On the Length of Maximum Sum-Rank Distance Codes

Abstract: The classical theory of arcs in finite projective spaces provides a powerful geometric framework for studying maximum distance separable (MDS) codes and their possible lengths. In this paper, we investigate how this perspective extends to maximum sum-rank distance (MSRD) codes, whose geometric description involves systems of linear sets rather than individual projective points. Our main goal is to understand how the block structure of an MSRD code constrains its length. In dimension three, we determine the possible intersection patterns of lines with the associated linear sets and obtain explicit formulas for their characters and the weight distribution of the corresponding codes. These results lead to new length bounds and show that the block structure comes at a cost: the presence of a block of rank at least two reduces the maximum possible total length by at least $q-1$, and sufficiently long MSRD codes must be classical, that is, MDS codes over $\mathbb F_{q<sup>m}$. We also prove that three-dimensional one-weight MSRD codes do not exist. In arbitrary dimension, we interpret shortening geometrically as projection and establish a general projection bound that extends classical counting arguments for arcs. This yields restrictions depending explicitly on the block profile and, for full-length blocks, identifies parameter ranges in which linearized Reed--Solomon codes attain the maximum possible number of blocks. Finally, we construct an exceptional three-dimensional MSRD code over $\mathbb F_9$ with three blocks of length two and propose a sum-rank analogue of the MDS conjecture.