# Almost designs and their links with balanced incomplete block designs

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## Abstract

Almost designs (*t*-adesigns) were proposed and discussed by Ding as a certain generalization of combinatorial designs related to almost difference sets. Unlike *t*-designs, it is not clear whether *t*-adesigns need also be \((t-1)\)-designs or \((t-1)\)-adesigns. In this paper we discuss a particular class of 3-adesigns, i.e., 3-adesigns coming from certain strongly regular graphs and tournaments, and find that these are also 2-designs. We construct several classes of these, and discuss some of the restrictions on the parameters of such a class. We also construct several new classes of 2-adesigns, and discuss some of their properties as well.

## Keywords

Almost difference set Difference set Strongly regular graph*t*-adesign Tournament

## Mathematics Subject Classification

05B05 05E30## Notes

### Acknowledgements

The authors are very grateful to the three anonymous referees and to the Coordinating Editor for all of their detailed comments that greatly improved the quality and the presentation of this paper.

## References

- 1.Assmus E.F., Key J.D.: Designs and Their Codes, vol. 103. Cambridge University Press, Cambridge (1992).CrossRefzbMATHGoogle Scholar
- 2.Behbahani M.: On strongly regular graphs. PhD thesis, Concordia University (2009).Google Scholar
- 3.Beth T., Jungnickel D., Lenz H.: Design Theory, vol. I, 2nd edn. Cambridge University Press, Cambridge (1999).CrossRefzbMATHGoogle Scholar
- 4.Bose R.C.: On the construction of balanced incomplete block designs. Ann. Eugen.
**9**, 353–399 (1939).MathSciNetCrossRefGoogle Scholar - 5.Cameron P.J., van Lint J.H.: Designs, Graphs, Codes and Their Links. London Mathematical Society Student Texts. LMSSTCambridge University Press, Cambridge (1991).CrossRefzbMATHGoogle Scholar
- 6.Cusick T.W., Ding C., Renvall A.: Stream Ciphers and Number Theory, vol. 55. North-Holland Mathematical LibraryNorth-Holland Publishing Co., Amsterdam (1998).CrossRefzbMATHGoogle Scholar
- 7.Davis J.A.: Almost difference sets and reversible divisible difference sets. Arch. Math. (Basel)
**59**(6), 595–602 (1992).MathSciNetCrossRefzbMATHGoogle Scholar - 8.Dembowski P.: Finite Geometries. Springer, New York (1968).CrossRefzbMATHGoogle Scholar
- 9.Ding C.: The differential cryptanalysis and design of natural stream ciphers. In: Fast Software Encryption, pp. 101–115. Springer, Berlin (1994).Google Scholar
- 10.Ding C.: Codes from Difference Sets. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2015).Google Scholar
- 11.Ding C., Yin J.: Constructions of almost difference families. Discret. Math.
**308**(21), 4941–4954 (2008).MathSciNetCrossRefzbMATHGoogle Scholar - 12.Ding C., Helleseth T., Martinsen H.: New families of binary sequences with optimal three-level autocorrelation. IEEE Trans. Inform. Theory
**47**(1), 428–433 (2001).MathSciNetCrossRefzbMATHGoogle Scholar - 13.Ding C., Pott A., Wang Q.: Skew Hadamard difference sets from Dickson polynomials of order 7. J. Comb. Designs
**23**(10), 436–461 (2015).MathSciNetCrossRefzbMATHGoogle Scholar - 14.Fisher R.A.: An examination of the different possible solutions of a problem in incomplete blocks. Ann. Eugen.
**10**, 52–75 (1940).MathSciNetCrossRefGoogle Scholar - 15.Goethals J., Seidel J.: Orthogonal matrices with zero diagonal. Can. J. Math.
**19**, 1001–1010 (1967).MathSciNetCrossRefzbMATHGoogle Scholar - 16.Golomb S.W., Gong G.: Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar. Cambridge University Press, Cambridge (2005).CrossRefzbMATHGoogle Scholar
- 17.Hirschfeld J.W.P.: Projective Geometries Over Finite Feilds, 2nd edn. Oxford University Press, Oxford (1998).zbMATHGoogle Scholar
- 18.Horsley D.: Generalizing Fisher’s inequality to coverings and packings. Combinatorica
**37**(4), 673–696 (2017).MathSciNetCrossRefzbMATHGoogle Scholar - 19.Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRefzbMATHGoogle Scholar
- 20.Johnson S.: A new upper bound for error-correcting codes. IRE Trans. Inform. Theory
**8**, 203–207 (1962).MathSciNetCrossRefzbMATHGoogle Scholar - 21.Liu H., Ding C.: Infinite families of 2-designs from ${GA}_{1}(q)$ actions. arXiv:1707.02003v1 (2017).Google Scholar
- 22.Michel J.: New partial geometric difference sets and partial geometric difference families. Acta Math. Sin.
**33**(5), 591–606 (2017).MathSciNetCrossRefzbMATHGoogle Scholar - 23.Michel J., Ding B.: A generalization of combinatorial designs and related codes. Des. Codes Cryptogr.
**82**(3), 511–529 (2016).MathSciNetCrossRefzbMATHGoogle Scholar - 24.Neumaier A.: $t\frac{1}{2}$-designs. J. Comb. Theory (A)
**78**, 226–248 (1980).CrossRefzbMATHGoogle Scholar - 25.Pasechnik D.: Skew-symmetric association schemes with two classes and strongly regular graphs of type $l_{2n-1}(4n-1)$. Acta Appl. Math.
**29**(1), 129–138 (1992).MathSciNetCrossRefzbMATHGoogle Scholar - 26.Reid K., Brown E.: Doubly regular tournaments are equivalent to skew hadamard matrices. J. Comb. Theory (A)
**12**(3), 332–338 (1972).MathSciNetCrossRefzbMATHGoogle Scholar - 27.Schonheim J.: On coverings. Pac. J. Math.
**14**, 1405–1411 (1964).CrossRefzbMATHGoogle Scholar - 28.Stinson D.R.: Combinatorial Designs: Constructions and Analysis. Springer, New York (2003).zbMATHGoogle Scholar
- 29.Storer T.: Cyclotomy and Difference Sets, pp. 65–72. Markham, Chicago (1967).zbMATHGoogle Scholar
- 30.Wang X., Wu D.: The existence of almost difference families. J. Stat. Plan. Inference
**139**(12), 4200–4205 (2009).MathSciNetCrossRefzbMATHGoogle Scholar - 31.Wilson R.M.: Cyclotomy and difference families in elementary abelian groups. J. Number Theory
**4**, 17–47 (1972).MathSciNetCrossRefzbMATHGoogle Scholar - 32.Zhang Y., Lei J.G., Zhang S.P.: A new family of almost difference sets and some necessary conditions. IEEE Trans. Inform. Theory
**52**(5), 2052–2061 (2006).MathSciNetCrossRefzbMATHGoogle Scholar