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This is from a note I found enter image description here

I have one question, could t=3?

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  • $\begingroup$ Isn't that the case for the array you show? Within each $N\times 3$ subarray, each of the tuples appears equally often. A bunch of small cases with $t=3$ are discussed here. A quick search turns up several papers discussing methods for construction of arrays of strength 3. $\endgroup$ – Glen_b Sep 26 '16 at 9:46
  • $\begingroup$ @Glen_b the note actually says that t cannot be 3, t can only be 2. I think it might relate to the definition of "t -tuple", is it actually tuples appear in the row or all possibile tuples? $\endgroup$ – whoisit Sep 26 '16 at 9:57
  • $\begingroup$ I can't understand what you're asking me there. However, it's definitely the case that you can get asymmetric orthogonal arrays of strength three. Here's one of the papers that offer constructions. Section 4.7 of the book Fractional Factorial plans by Dey and Mukherjee discusses them as well. In general Hadamard matrices are used to construct them. $\endgroup$ – Glen_b Sep 26 '16 at 10:04
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    $\begingroup$ It's not possible to resolve a statement we have no context for; perhaps there some condition you're failing to mention. As your question stands the answer is "Yes". The only reason that's not currently posted as an answer is "yes" is a bit brief... and I fear that as soon as I answer it you'll change the question. $\endgroup$ – Glen_b Sep 26 '16 at 10:10

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