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Cooperative repair model is an available technology to deal with multiplenode failures in distributed storage systems. Recently, explicit constructionsof cooperative MSR codes were given by Ye (IEEE Transactions on InformationTheory, 2020) with sub-packetization level $(d-k+h)(d-k+1)^n$. Specifically,the sub-packetization level is $(h+1)2^n$ when $d=k+1$. In this paper, wepropose a new cooperative repair scheme by means of the inter-instance andintra-instance pairing inherited from the perfect code which reduces thesub-packetization to $2^n$ when $(h+1)|2^n$ and $(2\ell+1)2^n$ when$h+1=(2\ell+1)2^m$ for $m\ge 0$, $\ell\ge 1$ with $d=k+1$ helper nodes. That isto say, the sub-packetization is $h + 1 $ times or $2^m$ times less than Ye's.It turned out to be the best result so far known.

A New Cooperative Repair Scheme with k + 1 Helper Nodes for (n, k) Hadamard MSR codes with Small Sub-packetization