The index of a string consisting of 4 blocks

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Generalized Fibonacci cube Qd(f), introduced by Ilić, Klavžar and Rho, is the graph obtained from the d-hypercube Qd by removing all vertices that contain f as a substring. The smallest integer d such that Qd(f) is not an isometric subgraph of Qd is called the index of f. A non-extendable sequence of contiguous equal digits in a string μ is called a block of μ. The question that determine the index of a string consisting of at most 3 blocks is solved by Ilić, Klavžar and Rho. This question is further studied and the index of a string consisting of 4 blocks is determined, and the necessity of a string being good is also given for the strings with even blocks.