each positive integer ℓ, we endow the 2^2 ℓ pairs of binary sequences of length ℓ with uniform probability measure and study the distribution of their crosscorrelation demerit factors. Sarwate showed that the mean value is always $1$ regardless of length ℓ. We develop a method for finding an exact formula for the pth central moment (for any positive integer p) as a function of ℓ. Formulae for the variance and third central moment (p=2 and $3$) are then obtained by hand calculations, while the fourth through sixth central moments are obtained by computer-assisted calculations. Our theory also shows that all the central moments must be strictly positive for p≥ 2 and ℓ ≥ 3.
ER -