Constructing (h, d) cooperative MSR codes with sub-packetization (d-k+h) (d-k+1) ^ n/2

We address the multi-node failure repair challenges for MDS array codes. Presently, two primary models are employed for multi-node repairs: the centralized model where all failed nodes are restored in a singular data center, and the cooperative model where failed nodes acquire data from auxiliary nodes and collaborate amongst themselves for the repair process. This paper focuses on the cooperative model, and we provide explicit constructions of optimal MDS array codes with d helper nodes under this model. The sub-packetization level of our new codes is (d-k+h) (d-k+1) ^ n/2 where h is the number of failed nodes, k the number of information nodes and n the code length. This improves upon recent constructions given by Liu et al. (IEEE Transactions on Information Theory, Vol. 69, 2023).