# Parameter Inequalities for Orthogonal Arrays with Mixed Levels

## Summary (1 min read)

### 1 Introduction

- The authors will use |A| to denote the cardinality of a multiset A, taking into account how often each of its element occurs.
- (Such arrays are also often referred to as asymmetric orthogonal arrays.).

### 2 A bound for mixed-level arrays

- (1) Note that this is the dual of the Plotkin bound in the sense that upper bounds on the minimum distance of codes yields lower bounds on the size of orthogonal arrays.
- In addition to being much easier to compute than Rao’s bound, the inequality (1) leads to a larger minimum value for N than Rao’s bound for a number of parameter combinations.
- The authors will adapt his method to provide the following generalization of Theorem 2.1 for orthogonal arrays with mixed levels.
- The authors have not been able to determine the exact conditions under which the bound given in (2) is superior to Rao’s bounds.
- As is the case for symmetric orthogonal arrays, it is possible that even though the bound in (2) is greater than Rao’s bound, both still lead to the same minimum value of N .

### If t is odd, then

- (20) The proof of Theorem 3.1 indicated in Rao’s paper is valid only for simple orthogonal arrays (i.e. arrays that have no repeated columns) and is carried out in Beder [1].
- Bierbrauer [2] provided a proof for Rao’s inequalities in the symmetric case that is based on his technique for developing the lower bound in Theorem 2.1.
- The authors will extend Bierbrauer’s method to prove Rao’s inequalities for arrays with mixed levels.
- This proof also does not require simplicity.
- Thus for z ∈ Z0, the functions fz are pairwise orthogonal in L2(P ) and therefore linearly independent.

### 4 Concluding remarks

- The lower bound in (2) appears to be sharper than Rao’s bounds for a significant number of parameter combinations.
- Whether a mixed-level orthogonal array for a given parameter combination and having the minimum size prescribed by the bound actually exists is a question that has yet to be answered.
- Both Rao’s bounds and the linear programming bound for mixed-level orthogonal arrays developed in Sloane and Stufken [7] are much more difficult to compute than the one developed in this paper.
- A general comparison of the different bounds awaits further investigation.

### Acknowledgments

- I would like to thank the referees for a number of suggestions that greatly improved this paper.
- Thanks also to Jay Beder for his helpful comments.

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##### References

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### "Parameter Inequalities for Orthogon..." refers background in this paper

...An alternate proof that does not require the array to be simple is suggested in Hedayat et al. [ 3 ]....

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...Various other bounds have been developed since then; see Hedayat et al. [ 3 ] for a survey of the dierent inequalities that are known thus far....

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441 citations

### "Parameter Inequalities for Orthogon..." refers background in this paper

...1 Introduction Among the first lower bounds on the size of an orthogonal array were those developed by Rao [5] for symmetric orthogonal arrays, and in Rao [6] for orthogonal arrays with mixed levels....

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179 citations

### "Parameter Inequalities for Orthogon..." refers background in this paper

...It is a well-known fact that the characters of G form an orthonormal basis of L2(G) (see [4])....

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32 citations

### "Parameter Inequalities for Orthogon..." refers background in this paper

...The proof is based on Bierbrauer’s proof in the symmetric case....

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...Bierbrauer [2] outlined an algebraic proof in which he regards the orthogonal array as a multiset on the group G = (Zs)....

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...We will extend Bierbrauer’s method to prove Rao’s inequalities for arrays with mixed levels....

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...In the same paper, Bierbrauer also demonstrated how his method could be used to prove Rao’s inequalities for symmetric orthogonal arrays....

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...We will then prove a generalization of Bierbrauer’s inequality for mixed-level orthogonal arrays....

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