Two conjectures on the largest minimum distances of binary self-orthogonal codes with dimension 5

Abstract

The purpose of this paper is to solve the two conjectures on the largest minimum distance d so (n, 5) of a binary self-orthogonal [n, 5] code proposed by Kim and Choi (IEEE Trans. Inf. Theory, 2022). The determination of d so (n, k) has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogonal codes as the dimension k increases. Recently, Kim et al. ( 2021) considered the shortest self-orthogonal embedding of a binary linear code, and many binary optimal self-orthogonal [n, k] codes were constructed for k = 4, 5. Kim and Choi (2022) improved some results of Kim et al. ( 2021) and made two conjectures on d so (n, 5). In this paper, we develop a general method to determine the exact value of d so (n, k) for k = 5, 6 and show that the two conjectures made by Kim and Choi (2022) are true.