A note on $\overline{2}$-separable codes and $B_2$ codes

TL;DR

This paper provides a simplified information-theoretic proof for bounds on the rates of $ar{2}$-separable and $B_2$ codes, improving or simplifying previous results for certain cases.

Contribution

It introduces a straightforward proof technique for upper bounds on code rates, extending to $B_2$ codes with simpler methods that could aid future research.

Findings

Improved upper bounds on $ar{2}$-separable codes for $q extgreater= 13$.

Simpler derivation of known results for $q=2$.

Extension of bounds to $B_2$ codes using accessible tools.

Abstract

We derive a simple proof, based on information theoretic inequalities, of an upper bound on the largest rates of $q$-ary $\overline{2}$-separable codes that improves recent results of Wang for any $q\geq 13$. For the case $q=2$, we recover a result of Lindstr\"om, but with a much simpler derivation. The method easily extends to give bounds on $B_2$ codes which, although not improving on Wang's results, use much simpler tools and might be useful for future applications.