This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension of a rank-metric code with given specified Ferrers diagram shape and rank distance. While the conjecture in its generality is wide open, several cases have been established in the literature. This paper contributes further cases of Ferrers diagrams and ranks for which the conjecture holds true. In addition, the proportion of maximal Ferrers diagram codes within the space of all rank-metric codes with the same shape and dimension is investigated. Special attention is being paid to MRD codes. It is shown that for growing field size the limiting proportion depends highly on the Ferrers diagram. For instance, for [m × 2]-MRD codes with rank 2 this limiting proportion is close to 1/e. of MRD codes that are not equivalent to Gabidulin codes and not necessarily F q m -linear. Most notably, in [27] Sheekey presents a construction of MRD-codes that are not equivalent to Gabidulin codes. Further contributions have been made by de la Cruz et al. [6] and Trombetti/Zhou [33].A very straightforward construction of good subspace codes with the aid of rank-metric codes is the lifting construction [19]: to each matrix M in the given rank-metric code one associates the row space of the matrix (I | M ), where I is the identity matrix of suitable size. While this simple construction leads to subspace codes with good distance, it usually does not produce large codes. A remedy has been introduced by Etzion/Silberstein [9]: obviously a matrix of the form (I | M ) ∈ F m×(m+n) is in reduced row echelon form (RREF) with pivot indices 1, . . . , m. This observation has led to the multilevel construction, where for each level a rank-metric code in F m×n is used to construct a subspace code in F m+n with all representing m × (m + n)-matrices being in RREF with a fixed set of general pivot indices. For this to work out properly, the matrices in the given rank-metric code have to be supported by the Ferrers diagram associated with the list of pivot indices; see [9] and Remark 2.5 later in this paper. As a result, the multilevel construction leads to the task of constructing large Ferrers diagram codes with a given rank distance. In [9] the authors provide an upper bound for the dimension of a rank-metric code supported by a given Ferrers diagram F and with a given rank distance δ. In this paper, codes attaining this bound will be called maximal [F; δ]-codes. To this day, it is not clear whether maximal [F; δ]-codes exist for all pairs (F; δ) and all finite fields. Several cases have been settled by Etzion et al. [8,9] and Gorla/Ravagnani [16] and, more recently, by Liu et al. [20] and Zhang/Ge [35], but the general case remains widely open. In [2] Ballico studies the existence of maximal [F; δ]-codes over number fields.In this paper we survey some of these results and extend them to further classes of pairs...

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Let R be a finite principal left ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module R n is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combination with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal left ideal rings and show that the total ordering of the ring elements has to respect containment of ideals in order for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.

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