(1+ε )-optimal MDS codes: contacting any set of helper nodes smaller than n-1

$${\textbf {(1}}+ \varvec{\epsilon } {\textbf {)}}$$ -optimal Maximum Distance Separable ( $${\textbf {1}}+\varvec{\epsilon } {\textbf {)}}$$ -optimal MDS) codes, which are a special kind of $${\textbf {(n, k)}}$$ MDS codes, can repair a single failed node by downloading slightly sub-optimal amount of data from $${\textbf {d}}$$ helper nodes where $${\textbf {d}} \varvec{\in } {\textbf {[k, n)}}$$ , and have small sub-packetization level. Some $${\textbf {(1}}+\varvec{\epsilon } {\textbf {)}}$$ -optimal MDS codes have been constructed in the literatures. However, for all the existing $${\textbf {(1}}+\varvec{\epsilon } {\textbf {)}}$$ -optimal MDS codes with the number of helper nodes $${\textbf {d}}$$ smaller than $${\textbf {n}}-{\textbf {1}}$$ , a few compulsory nodes need to be contacted when repairing a failed node. In this paper, we provide two explicit $${\textbf {(1}}+\varvec{\epsilon } {\textbf {)}}$$ -optimal MDS codes contacting any set of $${\textbf {d}}$$ helper nodes with $${\textbf {d}}$$ smaller than $${\textbf {n}}-{\textbf {1}}$$ for the first time.