On dually QMRD and AMRD codes
Talk by Prof. Dr. Javier de la Cruz
Date: 07.12.16 Time: 15.00 - 16.00 Room: Y27H12
Due to the multiple applications in network coding and cryptography, there is a steady stream of work that focuses on general properties of codes with the rank metric. The MRD codes, having the largest distance, have been the most studied. Existence of MRD codes was established in [3] and [4] for all 1 ≤ n ≤ m and 1 ≤ d ≤ n. Recently, constructions of MRD codes which are not equivalent to the previous ones appeared [2]. In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for rank metric codes. The rank defect of a code C measures how far C is away from being a MRD code. Based on this concept we define two classes of codes fairly close to the MRD codes, which we call dually QMRD and dually AMRD codes. We show interesting properties of these classes of codes and relationships between them. Moreover, we show that the rank distribution of a rank metric code of positive rank defect is completely determined by its small rank weights.Much of this talk is based on a join work with E. Gorla, H. López and A. Ravagnani.