New 2-designs over finite fields from derived and residual designs

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February 2019, 13(1): 165-170. doi: 10.3934/amc.2019010

New 2-designs over finite fields from derived and residual designs

Michael Braun ^1,, , Michael Kiermaier ^2,3, and Reinhard Laue ^2,3,

1.	Faculty of Computer Science, University of Applied Sciences Darmstadt, Schoefferstr. 8b, 64295 Darmstadt, Germany
2.	Mathematisches Institut, Universität Bayreuth, 95447 Bayreuth, Germany
3.	Institut für Informatik, Universität Bayreuth, 95447 Bayreuth, Germany

* Corresponding author: Michael Braun

Received June 2018 Published December 2018

Based on the existence of designs for the derived and residual parameters of admissible parameter sets of designs over finite fields we obtain a new infinite series of designs over finite fields for arbitrary prime powers $q$ with parameters $2\text{-}(8,4,\frac{(q^6-1)(q^3-1)}{(q^2-1)(q-1)};q)$ as well as designs with parameters $2\text{-}(10,4,85λ;2)$, $2\text{-}(10,5,765λ;2)$, $2\text{-}(11,5,6205λ;2)$, $2\text{-}(11,5,502605λ;2)$, and $2\text{-}(12,6,423181λ;2)$ for $λ = 7,12,19,21,22,24,31,36,42,43,48,49,55,60,63$.

Keywords: q-analog, designs over finite fields, residual design, derived design.

Mathematics Subject Classification: Primary: 51E20; Secondary: 05B05, 05B25.

Citation: Michael Braun, Michael Kiermaier, Reinhard Laue. New 2-designs over finite fields from derived and residual designs. Advances in Mathematics of Communications, 2019, 13 (1) : 165-170. doi: 10.3934/amc.2019010

References:

[1]	M. Braun, Designs over the binary field from the complete monomial group, Australas. J. Combin., 67 (2017), 470-475.
[2]	M. Braun, Some new designs over finite fields, Bayreuth. Math. Schr., 74 (2005), 58-68.
[3]	M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wassermann, Existence of q-analogs of steiner systems, Forum Math. Pi, 4 (2016), e7, 14pp. doi: 10.1017/fmp.2016.5.
[4]	M. Braun, A. Kerber and R. Laue, Systematic construction of q-analogs of designs, Des. Codes Cryptogr., 34 (2005), 55-70. doi: 10.1007/s10623-003-4194-z.
[5]	M. Braun, M. Kiermaier, A. Kohnert and R. Laue, Large sets of subspace designs, J. Combin. Theory Ser. A, 147 (2017), 155-185. doi: 10.1016/j.jcta.2016.11.004.
[6]	M. Braun, A. Kohnert, P. R. J. Östergård and A. Wassermann, Large sets of t-designs over finite fields, J. Combin. Theory Ser. A, 124 (2014), 195-202. doi: 10.1016/j.jcta.2014.01.008.
[7]	S. Braun, Construction of q-analogs of combinatorial designs, ALCOMA 2010, Thurnau, 2010.
[8]	M. Braun, M. Kiermaier and A. Wassermann, q-analogs of designs: subspace designs, In M. Greferath, M.O. Pavčević, N. Silberstein, and M.A. Vázquez-Castro, editors, Network Coding and Subspace Designs, Springer International Publishing, (2018), 171-211.
[9]	M. Braun, M. Kiermaier and A. Wassermann, Computational methods in subspace designs, In M. Greferath, M.O. Pavčević, N. Silberstein, and M.A. Vázquez-Castro, editors, Network Coding and Subspace Designs, Springer International Publishing, (2018), 213-244.
[10]	T. Itoh, A new family of 2-designs over $ GF(q)$ admitting $ SL_m(q^l)$, Geom. Dedicata, 69 (1998), 261-286. doi: 10.1023/A:1005057610394.
[11]	M. Kiermaier and R. Laue, Derived and residual subspace designs, Adv. Math. Commun., 9 (2015), 105-115. doi: 10.3934/amc.2015.9.105.
[12]	M. Kiermaier, R. Laue and A. Wassermann, A new series of large sets of subspace designs over the binary field, Des. Codes Cryptogr., 86 (2018), 251-268. doi: 10.1007/s10623-017-0349-1.
[13]	E. Kramer and D. Mesner, t-designs on hypergraphs, Discrete Math., 15 (1976), 263-296.
[14]	M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small 2-designs over $ GF(q)$, J. Combin. Des., 3 (1995), 61-77. doi: 10.1002/jcd.3180030108.
[15]	H. Suzuki, 2-designs over $ GF(2^m)$, Graph. Combinator., 6 (1990), 293-296. doi: 10.1007/BF01787580.
[16]	H. Suzuki, On the inequalities of t-designs over a finite field, Eur. J. Comb., 11 (1990), 601-607. doi: 10.1016/S0195-6698(13)80045-5.
[17]	H. Suzuki, 2-designs over $ GF(q)$, Graph. Combinator., 8 (1992), 381-389. doi: 10.1007/BF02351594.
[18]	S. Thomas, Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242. doi: 10.1007/BF00150939.
[19]	A. Wassermann, Finding simple t-designs with enumeration techniques, J. Combin. Des., 6 (1998), 79-90. doi: 10.1002/(SICI)1520-6610(1998)6:2<79::AID-JCD1>3.0.CO;2-S.

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References:

[1]	M. Braun, Designs over the binary field from the complete monomial group, Australas. J. Combin., 67 (2017), 470-475.
[2]	M. Braun, Some new designs over finite fields, Bayreuth. Math. Schr., 74 (2005), 58-68.
[3]	M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wassermann, Existence of q-analogs of steiner systems, Forum Math. Pi, 4 (2016), e7, 14pp. doi: 10.1017/fmp.2016.5.
[4]	M. Braun, A. Kerber and R. Laue, Systematic construction of q-analogs of designs, Des. Codes Cryptogr., 34 (2005), 55-70. doi: 10.1007/s10623-003-4194-z.
[5]	M. Braun, M. Kiermaier, A. Kohnert and R. Laue, Large sets of subspace designs, J. Combin. Theory Ser. A, 147 (2017), 155-185. doi: 10.1016/j.jcta.2016.11.004.
[6]	M. Braun, A. Kohnert, P. R. J. Östergård and A. Wassermann, Large sets of t-designs over finite fields, J. Combin. Theory Ser. A, 124 (2014), 195-202. doi: 10.1016/j.jcta.2014.01.008.
[7]	S. Braun, Construction of q-analogs of combinatorial designs, ALCOMA 2010, Thurnau, 2010.
[8]	M. Braun, M. Kiermaier and A. Wassermann, q-analogs of designs: subspace designs, In M. Greferath, M.O. Pavčević, N. Silberstein, and M.A. Vázquez-Castro, editors, Network Coding and Subspace Designs, Springer International Publishing, (2018), 171-211.
[9]	M. Braun, M. Kiermaier and A. Wassermann, Computational methods in subspace designs, In M. Greferath, M.O. Pavčević, N. Silberstein, and M.A. Vázquez-Castro, editors, Network Coding and Subspace Designs, Springer International Publishing, (2018), 213-244.
[10]	T. Itoh, A new family of 2-designs over $ GF(q)$ admitting $ SL_m(q^l)$, Geom. Dedicata, 69 (1998), 261-286. doi: 10.1023/A:1005057610394.
[11]	M. Kiermaier and R. Laue, Derived and residual subspace designs, Adv. Math. Commun., 9 (2015), 105-115. doi: 10.3934/amc.2015.9.105.
[12]	M. Kiermaier, R. Laue and A. Wassermann, A new series of large sets of subspace designs over the binary field, Des. Codes Cryptogr., 86 (2018), 251-268. doi: 10.1007/s10623-017-0349-1.
[13]	E. Kramer and D. Mesner, t-designs on hypergraphs, Discrete Math., 15 (1976), 263-296.
[14]	M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small 2-designs over $ GF(q)$, J. Combin. Des., 3 (1995), 61-77. doi: 10.1002/jcd.3180030108.
[15]	H. Suzuki, 2-designs over $ GF(2^m)$, Graph. Combinator., 6 (1990), 293-296. doi: 10.1007/BF01787580.
[16]	H. Suzuki, On the inequalities of t-designs over a finite field, Eur. J. Comb., 11 (1990), 601-607. doi: 10.1016/S0195-6698(13)80045-5.
[17]	H. Suzuki, 2-designs over $ GF(q)$, Graph. Combinator., 8 (1992), 381-389. doi: 10.1007/BF02351594.
[18]	S. Thomas, Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242. doi: 10.1007/BF00150939.
[19]	A. Wassermann, Finding simple t-designs with enumeration techniques, J. Combin. Des., 6 (1998), 79-90. doi: 10.1002/(SICI)1520-6610(1998)6:2<79::AID-JCD1>3.0.CO;2-S.

Figure 1. Connections of parameters

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Table 1. $2\text{-}(9,k,\lambda;2)$ designs for $k\in\{3,4\}$

$t\text{-}(n,k,\lambda;q)$	$G$	$\|A_{t,k}^G\|$	$\lambda$
$2\text{-}(9,3,\lambda;2)$	$N(3,2^3)$	$31\!\times\!529$	$21$, $22$, $42$, $43$, $63$
	$N(8,2)\!\times\! 1$	$28\!\times\!408$	$7$, $12$, $19$, $24$, $31$, $36$, $43$, $48$,
			$55$, $60$
	$M(3,2^3)$	$40\!\times\!460$	$49$
$2\text{-}(9,4,\lambda;2)$	$N(9,2)$	$11\!\times\!725$	$21$, $63$, $84$, $126$, $147$, $189$,
			$210$, $252$, $273$, $315$, $336$, $378$,
			$399$, $441$, $462$, $504$, $525$, $567$,
			$588$, $630$, $651$, $693$, $714$, $756$,
			$777$, $819$, $840$, $882$, $903$, $945$,
			$966$, $1008$, $1029$, $1071$, $1092$,
			$1134$, $1155$, $1197$, $1218$, $1260$,
			$1281$, $1323$

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$t\text{-}(n,k,\lambda;q)$	$G$	$\|A_{t,k}^G\|$	$\lambda$
$2\text{-}(9,3,\lambda;2)$	$N(3,2^3)$	$31\!\times\!529$	$21$, $22$, $42$, $43$, $63$
	$N(8,2)\!\times\! 1$	$28\!\times\!408$	$7$, $12$, $19$, $24$, $31$, $36$, $43$, $48$,
			$55$, $60$
	$M(3,2^3)$	$40\!\times\!460$	$49$
$2\text{-}(9,4,\lambda;2)$	$N(9,2)$	$11\!\times\!725$	$21$, $63$, $84$, $126$, $147$, $189$,
			$210$, $252$, $273$, $315$, $336$, $378$,
			$399$, $441$, $462$, $504$, $525$, $567$,
			$588$, $630$, $651$, $693$, $714$, $756$,
			$777$, $819$, $840$, $882$, $903$, $945$,
			$966$, $1008$, $1029$, $1071$, $1092$,
			$1134$, $1155$, $1197$, $1218$, $1260$,
			$1281$, $1323$

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