On the resolutions of cyclic Steiner triple systems with small parameters

Year 2016, , 201 - 208, 09.08.2016

Svetlana Topalova

https://doi.org/10.13069/jacodesmath.47635

Abstract

The paper presents useful invariants of resolutions of cyclic $STS(v)$ with $v\le 39$, namely of all resolutions of cyclic $STS(15)$, $STS(21)$ and $STS(27)$, of the resolutions with nontrivial automorphisms of cyclic $STS(33)$ and of resolutions with automorphisms of order $13$  of cyclic $STS(39)$.

References

• T. Baicheva, S. Topalova, Classification results for (v, k, 1) cyclic difference families with small parameters, Mathematics of Distances and Applications. M. Deza, M. Petitjean, K. Markov (Eds.), in International book series: Information Science and Computing, 25 (2012), 24–30.
• T. Beth, D. Jungnickel, H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1993.
• C. J. Colbourn, J. H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 2007.
• C. J. Colbourn, J. H. Dinitz, D. R. Stinson, Applications of Combinatorial Designs to Communications, Cryptography, and Networking, Surveys in Combinatorics, 1999, Edited by J. D. Lamb and D. A. Preece, London Mathematical Society Lecture Note Series 267 (1999), 37–100.
• C. J. Colbourn, S. S. Magliveras, R. A. Mathon, Transitive Steiner and Kirkman triple systems of order 27, Math. Comp. 58(197) (1992) 441–449.
• M. J. Colbourn, R. A. Mathon, On cyclic Steiner 2-designs. Topics on Steiner systems, Ann. Discret Math. 7 (1980) 215–253.
• M. Genma, M. Mishima, M. Jimbo, Cyclic resolvability of cyclic Steiner 2-designs, J. Combin. Des. 5(3) (1997) 177–187.
• A. Gruner, M. Huber, New combinatorial construction techniques for low-density parity-check codes and systematic repeat-accumulate codes, IEEE Trans. Commun. 60(9) (2012) 2387–2395.
• M. Huber, Combinatorial designs for authentication and secrecy codes, Found. Trends Commun. Inf. Theory 5(6) (2008) 581–675.
• N. L. Johnson, Two-transitive parallelisms, Des. Codes Cryptogr. 22(2) (2001) 179–189.
• S. J. Johnson, S. R. Weller, Resolvable 2-designs for regular low-density parity-check codes, IEEE Trans. Commun. 51(9) (2003) 1413–1419.
• S. Kageyama, A survey of resolvable solutions of balanced incomplete block designs, Int. Stat. Rev. 40(3) (1972) 269–273.
• P.Kaski, P.Östergård, Classification Algorithms for Codes and Designs, Springer, Berlin, 2006.
• C. Lam, Y. Miao, On cyclically resolvable cyclic steiner 2-designs, J. Combin. Theory Ser. A 85(2) (1999) 194–207.
• C. W. H. Lam, Y. Miao, Cyclically resolvable cyclic Steiner triple systems of order 21 and 39, Discrete Math. 219(1-3) (2000) 173–185.
• R. A. Mathon, K. T. Phelps, A. Rosa, A class of Steiner triple systems of order 21 and associated Kirkman systems, Math. Comp. 37(155) (1981) 209–222.
• M. Mishima, M. Jimbo, Some types of cyclically resolvable cyclic Steiner 2-designs, Congr. Numer. 123 (1997) 193–203.
• M. Mishima, M. Jimbo, Recursive constructions for cyclic quasiframes and cyclically resolvable cyclic Steiner 2-designs, Discrete Math. 211(1-3) (2000) 135–152.
• N. V. Semakov, V. A. Zinoviev, Equidistant q-ary codes with maximal distance and resolvable balanced incomplete block designs, Probl. Peredachi. Inf. 4(2) (1968) 3–10.
• T. Etzion, N. Silberstein, Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory 59(2) (2013) 1004–1017.
• D. Stinson, Combinatorial Designs: Constructions and Analysis, Springer, New York, 2004.
• V. D. Tonchev, Combinatorial Configurations, Longman Scientific and Technical, New York, 1988.
• V. D. Tonchev, Steiner systems for two-stage disjunctive testing, J. Comb. Optim. 15(1) (2008) 1–6.
• V. D. Tonchev, S. A. Vanstone, On Kirkman triple systems of order 33, Discrete Math. 106/107 (1992) 493–496.