# American Institute of Mathematical Sciences

February  2020, 14(1): 69-88. doi: 10.3934/amc.2020006

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

 1 Faculty of Mathematics and Statistics, University of St. Gallen, Dufourstrasse 50, 9000 St. Gallen, Switzerland 2 Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

* Corresponding author: Alessandro Neri

Received  August 2018 Revised  February 2019 Published  August 2019

Fund Project: The second author is supported by Swiss National Science Foundation grant no. 169510

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.

Citation: Anna-Lena Horlemann-Trautmann, Alessandro Neri. A complete classification of partial MDS (maximally recoverable) codes with one global parity. Advances in Mathematics of Communications, 2020, 14 (1) : 69-88. doi: 10.3934/amc.2020006
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