On dually almost MRD codes

In this paper we define and study a family of codes which are coming close to MRD codes. Thus we call them AMRD codes (almost MRD). An AMRD code is a code with rank defect equal to 1. These codes can be viewed as q-analogs of classical AMDS codes as considered in [3], [7], [9]. AMRD codes whose duals are AMRD as well are called dually AMRD. These codes have important symmetry properties. For instance, the number of codewords of minimum rank in C and C⊥ are equal if the size of the matrices divides the dimension of C[4]. Necessary and sufficient conditions are given for codes to be dually AMRD and we give a construction of such codes of minimal dimension. We also construct self-dual AMRD codes. Furthermore, we show that dually AMRD codes coincide with codes of rank defect one and maximum 2-generalized weight if the size of the matrices divides the dimension. The results may be seen as q-analogs of results established in classical coding theory [7], [9].