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

10/12/2022

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by   Minjia Shi, et al.

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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.