A mathematical concept has a corresponding relationship with its existence. It must first define the concept accurately, and then study its existence and the uniqueness of the solution. The author repeatedly changed the definition of LHD concepts to adapt to the existence theorem, just proved that they had the existence theorem before, then this theorem is surely not derived by himself. Because the author's LHD definition domain is different from that of the literature, its existence does not mean the existence of LHD defined by the literature. This determined that they cannot prove the theorem correctly.

1 A Direct Comparison of Two Existence Theorems of He and Lin

1.1 the existence theorem in He(2009), 1.2 The Existence Theorem in Lin(2008)

2. Amazing similarity

Mathematical concepts correspond to their existence theorems. It must first define the concept accurately, and then study its existence and the uniqueness of the solution. The author repeatedly changed the concept of LHD to adapt to the existence theorem, exactly proven that they first had the existence theorem, and then pieced together the proof method. Such theorems must not be derived by themselves.

Because the author's LHD is a special case of the LHD definition, its existence does not mean the existence of the OLHD defined by the literature. This determines that they cannot prove the theorem.

The necessary conditions for existence in Lin (2008, 2010)’s existence theorem are exactly the same as He (2009). The key parameter 4k+2 and the symbols used are exactly the same as those of He (2009). The three proof steps described in the introduction are extracted from the proof of He (2009).

Lin (2009) does not include these OLHDs that number of the run has the form of 4k+2, that shows that she had known the existence conditions of OLHD from He (2009) but had not formed her own method of proof at that time, that not -so-correct proof method of the existence theorem 2.5 in Lin (2008) was obtained by the reverse analysis from He (2009).

3 “run size n cannot be two or three” is a false proposition.

The run size n cannot be 2 and 3. It is correct only in the case of zero correlation design, He (2009) have a proof, "When n=3, there are only 6 vectors in S, and the correlation coefficient between any two vectors in S is not zero, so n cannot be 3." In Lin(2008), the author used a reason: " it is easy to verify" which replaced the proof in He(2009).. In Lin(2010), the author said:"Trivially, run size n cannot be two or three." Also didn't give a reason. In fact, in Latin space, under Rao's definition, OA_II(3,3,3,2) exists(Hedayat ea al (1999) p.133). If n=2 and b is a real number, then ((b,0)^T,(0,b)^T) not only satisfies Hedayat's orthogonal definition; And its inner product is zero, it also satisfies the orthogonal definition of algebra, but not a zero correlation. So, In Lin's theorem, k in 4k+2 cannot be 0.

4 Lin's Proof logic of the Necessity Part Was Seriously out of Specification

The task proving this proposition should be to prove that any two members of the permutation set are not orthogonal if n=4k+2. Didn't Lin answer the question.

She specified a and b "be first and second columns", and "Without loss of generality, we assume that a=...". Because this vector a is a special structure that has been carefully arranged, which is not general, and the author did not say how to continue her reasoning when a vector is not this structure. So, “Without loss of generality” was used improper here. Even if Lin proved that a and b are not orthogonal, one cannot conclude that any two elements of the permutation set can not be orthogonal. Lin proved at most that a is not orthogonal to its other permutations, but did not proved between other permutations are not orthogonal to each other. Take n=6 for example, there are (6!=)720 permutations in the permutation set. Even if a=(-1,-3,-5,1,3,5) is not orthogonal to all other 719 permutations, it can not naturally conclude that 258,121 pairs of vectors of the remaining 719 permutations are all no orthogonal. Lin caused a contradiction using a specifying vector a, immediately established conclusions and ended the proof process, which is incomplete induction. Using a specifying member of a set to introduce a contradictions, can other member of the set introduce the same contradiction? This is exactly what needs to be proved, but the author has not proved it. After proving propositional logic with an instance, it must be proved that same result can be obtained for any member of the set. Only this way is Complete Induction. Just as we cannot using "Without loss of generality" to specify an integer to prove a formula of the sum of squares of n natural numbers from 1 to n, that must be proved the formula holds for any positive integer n. Mathematical induction (Complete induction) is usually used. If "Without loss of generality" can be used like Lin, you can also easily prove many difficult problems like the Goldbach conjecture and to trisect an any-angle..

5 The Key Logical Premise in Lin's Proof Was False

In Lin's proof, the most important logical premise was that "both 2b_i and 2b_i+n/2 are odd, i=1,...,n/2", This is only valid if spacing between levels is odd; but invalid if spacing between levels is even number. Reasoning cannot to be continue, and the contradictions that Lin wants cannot be obtained. The author did not explain how to deal with the case where the spacing is not 1. Because the result of her LHD linear transformation is not in the domain, she has no way to complete the proof. This is the mystery why the author changed the definition of LHD.

The essential part of the author's proof is missing at least three steps, so the existing proof does not conform to the logical norm. If the author does not know in advance that the proposition is true, can they just believe their theorem like this? This proved that the theorem does not belong to the author.

6. The Sufficiency Part of the Theorem Has Not Been Effectively Proved

The construction of orthogonal hypercube designs is a kind of NP hard problem. How many columns exist in an orthogonal design is unknown, so we can't discuss its existence generally.

An experimental design D is called an orthogonal design that distinguishes from a general orthogonal matrix, it is required that not only column-orthogonal, and the marginal distribution is uniform, its joint distribution is also balanced. The OLHDs constructed by Lin in the proof is very unbalanced.

This Figure showed a result for stacking an OLHD whose run size is five four times with O₂. The result of continuing to stack is that four rays will be indefinitely extended, and we call It a "×"-distribution. Experimental designs can not accept such a class of designs. In addition to their imbalance, the collinear line of some quadric surfaces (such as paraboloid and saddle surfaces) are two crossed straight lines. If the real law of a process is a paraboloid or saddle surface, and sampling on two crossed straight lines, then the curved surface will inevitably be judged as a plane, unable to distinguish the truth or false of the "linear hypothesis". Such an experiment cannot find an optimized area, and the Optimized solution cannot be formed. The entire experiment fails and gives wrong conclusions. In industrial experiments, the mechanism of the process is usually unknown, and quadratic functions are often used to fit the experimental data, and building forecasting models and looking for optimization solutions, which will encounter such problems. In the actual experiment process, a completely linear process is rarely seen. If an X-shaped experimental point distribution is used as a sampling scheme, failure is a high probability event. The high failure rate is one of the reasons why orthogonal experimental design is difficult to promote in industrial experiments.

Some methods can improve its uniformity, but there is no guarantee that it exists for any number of runs, see Discuss with Dr.C.D.Lin how to construct orthogonal hypercube designs with the stacking method (in Chinese).

Thus, at present, the proposition has not been effectively proved.