Six Families of Binary Codes Arising from Ding's Conjectures

2026-09-29

AI Summary

This study addresses the six remaining open conjectures proposed by Ding concerning binary linear codes derived from Boolean functions. By integrating combinatorics, finite field theory, and Boolean function analysis, we systematically investigate the weight distribution properties of specific families of binary codes. Through the derivation of general upper bounds on weights and the construction of counterexamples to refine the original conjectures, we precisely determine several parameter cases. The principal contributions include confirming or refuting the relevant conjectures, discovering an infinite family of five-weight codes, and completely resolving Conjecture 37. These findings provide essential theoretical support for advancing research in coding theory.

Abstract

Ding \cite{Ding2016} proposed ten conjectures on binary linear codes arising from Boolean functions. Four of them, namely Conjectures 38--41, were subsequently proved by Göloğlu and Krasnayová \cite{GologluKrasnayova2019}. In this paper, we investigate the remaining six conjectures, namely Conjectures 19, 27, 30, 33, 34, and 37. For Conjectures~19 and~27, we obtain common weight restrictions and several infinite five-weight families. For Conjecture~30, we prove that every admissible code has three, four, or five nonzero weights, and an explicit four-weight example disproves the original ``three or five weights'' assertion. For Conjecture~33, an infinite five-weight family is obtained. For Conjecture~34, we obtain a general $(2h+1)$-weight upper bound and give an explicit six-weight counterexample, showing that the original ``three or five weights'' assertion is false in general, where $h$ is a positive integer. The case $h=3$ with $3\nmid m$ is also completely determined. Finally, Conjecture~37 is completely resolved by combining the known results of Ahmadi and Shafaeiabr \cite{AhmadiShafaeiabr2023} with the treatment of the two remaining classes $(a)$ and $(b)$.

Problem

• Binary linear codes
• Boolean functions
• Ding's conjectures
• Weight distribution

Innovation

• Binary linear codes
• Boolean functions
• Ding's conjectures
• Weight distribution
• Counterexamples