Two Conjectures on the Largest Minimum Distances of Binary Self-Orthogonal Codes With Dimension 5
Minjia Shi, Shitao Li, Jon-Lark Kim
Abstract
The purpose of this paper is to solve the two conjectures on the largest minimum distance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d_{so}(n,5)$ </tex-math></inline-formula> of a binary self-orthogonal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n, 5]$ </tex-math></inline-formula> code proposed by Kim and Choi (2022). The determination of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d_{so}(n,k)$ </tex-math></inline-formula> has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogonal codes as the dimension <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> increases. Recently, Kim et al. (2021) considered the shortest self-orthogonal embedding of a binary linear code, and many binary optimal self-orthogonal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n,k]$ </tex-math></inline-formula> codes were constructed for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k=4,5$ </tex-math></inline-formula> . Kim and Choi (2022) improved some results of Kim et al. (2021) and made two conjectures on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d_{so}(n,5)$ </tex-math></inline-formula> . In this paper, we develop a general method to determine the exact value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d_{so}(n,k)$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k=5,6$ </tex-math></inline-formula> and show that the two conjectures made by Kim and Choi (2022) are true.