Optimal Trace Codes and Their Self-Orthogonality

The primary objective of this paper is the construction of optimal codes with self-orthogonality that can be used to construct quantum codes. Recently, Ding and Heng explored subfield codes, which can be viewed as trace codes. In this paper, we focus on investigating self-orthogonal optimal trace codes. First, we provide a novel description of trace codes by choosing suitable defining sets. Second, we determine the parameters of the codes and their trace codes whose defining sets are disjoint union of some affine subspaces in both non-projective cases and projective-cases. This result extends the main findings in (Hu, Li, Zeng, Wang, Tang, IEEE Trans. Inform. Theory, 68(7): 4408-4421, 2022). Third, we compute the parameters of trace codes for MacDonald codes, including the first order Reed-Muller codes and simplex codes as special cases. Finally, we examine their self-orthogonality and distance-optimality to find several classes of self-orthogonal Griesmer codes. Additionally, we resolve a problem proposed by Ding and Heng as a byproduct.