## Description

An orthogonal array, or OA\(_{\lambda}(t,n,q)\), of strength \(t\) with \(q\) levels and \(n\) constraints is a set of \(q\)-ary strings such that any subset of \(t\) coordinates contains every length-\(t\) string an equal number of times \(\lambda\), which is the index of the array.

## Notes

See [4] for a book on orthogonal arrays.

## Parents

- \(q\)-ary code — There is a relation between \(q\)-ary codes and orthogonal arrays which is phrased in terms of the codes' dual distance [5; Thm. 4.5][4; Thm. 4.9].
- \(t\)-design — Orthogonal arrays are designs on Hamming space \(GF(q)^n\) (a.k.a. the Hamming association scheme) [6–8][9; Exam. 1]; see also Ref. [10].

## Children

- Perfect binary code — Perfect distance-three binary codes of length \(n =2^m-1\) are equivalent to binary orthogonal arrays of strength \(t = 2^{m-1}-1\) [5,11,12].
- Maximum distance separable (MDS) code — An MDS code is an OA\(_{1}(k,n,q)\) [13; Thm. 3.3.19].
- \(q\)-ary sharp configuration

## Cousins

- Golay code — The extended Golay code is an orthogonal array of strength 7 [9; Exam. 1]
- Binary code — An \((n,K)\) binary code with dual distance \(d^{\perp}\) is an OA\(_{K/2^{d^{\perp}-1}}(d^{\perp}-1,n,2)\) [14][15; Ch. 5].
- Reed-Muller (RM) code — RM codes are related to orthogonal arrays [16; Exam. 10.57].
- Mixed code — Orthogonal arrays generalized to mixed alphabets are called mixed-level orthogonal arrays [17,18], (see [4; Ch. 9]). See Ref. [19] for bounds on mixed orthogonal arrays.
- Perfect-tensor code — Orthogonal arrays and \(d\)-uniform quantum states are related [20,21].

## References

- [1]
- C. R. Rao, Hypercubes of strength d leading to confounded designs in factorial experiments. Bull. Calcutta Math. Soc., 38, 67-78.
- [2]
- C. R. Rao, “Factorial Experiments Derivable from Combinatorial Arrangements of Arrays”, Supplement to the Journal of the Royal Statistical Society 9, 128 (1947) DOI
- [3]
- C. R. Rao, “On a Class of Arrangements”, Proceedings of the Edinburgh Mathematical Society 8, 119 (1949) DOI
- [4]
- A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays (Springer New York, 1999) DOI
- [5]
- P. Delsarte, “Four fundamental parameters of a code and their combinatorial significance”, Information and Control 23, 407 (1973) DOI
- [6]
- Delsarte, Philippe. "An algebraic approach to the association schemes of coding theory." Philips Res. Rep. Suppl. 10 (1973): vi+-97.
- [7]
- Ph. Delsarte, “Hahn Polynomials, Discrete Harmonics, andt-Designs”, SIAM Journal on Applied Mathematics 34, 157 (1978) DOI
- [8]
- V. I. Levenshtein, “Universal bounds for codes and designs,” in Handbook of Coding Theory 1, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp.499-648.
- [9]
- P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory”, IEEE Transactions on Information Theory 44, 2477 (1998) DOI
- [10]
- G. Kuperberg, S. Lovett, and R. Peled, “Probabilistic existence of regular combinatorial structures”, (2017) arXiv:1302.4295
- [11]
- T. Etzion and A. Vardy, “Perfect binary codes: constructions, properties, and enumeration”, IEEE Transactions on Information Theory 40, 754 (1994) DOI
- [12]
- P. R. J. Ostergard, O. Pottonen, and K. T. Phelps, “The Perfect Binary One-Error-Correcting Codes of Length 15: Part II—Properties”, IEEE Transactions on Information Theory 56, 2571 (2010) arXiv:0903.2749 DOI
- [13]
- P. R. J. Östergård, "Construction and Classification of Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [14]
- Delsarte, Philippe. "Bounds for unrestricted codes, by linear programming." (1972).
- [15]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [16]
- Combinatorial Designs (Springer-Verlag, 2004) DOI
- [17]
- Addelman, S., & Kempthorne, O. (1961b). Orthogonal Main-Effect Plans. Technical Report ARL 79, Aeronautical Research Lab., Wright-Patterson Air Force Base, Ohio, Nov. 1961.
- [18]
- C. R. RAO, “Some Combinatorial Problems of Arrays and Applications to Design of Experiments††Paper read at the International Symposium on Combinatorial Mathematics and its Applications, Fort Collins, Colorado, September 1971.”, A Survey of Combinatorial Theory 349 (1973) DOI
- [19]
- N. J. A. Sloane and J. Stufken, “A linear programming bound for orthogonal arrays with mixed levels”, Journal of Statistical Planning and Inference 56, 295 (1996) DOI
- [20]
- D. Goyeneche and K. Życzkowski, “Genuinely multipartite entangled states and orthogonal arrays”, Physical Review A 90, (2014) arXiv:1404.3586 DOI
- [21]
- D. Goyeneche et al., “Entanglement and quantum combinatorial designs”, Physical Review A 97, (2018) arXiv:1708.05946 DOI

## Page edit log

- Victor V. Albert (2024-01-09) — most recent

## Cite as:

“Orthogonal array (OA)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/orthogonal_array