Designs, Codes and Cryptography

, Volume 81, Issue 2, pp 365–391

# On generalized Howell designs with block size three

• R. Julian R. Abel
• Robert F. Bailey
• Andrea C. Burgess
• Peter Danziger
• Eric Mendelsohn
Article

## Abstract

In this paper, we examine a class of doubly resolvable combinatorial objects. Let $$t, k, \lambda , s$$ and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-$$\mathrm {GHD}_{k}(s,v;\lambda )$$, is an $$s\times s$$ array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than $$\lambda$$ cells. Particular instances of the parameters correspond to Howell designs, doubly resolvable balanced incomplete block designs (including Kirkman squares), doubly resolvable nearly Kirkman triple systems, and simple orthogonal multi-arrays (which themselves generalize mutually orthogonal Latin squares). Generalized Howell designs also have connections with permutation arrays and multiply constant-weight codes. In this paper, we concentrate on the case that $$t=2$$, $$k=3$$ and $$\lambda =1$$, and write $$\mathrm {GHD}(s,v)$$. In this case, the number of empty cells in each row and column falls between 0 and $$(s-1)/3$$. Previous work has considered the existence of GHDs on either end of the spectrum, with at most 1 or at least $$(s-2)/3$$ empty cells in each row or column. In the case of one empty cell, we correct some results of Wang and Du, and show that there exists a $$\mathrm {GHD}(n+1,3n)$$ if and only if $$n \ge 6$$, except possibly for $$n=6$$. In the case of two empty cells, we show that there exists a $$\mathrm {GHD}(n+2,3n)$$ if and only if $$n \ge 6$$. Noting that the proportion of cells in a given row or column of a $$\mathrm {GHD}(s,v)$$ which are empty falls in the interval [0, 1 / 3), we prove that for any $$\pi \in [0,5/18]$$, there is a $$\mathrm {GHD}(s,v)$$ whose proportion of empty cells in a row or column is arbitrarily close to $$\pi$$.

## Keywords

Generalized Howell designs Triple systems Doubly resolvable designs

## Mathematics Subject Classification

Primary 05B07 05B15 Secondary 05B40 94B25

## Notes

### Acknowledgments

The authors would like to thank Esther Lamken for a number of useful comments and in particular for suggesting the intransitive starter-adder method, which was used for several of the smaller GHDs in this paper. R. F. Bailey is supported by the Vice-President (Grenfell Campus) Research Fund, Memorial University of Newfoundland. A. C. Burgess and P. Danziger are supported by an NSERC Discovery Grant.

## References

1. 1.
Abel R.J.R.: Existence of five MOLS of orders 18 and 60. J. Comb. Des. 23, 135–139 (2015).Google Scholar
2. 2.
Abel R.J.R., Bennett F.E.: Existence of 2 SOLS and 2 ISOLS. Discret. Math. 312, 854–867 (2012).Google Scholar
3. 3.
Abel R.J.R., Bennett F.E., Ge G.: The existence of four HMOLS with equal sized holes. Des. Codes Cryptogr. 26, 7–31 (2002).Google Scholar
4. 4.
Abel R.J.R., Chan N., Colbourn C.J., Lamken E.R., Wang C., Wang J.: Doubly resolvable nearly Kirkman triple systems. J. Comb. Des. 21, 342–358 (2013).Google Scholar
5. 5.
Abel R.J.R., Lamken E.R., Wang J.: A few more Kirkman squares and doubly near resolvable BIBDs with block size 3. Discret. Math. 308, 1102–1123 (2008).Google Scholar
6. 6.
Anderson B.A., Schellenberg P.J., Stinson, D.R.: The existence of Howell designs of even side. J. Comb. Theory Ser. A 36, 23–55 (1984).Google Scholar
7. 7.
Arhin J.: On the construction and structure of SOMAs and related partial linear spaces. Ph.D. Thesis, University of London (2006).Google Scholar
8. 8.
Arhin J.: Every $$\text{ SOMA }(n-2, n)$$ is Trojan. Discret. Math. 310, 303–311 (2010).Google Scholar
9. 9.
Bailey R.F., Burgess A.C.: Generalized packing designs. Discret. Math. 313, 1167–1190 (2013).Google Scholar
10. 10.
Bennett F.E., Colbourn C.J., Zhu L.: Existence of three HMOLS of types $$h^n$$ and $$2^n3^1$$. Discret. Math. 160, 49–65 (1996).Google Scholar
11. 11.
Brickell E.F.: A few results in message authentication. Congr. Numer. 43, 141–154 (1984).Google Scholar
12. 12.
Brouwer A.E., van Rees G.H.J.: More mutually orthogonal Latin squares. Discret. Math. 39, 263–281 (1982).Google Scholar
13. 13.
Burgess A., Danziger P., Mendelsohn E., Stevens B.: Orthogonally resolvable cycle decompositions. J. Comb. Des. 23, 328–351 (2015).Google Scholar
14. 14.
Cameron P.J.: A generalisation of $$t$$-designs. Discret. Math. 309, 4835–4842 (2009).Google Scholar
15. 15.
Chee Y.M., Cherif Z., Danger J.-L., Guilley S., Kiah H.M., Kim J.-L., Solé P., Zhang X.: Multiply constant-weight codes and the reliability of loop physically unclonable functions. IEEE Trans. Inf. Theory 60, 7026–7034 (2014).Google Scholar
16. 16.
Chee Y.M., Kiah H.M., Zhang H., Zhang X.: Constructions of optimal and near-optimal multiply constant-weight codes. arXiv:1411.2513 (2014).
17. 17.
Chu W., Colbourn C.J., Dukes P.: Constructions for permutation codes in powerline communications. Des. Codes Cryptogr. 32, 51–64 (2004).Google Scholar
18. 18.
Colbourn C.J., Dinitz J.H. (eds.): The CRC Handbook of Combinatorial Designs, 2nd edn. CRC Press, Boca Raton (2007).Google Scholar
19. 19.
Colbourn C.J., Kaski P., Östergård P.R.J., Pike D.A., Pottonen O.: Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19. Discret. Math. 311, 827–834 (2011).Google Scholar
20. 20.
Colbourn C.J., Kløve T., Ling A.C.H.: Permutation arrays for powerline communication and mutually orthogonal Latin squares. IEEE Trans. Inf. Theory 50, 1289–1291 (2004).Google Scholar
21. 21.
Colbourn C.J., Lamken E.R., Ling A.C.H., Mills W.H.: The existence of Kirkman squares-doubly resolvable $$(v, 3, 1)$$-BIBDs. Des. Codes Cryptogr. 26, 169–196 (2002).Google Scholar
22. 22.
Curran D.G., Vanstone S.A.: Doubly resolvable designs from generalized Bhaskar Rao designs. Discret. Math. 73, 49–63 (1988).Google Scholar
23. 23.
Deza M., Vanstone S.A.: Bounds for permutation arrays. J. Stat. Plan. Inference 2, 197–209 (1978).Google Scholar
24. 24.
Dinitz J.H., Stinson D.R.: Room squares and related designs. In: Dinitz J.H., Stinson D.R. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 137–204. Wiley, New York (1992).Google Scholar
25. 25.
Du J., Abel R.J.R., Wang J.: Some new resolvable GDDs with $$k=4$$ and doubly resolvable GDDs with $$k=3$$. Discret. Math. 338, 2105–2118 (2015).Google Scholar
26. 26.
Etzion T.: Optimal doubly constant weight codes. J. Comb. Des. 16, 137–151 (2008).Google Scholar
27. 27.
Fuji-Hara R., Vanstone S.A.: On the spectrum of doubly resolvable Kirkman systems. Congr. Numer. 28, 399–407 (1980).Google Scholar
28. 28.
Fuji-Hara R., Vanstone S.A.: Transversal designs and doubly resolvable designs. Eur. J. Comb. 1, 219–223 (1980).Google Scholar
29. 29.
Fuji-Hara R., Vanstone S.A.: The existence of orthogonal resolutions of lines in $$\text{ AG }(n, q)$$. J. Comb. Theory Ser. A 45, 139–147 (1987).Google Scholar
30. 30.
Huczynska S.: Powerline communication and the 36 officers problem. Phil. Trans. R. Soc. A 364, 3199–3214 (2006).Google Scholar
31. 31.
Kirkman, T.P.: Note on an unanswered prize question. Cambridge Dublin Math. J. 5, 255–262 (1850)Google Scholar
32. 32.
Kotzig A., Rosa A.: Nearly Kirkman systems. In: Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic University, Boca Raton, FL, 1974), pp. 607–614. Congressus Numerantium, No. X, Utilitas Math., Winnipeg (1974).Google Scholar
33. 33.
Lamken E.R.: The existence of doubly resolvable $$(v,3,2)$$-BIBDs. J. Comb. Theory Ser. A 72, 50–76 (1995).Google Scholar
34. 34.
Lamken E.R.: Designs with mutually orthogonal resolutions and decompositions of edge-colored complete graphs. J. Comb. Des. 17, 425–447 (2009).Google Scholar
35. 35.
MacNeish H.: Euler squares. Ann. Math. 23(2), 221–227 (1922).Google Scholar
36. 36.
Moore E.H.: Tactical Memoranda I–III. Am. J. Math. 18, 264–303 (1896).Google Scholar
37. 37.
Mathon R., Vanstone S.A.: On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays. Discret. Math. 30, 157–172 (1980).Google Scholar
38. 38.
Mullin R.C., Wallis W.D.: The existence of Room squares. Aequ. Math. 1, 1–7 (1975).Google Scholar
39. 39.
Phillips N.C.K., Wallis W.D.: All solutions to a tournament problem. Congr. Numer. 114, 193–196 (1996).Google Scholar
40. 40.
Room T.G.: A new type of magic square. Math. Gaz. 39, 307 (1955).Google Scholar
41. 41.
Rosa A.: Generalized Howell designs. In: Second International Conference on Combinatorial Mathematics. Ann. N. Y. Acad. Sci. 319, 484–489 (1979).Google Scholar
42. 42.
Rosa A., Vanstone S.A.: Starter-adder techniques for Kirkman squares and Kirkman cubes of small sides. Ars Comb. 14, 199–212 (1982).Google Scholar
43. 43.
Smith P.: A doubly divisible nearly Kirkman system. Discret. Math. 18, 93–96 (1977).Google Scholar
44. 44.
Soicher L.H.: On the structure and classification of SOMAs: generalizations of mutually orthogonal Latin squares. Electron. J. Comb. 6(1), R32 (1999).Google Scholar
45. 45.
Stinson D.R.: The existence of Howell designs of odd side. J. Comb. Theory Ser. A 32, 53–65 (1982).Google Scholar
46. 46.
Todorov D.T.: Four mutually orthogonal Latin squares of order 14. J. Comb. Des. 20, 363–367 (2012).Google Scholar
47. 47.
Vanstone S.A.: Doubly resolvable designs. Discret. Math. 29, 77–86 (1980).Google Scholar
48. 48.
Vanstone S.A.: On mutually orthogonal resolutions and near resolutions. Ann. Discret. Math. 15, 357–369 (1982).Google Scholar
49. 49.
Wang C., Du B.: Existence of generalized Howell designs of side $$n+1$$. Utilitas Math. 80, 143–159 (2009).Google Scholar
50. 50.
Yan J., Yin J.: Constructions of optimal $$\text{ GDRP }(n,\lambda ; v)\text{ s }$$ of type $$\lambda ^{1}\mu ^{m-1}$$. Discret. Appl. Math. 156, 2666–2678 (2008).Google Scholar

## Authors and Affiliations

• R. Julian R. Abel
• 1
• Robert F. Bailey
• 2
• Andrea C. Burgess
• 3
• Peter Danziger
• 4
• Eric Mendelsohn
• 4
1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
2. 2.Division of Science (Mathematics), Grenfell CampusMemorial University of NewfoundlandCorner BrookCanada
3. 3.Department of Mathematical SciencesUniversity of New BrunswickSaint JohnCanada