Skip to main content
SpringerLink
Log in

D-Optimal Matrices of Orders 118, 138, 150, 154 and 174

  • Conference paper
Algebraic Design Theory and Hadamard Matrices

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 133))

Abstract

We construct supplementary difference sets (SDSs) with parameters (59; 28, 22; 21), (69; 31, 27; 24), (75; 36, 29; 28), (77; 34, 31; 27) and (87; 38, 36; 31). These SDSs give D-optimal designs (DO-designs) of two-circulant type of orders 118,138,150,154 and 174. Until now, no DO-designs of orders 138,154 and 174 were known. While a DO-design (not of two-circulant type) of order 150 was constructed previously by Holzmann and Kharaghani, no such design of two-circulant type was known. The smallest undecided order for DO-designs is now 198. We use a novel property of the compression map to speed up some computations.

This paper is in final form and no similar paper has been or is being submitted elsewhere.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Brent, R.P.: Finding many D-optimal designs by randomised decomposition and switching. Australas J. Comb. 55, 15–30 (2013)

    MathSciNet  MATH  Google Scholar 

  2. -​​Djoković, D.Ž.: Cyclic \((v;r,s;\lambda )\) difference families with two base blocks and v ≤ 50. Ann. Comb. 15, 233–254 (2011)

    Google Scholar 

  3. -​​Djoković, D.Ž., Kotsireas, I.S.: New results on D-optimal matrices. J. Comb. Des. 20, 278–289 (2012)

    Google Scholar 

  4. -​​Djoković, D.Ž., Kotsireas, I.S.: Compression of periodic complementary sequences and applications. Des. Codes Crypt. 74, 365–377 (2015)

    Google Scholar 

  5. Ehlich, H.: Determinantenabschätzungen für binäre Matrizen. Math. Z. 83, 123–132 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fletcher, R.J., Koukouvinos, C., Seberry, J.: New skew-Hadamard matrices of order 4.59 and new D-optimal designs of order 2.59. Discret. Math. 286, 251–253 (2004)

    Google Scholar 

  7. Holzmann, W.H., Kharaghani, H.: A D-optimal design of order 150. Discret. Math 190, 265–269 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kharaghani, H., Orrick, W.: D-optimal matrices. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, pp. 296–298, 2nd edn. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  9. Wojtas, W.: On Hadamard’s inequality for the determinants of order non-divisible by 4. Colloq. Math. 12, 73–83 (1964)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to acknowledge generous support by NSERC. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET) and Compute/Calcul Canada. We thank a referee for his suggestions.

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    Dragomir Ž.-Doković

  2. Department of Physics & Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada, N2L 3C5

    Ilias S. Kotsireas

Authors
  1. Dragomir Ž.-Doković
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilias S. Kotsireas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilias S. Kotsireas .

Editor information

Editors and Affiliations

  1. School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, Arizona, USA

    Charles J. Colbourn

Additional information

Dedicated to Hadi Kharaghani on the occasion of his 70th birthday

Appendix: D-Optimal SDSs with v < 100

Appendix: D-Optimal SDSs with v < 100

We list here all D-optimal parameter sets \((v;r,s;\lambda )\) with v∕2 ≥ r ≥ s and v < 100 and for each of them (with two exceptions) we give one DO-design of 2c type by recording the two base blocks of the corresponding SDS. In the two exceptional cases we indicate by a question mark that such designs are not yet known. In particular, this means that DO-designs of order 2v < 200, with v odd, exist for all feasible orders (those for which 2v − 1 is a sum of two squares) except for v = 99. This list will be useful to interested readers as examples of such designs are spread out over many papers in the literature. For the benefit of the readers interested in binary sequences we mention that these SDSs give two binary sequences of length v with PAF +2, i.e., D-optimal matrices.

$$\displaystyle{\begin{array}{ll} (v;r,s;\lambda ) &\mbox{ Base blocks} \\ \hline (3;1,0;0)&\{0\},\emptyset \\ (5;1,1;0) &\{0\},\{0\} \\ (7;3,1;1) &\{0,1,3\},\{0\} \\ (9;3,2;1) &\{0,1,4\},\{0,2\} \\ (13;4,4;2) &\{0,1,4,6\},\{0,1,4,6\} \\ (13;6,3;3) &\{0,1,2,4,7,9\},\{0,1,4\} \\ (15;6,4;3) &\{0,1,2,4,6,9\},\{0,1,4,9\} \\ (19;7,6;4) &\{0,1,2,3,7,11,14\},\{0,2,5,6,9,11\} \\ (21;10,6;6) &\{0,1,2,3,4,6,8,11,12,16\},\{0,1,3,7,10,15\} \\ (23;10,7;6) &\{0,1,3,4,5,7,8,12,14,18\},\{0,1,2,7,9,12,15\} \\ (25;9,9;6) &\{0,1,2,4,7,11,14,15,20\},\{0,1,2,4,6,9,10,12,17\} \\ (27;11,9;7) &\{0,1,3,4,5,9,10,11,13,16,19\},\{0,1,2,4,8,12,15,17,22\} \\ (31;15,10;10) &\{0,1,2,3,5,6,7,11,13,15,16,18,23,24,27\},\{0,2,3,5,6,8,12,19,20,27\} \\ (33;13,12;9) &\{0,1,2,4,5,6,8,10,15,17,20,25,26\}, \\ &\{0,2,3,5,6,9,12,13,17,19,24,25\} \\ (33;15,11;10) &\{0,1,2,3,4,5,8,10,12,13,14,18,19,22,26\}, \\ &\{0,1,2,5,8,11,15,17,20,22,28\} \\ (37;16,13;11)) &\{0,1,2,3,4,7,8,11,13,15,16,18,23,24,27,33\}, \\ &\{0,1,2,4,8,10,13,14,18,20,21,23,32\} \\ (41;16,16;12) &\{0,1,2,3,5,7,8,9,13,18,19,22,23,26,32,34\}, \\ &\{0,1,3,4,6,8,11,13,15,16,17,23,24,27,30,36\} \\ (43;18,16;13) &\{0,1,2,3,4,7,9,11,12,13,16,19,22,24,25,29,30,36\}, \\ &\{0,1,2,4,5,6,9,14,16,17,20,24,26,31,33,39\} \\ (43;21,15;15) &\{0,1,2,3,4,5,6,7,11,12,13,14,17,20,24,25,28,30,31,34,39\}, \\ &\{0,2,3,4,7,9,12,14,16,22,24,30,31,34,39\} \\ (45;21,16;15) &\{0,1,2,3,5,6,8,10,12,13,14,20,21,22,25,28,29,32,34,35,42\}, \\ &\{0,1,2,4,5,6,10,11,14,16,19,22,29,31,33,40\} \\ (49;22,18;16) &\{0,1,2,3,4,5,6,9,11,13,14,19,20,21,23,26,27,30,35,38,40,42\}, \\ &\{0,1,3,4,5,8,9,13,15,19,21,24,26,27,30,37,43,44\} \\ (51;21,20;16) &\{0,2,4,5,6,9,11,12,13,18,19,21,22,26,27,28,30,33,38,39,41\}, \\ &\{0,1,2,4,5,6,9,10,12,14,17,22,24,25,28,31,35,37,41,42\} \\ (55;24,21;18) &\{0,1,2,3,6,8,10,11,13,14,17,19,20,21,24,26,28,29,33,34,40,41,43,44\}, \\ &\{0,1,2,3,6,7,9,11,12,15,19,21,25,29,34,36,37,38,40,45,50\} \\ (57;28,21;21) &\{0,1,2,3,4,5,8,9,10,11,13,16,17,19,21,22,23,24,27,31,34,36,37,38,41, \\ &43,49,50\},\{0,1,3,4,7,9,11,13,15,16,20,25,26,29,30,35,37,40,41,43,48\}\\ \end{array} }$$
$$\displaystyle{\begin{array}{ll} (v;r,s;\lambda ) &\mbox{ Base blocks} \\ \hline (59;28,22;21)&\{0,2,3,5,6,8,9,10,13,15,16,17,19,23,25,26,27,29,30,34,38,39,41,43,44, \\ &45,53,56\},\{0,1,2,3,5,7,8,10,12,13,19,20,22,24,28,32,33,37,38,44,45,51\} \\ (61;25,25;20) &\{0,2,4,7,8,9,10,12,13,18,20,23,24,25,26,29,32,33,34,38,41,44, \\ &48,51,52\},\{0,1,2,4,6,7,8,12,13,14,15,16,19,23,29,30,32,34,36, \\ &39,41,44,49,50,53\} \\ (63;27,25;21) &\{0,1,2,3,5,7,10,11,12,15,18,21,23,24,25,26,31,32,36,37,40,43,44,47, \\ &49,51,53\},\{0,2,4,6,7,8,9,10,11,12,16,20,21,24,27,30,33,38,39,40,45, \\ &47,55,56,60\} \\ (63;29,24;22) &\{0,1,2,3,4,6,7,11,12,13,14,20,21,22,25,26,27,30,33,35,36, \\ &38,39,42,46,48,50,53,57\},\ \{0,1,3,5,7,8,10,11,13,14,16,18, \\ &19,23,30,33,34,35,39,40,48,52,54,56\} \\ (69;31,27;24) &\{0,1,3,4,6,9,10,11,13,14,17,18,20,22,26,28,29,32,33,34,39, \\ &41,43,45,46,48,51,59,60,62,63\},\ \{0,2,3,4,8,9,10,11,12,15, \\ &16,17,21,25,26,32,33,35,36,37,39,41,46,51,54,57,59\} \\ (73;31,30;25) &\{0,1,2,3,4,5,7,9,11,12,16,17,21,22,25,26,30,32,34,37,38,43,44,45,46, \\ &49,52,54,56,59,62\},\ \{0,1,3,4,7,8,9,11,15,16,17,18,21,23,26,27,28,29, \\ &31,33,40,42,46,47,50,53,56,62,63,65\} \\ (73;36,28;28) &\{0,1,3,4,6,7,9,10,12,13,14,15,19,20,21,25,27,28,29,30,31, \\ &36,38,39,41,42,43,46,50,51,54,55,57,59,61,63\}, \\ &\{0,1,4,6,7,11,13,14,18,20,21,22,23,24,26,30,31,35,38,40,48, \\ &51,53,54,58,59,63,65\} \\ (75;36,29;28) &\{0,1,2,3,4,5,8,9,10,12,13,16,17,19,22,25,27,28,30,32,33,34, \\ &38,40,42,44,47,49,51,54,57,60,61,65,66,67\}, \\ &\{0,1,2,4,5,6,7,9,10,12,16,17,21,24,25,30,31,32,35,38,39, \\ &41,43,45,51,52,61,63,64\} \\ (77;34,31;27) &\{0,2,3,4,5,6,9,10,12,14,17,19,22,23,24,26,29,30,32,33,36, \\ &37,39,44,45,48,50,54,58,60,61,63,69,71\}, \\ &\{0,1,2,4,5,6,9,10,12,14,17,20,21,22,23,24,28,29,35,38,40, \\ &44,45,49,51,52,53,54,60,64,65\} \\ (79;37,31;29) &\{0,1,2,3,4,5,6,9,12,13,14,16,18,23,24,30,31,32,33,35,38,39, \\ &40,44,46,48,52,53,56,57,58,61,64,67,69,72,73\}, \\ &\{0,1,3,4,6,8,10,11,13,14,15,17,21,22,27,28,30,32,33,34, \\ &37,44,46,47,50,52,53,55,65,69,75\} \\ (85;36,36;30) &\{0,1,2,3,5,6,8,9,12,13,15,22,24,26,28,29,33,34,35,36,38,40, \\ &41,46,48,49,51,52,56,57,60,66,70,75,78,80\},\ \\ &\{0,2,3,4,5,6,8,11,12,17,18,19,20,21,22,25,29,31,33,36,37, \\ &38,42,43,46,47,55,57,58,61,64,66,68,73,74,81\} \\ (85;39,34;31) &? \\ (87;38,36;31) &\{0,1,2,3,4,5,6,8,10,12,16,18,22,23,24,25,32,33,36,37,38,39, \\ &43,46,47,50,54,56,57,61,62,63,66,69,71,74,80,83\}, \\ &\{0,1,2,5,6,8,10,11,13,17,18,19,21,23,24,26,27,29,33,36,38, \\ &40,43,45,48,49,51,52,53,54,58,65,66,69,77,78\}\\ \end{array} }$$
$$\displaystyle{\begin{array}{ll} (v;r,s;\lambda ) &\mbox{ Base blocks} \\ \hline (91;45,36;36)&\{0,2,4,5,6,8,9,10,11,12,13,14,17,18,19,21,24,25,27,30,33,34, \\ &35,36,37,38,44,45,47,48,51,52,56,57,59,64,66,67,69,71,74,75, \\ &80,84,85\},\ \{0,2,4,6,9,10,11,14,15,16,20,22,24,27,29,31,32, \\ &34,37,38,46,49,50,51,52,53,60,63,64,66,69,70,72,76,77,85\} \\ (93;42,38;34) &\{0,1,4,5,6,7,8,10,15,16,17,19,22,23,26,29,30,32,33,34,35,38, \\ &40,41,45,46,47,49,53,55,60,63,65,66,70,72,73,74,77,80,82,84\},\ \\ &\{0,1,2,3,4,6,8,10,11,12,13,15,16,22,24,26,27,30,31,32,35, \\ &40,44,47,48,49,52,53,54,60,62,64,67,70,73,82,83,88\} \\ (93;45,37;36) &\{0,2,3,4,6,7,8,9,11,13,14,16,18,19,20,22,23,24,26,31,34,35, \\ &37,38,39,41,43,44,47,52,53,55,59,62,63,64,66,69,70,74,75,76, \\ &81,83,86\},\ \{0,1,2,3,6,7,10,11,12,15,16,18,19,20,26,28,29,30, \\ &33,36,40,42,51,52,53,55,57,58,60,65,66,74,77,79,80,85,87\} \\ (97;46,39;37) &\{0,1,2,4,6,7,8,9,11,12,14,15,17,21,22,24,25,26,28,29,34,35, \\ &36,38,44,45,47,49,51,52,53,55,57,63,64,67,68,69,73,76,78,81, \\ &82,83,86,94\},\ \{0,1,2,3,6,8,11,12,16,17,18,20,23,25,27,28,29,30, \\ &36,37,38,41,44,45,49,51,57,60,61,62,63,64,67,69,73,76,83,91,94\} \\ (99;43,42;36) &?\\ \end{array} }$$

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Ž.-Doković, D., Kotsireas, I.S. (2015). D-Optimal Matrices of Orders 118, 138, 150, 154 and 174. In: Colbourn, C. (eds) Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-17729-8_6

Download citation

Publish with us

Policies and ethics

Search

Navigation