Balanced Set Codes With Small Intersections

Abstract

Motivated by emerging applications in coding for molecular data storage, much attention has been paid to the intersecting set discrepancy problem, which aims to design a large family of subsets of a common labeled ground set with bounded pairwise intersection and bounded set discrepancy. In this paper, we study the maximum size of such families of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -subsets with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> elements ground set, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> -bounded intersections, and zero or one discrepancy, called as balanced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(t,k,v)$ </tex-math></inline-formula> set codes. By turning this problem into a graph edge-labeling problem, we are able to determine the maximum size of codes when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k=3,4$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t=2,3$ </tex-math></inline-formula> for a given ground set. The constructions are based on combinatorial designs, matching decompositions and edge coloring schemes. Furthermore, we improve the upper bound for balanced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(t,k,v)$ </tex-math></inline-formula> set codes with all integers <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\leq t &lt; k &lt; v$ </tex-math></inline-formula> . By the powerful probabilistic argument–Kahn’s Theorem, we show that the improved upper bound for any fixed integers <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \leq t &lt; k$ </tex-math></inline-formula> is asymptotically tight when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> goes to infinity.