On the maximum of the weighted binomial sum $2^{-r}{\sum }_{i=0}^r\left(\genfrac{}{}{0ex}{}{m}{i}\right)$

Abstract

The weighted binomial sum $f_m(r)=2^{-r}{\sum }_{i=0}^r\left(\genfrac{}{}{0ex}{}{m}{i}\right)$ arises in coding theory and information theory. We prove that,for $m\notin \{0,3,6,9,12\}$, the maximum value of $f_m(r)$ with $0⩽r⩽m$ 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 $m\to \infty$.

arXiv comment9 pages, 2 tables, hyperlinks and backrefs