Flag Codes from Planar Spreads in Network Coding

Abstract: In this paper we study a class of multishot network codes given by families of nested subspaces (flags) of a vector space $\mathbb{F}_q<sup>n$, being $q$ a prime power and $\mathbb{F}_q$ the finite field of $q$ elements. In particular, we focus on flag codes having maximum minimum distance (optimum distance flag codes). We explore the existence of these codes from spreads, based on the good properties of the latter ones. For $n=2k$, we show that optimum distance full flag codes with the largest size are exactly those that can be constructed from a planar spread. We give a precise construction of them as well as a decoding algorithm.