Abstract
In this paper, we study flag codes on the vector space {{mathbb {F}}}_q^n, being q a prime power and {{mathbb {F}}}_q the finite field of q elements. More precisely, we focus on flag codes that attain the maximum possible distance (optimum distance flag codes) and can be obtained from a spread of {{mathbb {F}}}_q^n. We characterize the set of admissible type vectors for this family of flag codes and also provide a construction of them based on well-known results about perfect matchings in graphs. This construction attains both the maximum distance for its type vector and the largest possible cardinality for that distance.
Highlights
Random network coding is introduced in [1] as a new method for attaining a maximum information flow by using a channel modelled as an acyclic-directed multigraph with possibly several senders and receivers
In this paper, we deal with the converse problem: given n and a divisor k of n, we look for conditions on the type vector of an optimum distance flag code on Fqn having a k-spread as a projected code
We deal with the inverse problem: given n and a divisor k of n, we look for conditions on the type vector of an optimum distance flag code on Fqn having a k-spread as a projected code
Summary
Random network coding is introduced in [1] as a new method for attaining a maximum information flow by using a channel modelled as an acyclic-directed multigraph with possibly several senders and receivers. In [2], the authors build optimum distance flag codes with a k-spread as a projected code fixing the full type vector, that is, In this paper, we deal with the converse problem: given n and a divisor k of n, we look for conditions on the type vector of an optimum distance flag code on Fqn having a k-spread as a projected code. 3, we determine the set of admissible type vectors for a flag code to attain the maximum possible distance and to have a k-spread as a projected code. We complete this section with an example of our construction for the admissible type (2, 4) on F62 having a 2-spread as the subspace code used at the first shot
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