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

Previous Article
Cyclic DNA codes over $ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$
AMC Home
This Issue
Next Article
Double circulant self-dual and LCD codes over Galois rings

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.

show all references

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

Figure Options

Download full-size image

Download as PowerPoint slide

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

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

Table Options

Download as excel

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

[1]	Michael Kiermaier, Reinhard Laue. Derived and residual subspace designs. Advances in Mathematics of Communications, 2015, 9 (1) : 105-115. doi: 10.3934/amc.2015.9.105
[2]	Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475
[3]	Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107-120. doi: 10.3934/amc.2012.6.107
[4]	Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459
[5]	Shengtian Yang, Thomas Honold. Good random matrices over finite fields. Advances in Mathematics of Communications, 2012, 6 (2) : 203-227. doi: 10.3934/amc.2012.6.203
[6]	Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022
[7]	Joseph H. Silverman. Local-global aspects of (hyper)elliptic curves over (in)finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 101-114. doi: 10.3934/amc.2010.4.101
[8]	Liren Lin, Hongwei Liu, Bocong Chen. Existence conditions for self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (1) : 1-7. doi: 10.3934/amc.2015.9.1
[9]	Uwe Helmke, Jens Jordan, Julia Lieb. Probability estimates for reachability of linear systems defined over finite fields. Advances in Mathematics of Communications, 2016, 10 (1) : 63-78. doi: 10.3934/amc.2016.10.63
[10]	David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35
[11]	Amin Sakzad, Mohammad-Reza Sadeghi, Daniel Panario. Cycle structure of permutation functions over finite fields and their applications. Advances in Mathematics of Communications, 2012, 6 (3) : 347-361. doi: 10.3934/amc.2012.6.347
[12]	Fatma-Zohra Benahmed, Kenza Guenda, Aicha Batoul, Thomas Aaron Gulliver. Some new constructions of isodual and LCD codes over finite fields. Advances in Mathematics of Communications, 2019, 13 (2) : 281-296. doi: 10.3934/amc.2019019
[13]	Nian Li, Qiaoyu Hu. A conjecture on permutation trinomials over finite fields of characteristic two. Advances in Mathematics of Communications, 2019, 13 (3) : 505-512. doi: 10.3934/amc.2019031
[14]	Hai Huyen Dam, Wing-Kuen Ling. Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank. Journal of Industrial & Management Optimization, 2019, 15 (1) : 97-112. doi: 10.3934/jimo.2018034
[15]	Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161
[16]	Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437
[17]	Ekkasit Sangwisut, Somphong Jitman, Patanee Udomkavanich. Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields. Advances in Mathematics of Communications, 2017, 11 (3) : 595-613. doi: 10.3934/amc.2017045
[18]	David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131
[19]	Marko Budišić, Stefan Siegmund, Doan Thai Son, Igor Mezić. Mesochronic classification of trajectories in incompressible 3D vector fields over finite times. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 923-958. doi: 10.3934/dcdss.2016035
[20]	Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010

2017 Impact Factor: 0.564

Tools

Metrics

Tools

Article outline

Figures and Tables

2017 Impact Factor: 0.564

Tools

Figures and Tables

2017 Impact Factor: 0.564

American Institute of Mathematical Sciences