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Maximum Size $t$-Intersecting Families and Anticodes

Xuan Wang, Tuvi Etzion, Denis Krotov, Minjia Shi

TL;DR

The paper addresses the maximum size of $t$-intersecting families and constant-weight anticodes over a nonbinary alphabet $q>2$, linking Erdős–Ko–Rado theory with the code-anticode bound. It identifies the canonical family ${\cal F}_q(t,k,n)$, proving asymptotic optimality and uniqueness for large $n$, and introduces two diameter-based anticodes ${\cal A}_q(t,k,n)$ and ${\cal A}'_q(t,k,n)$ with the same maximal size bound. A hierarchical framework ${\cal A}_q(t,\epsilon,k,n)$ is developed, showing how different anticodes dominate depending on $n$ and $q$ and revealing incomparability within the hierarchy, including an odd-diameter extension. The results illuminate the landscape of maximum-size anticodes and suggest future work on fully characterizing maximal anticodes within the hierarchy and extending these ideas to intersecting families more broadly.

Abstract

The maximum size of $t$-intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erdős-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding theory, were considered in a generalization of the well-known sphere-packing bound. In this work we consider the maximum size of $t$-intersecting families and their associated maximum size constant-weight anticodes over alphabet of size $q >2$. It is proved that the structure of the maximum size constant-weight anticodes with the same length, weight, and diameter, depends on the alphabet size. This structure implies some hierarchy of constant-weight anticodes.

Maximum Size $t$-Intersecting Families and Anticodes

TL;DR

The paper addresses the maximum size of -intersecting families and constant-weight anticodes over a nonbinary alphabet , linking Erdős–Ko–Rado theory with the code-anticode bound. It identifies the canonical family , proving asymptotic optimality and uniqueness for large , and introduces two diameter-based anticodes and with the same maximal size bound. A hierarchical framework is developed, showing how different anticodes dominate depending on and and revealing incomparability within the hierarchy, including an odd-diameter extension. The results illuminate the landscape of maximum-size anticodes and suggest future work on fully characterizing maximal anticodes within the hierarchy and extending these ideas to intersecting families more broadly.

Abstract

The maximum size of -intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erdős-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding theory, were considered in a generalization of the well-known sphere-packing bound. In this work we consider the maximum size of -intersecting families and their associated maximum size constant-weight anticodes over alphabet of size . It is proved that the structure of the maximum size constant-weight anticodes with the same length, weight, and diameter, depends on the alphabet size. This structure implies some hierarchy of constant-weight anticodes.

Paper Structure

This paper contains 4 sections, 17 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Maximum Size $t$-Intersecting Families and Anticodes
  3. A Hierarchy of Anticodes
  4. Conclusion, Discussion, and Future Research

Key Result

Theorem 1

If $n \geq (t+1)(k-t+1)$ then any $t$-intersecting family of $k$-subsets from an $n$-set contains at most $\binom{n-t}{k-t}$ subsets. The bound is attained by all $k$-subsets (of an $n$-set) that contain a fixed $t$-subset. If $n > (t+1)(k-t+1)$ this family is unique and if $n = (t+1)(k-t+1)$ there

Theorems & Definitions (32)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Definition 2
  • Definition 3
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Lemma 1
  • proof
  • ...and 22 more