On the maximum of the weighted binomial sum $2^{-r}\sum_{i=0}^r\binom{m}{i}$

TL;DR

This paper determines the maximum of a weighted binomial sum relevant to coding and information theory, showing where it occurs and its asymptotic behavior for large m.

Contribution

It proves the exact location of the maximum of the weighted binomial sum for most m and derives its asymptotic form as m grows large.

Findings

Maximum occurs at r=⌊m/3⌋+1 for most m

Maximum value asymptotic to (3/√(πm))(3/2)^m

Identifies exceptions at m=0,3,6,9,12

Abstract

The weighted binomial sum $f_m(r)=2^{-r}\sum_{i=0}^r\binom{m}{i}$ arises in coding theory and information theory. We prove that,for $m\not \in\{0,3,6,9,12\}$, the maximum value of $f_m(r)$ with $0\leqslant r\leqslant m$ occurs when $r=\lfloor m/3\rfloor+1$. We also show this maximum value is asymptotic to $\frac{3}{\sqrt{{\pi}m}}\left(\frac{3}{2}\right)^m$ as $m\to\infty$.