A Unified Treatment of Some Classic Combinatorial Inequalities Using the Variance Method
Douglas R. Stinson
TL;DR
The paper surveys a unified variance-method framework for deriving classical combinatorial inequalities arising in design and coding theory, using the fundamental nonnegativity of variance in the form $S_0 S_2 - S_1^2 \ge 0$ with $S_0=n$, $S_1=\sum a_i$, $S_2=\sum a_i^2$ and $S^* = \sum {a_i \choose 2}$. By specializing these aggregate sums to problem-specific incidences (and sometimes lower bounds on $S_1$), it recovers Fisher's inequality, the Plackett-Burman bound, the Second Johnson bound, the Stanton-Kalbfleisch bound, and the Mullin-Vanstone bound, along with a two-point sampling derandomization result. The notes further illustrate extensions of the variance method to strengthen bounds (via integer shifts $\ell$) and provide concrete applications to orthogonal arrays, pairwise balanced designs, and code bounds. Together, the work presents a cohesive, accessible approach for obtaining tight combinatorial constraints with direct implications for design theory and coding theory. The methods yield both classical and refined bounds, highlighting the method's flexibility and potential for new applications.
Abstract
The "variance method" has been used to prove many classical inequalities in design theory and coding theory. The purpose of this expository note is to review and present some of these inequalities in a unified setting. I will also discuss some examples from my own research where I have employed these techniques.