A complete classification of partial MDS (Maximally recoverable) codes with one global parity

We generalize the definition of partial MDS codes to locality blocks of various length and show that these codes are maximally recoverable. Then we focus on partial MDS codes with exactly one global parity. We derive a general construction for such codes by describing a suitable parity check matrix. Then we give a construction of generator matrices of such codes. Afterwards we show that all partial MDS codes with one global parity have a generator matrix (or parity check matrix) of this form. This gives a complete classification and hence also a sufficient and necessary condition on the underlying field size for the existence of such codes. This condition is related to the classical MDS conjecture. Moreover, we investigate the decoding of such codes and give some comments on partial MDS codes with more than one global parity

Random construction of partial MDS codes

This work deals with partial MDS (PMDS) codes, a special class of locally repairable codes, used for distributed storage systems. We first show that a known construction of these codes, using Gabidulin codes, can be extended to use any maximum rank distance code. Then we define a standard form for the generator matrices of PMDS codes and use this form to give an algebraic description of PMDS generator matrices. This implies that over a sufficiently large finite field a randomly chosen generator matrix in PMDS standard form generates a PMDS code with high probability. This also provides sufficient conditions on the field size for the existence of PMDS codes

Equivalence and characterizations of linear rank-metric codes based on invariants

We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. The same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism. These invariants give rise to easily computable criteria to check if two codes are inequivalent. We derive some concrete values and bounds for these dimension sequences for some known families of rank-metric codes, namely Gabidulin and (generalized) twisted Gabidulin codes. We then derive conditions on the length of the codes with respect to the field extension degree, such that codes from different families cannot be equivalent. Furthermore, we derive upper and lower bounds on the number of equivalence classes of Gabidulin codes and twisted Gabidulin codes, improving a result of Schmidt and Zhou for a wider range of parameters. In the end we use the aforementioned sequences to determine a characterization result for Gabidulin codes

On the genericity of maximum rank distance and Gabidulin codes

We consider linear rank-metric codes in Fqmn. We show that the properties of being maximum rank distance (MRD) and non-Gabidulin are generic over the algebraic closure of the underlying field, which implies that over a large extension field a randomly chosen generator matrix generates an MRD and a non-Gabidulin code with high probability. Moreover, we give upper bounds on the respective probabilities in dependence on the extension degree m