Flag Codes: Distance Vectors and Cardinality Bounds

Given Fq the finite field with q elements and an integer n > 2, a flag is a sequence of nested subspaces of Fnq and a flag code is a nonempty set of flags. In this context, the distance between flags is the sum of the corresponding subspace distances. Hence, a given flag distance value might be obtained by many different combinations. To capture such a variability, in the paper at hand, we introduce the notion of distance vector as an algebraic object intrinsically associated to a flag code that encloses much more information than the distance parameter itself. Our study of the flag distance by using this new tool allows us to provide a fine description of the structure of flag codes as well as to derive bounds for their maximum possible size once the minimum distance and dimensions are fixed.The authors received financial support of Ministerio de Ciencia e Innovación (PID2019-108668GB-I00). The second author is supported by Generalitat Valenciana and Fondo Social Europeo (ACIF/2018/196)