# American Institute of Mathematical Sciences

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

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

 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 $λ = 7,12,19,21,22,24,31,36,42,43,48,49,55,60,63$.

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. Google Scholar [2] M. Braun, Some new designs over finite fields, Bayreuth. Math. Schr., 74 (2005), 58-68. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] S. Braun, Construction of q-analogs of combinatorial designs, ALCOMA 2010, Thurnau, 2010.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] E. Kramer and D. Mesner, t-designs on hypergraphs, Discrete Math., 15 (1976), 263-296.Google Scholar [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. Google Scholar [15] H. Suzuki, 2-designs over $GF(2^m)$, Graph. Combinator., 6 (1990), 293-296. doi: 10.1007/BF01787580. Google Scholar [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. Google Scholar [17] H. Suzuki, 2-designs over $GF(q)$, Graph. Combinator., 8 (1992), 381-389. doi: 10.1007/BF02351594. Google Scholar [18] S. Thomas, Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242. doi: 10.1007/BF00150939. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] M. Braun, Designs over the binary field from the complete monomial group, Australas. J. Combin., 67 (2017), 470-475. Google Scholar [2] M. Braun, Some new designs over finite fields, Bayreuth. Math. Schr., 74 (2005), 58-68. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] S. Braun, Construction of q-analogs of combinatorial designs, ALCOMA 2010, Thurnau, 2010.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] E. Kramer and D. Mesner, t-designs on hypergraphs, Discrete Math., 15 (1976), 263-296.Google Scholar [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. Google Scholar [15] H. Suzuki, 2-designs over $GF(2^m)$, Graph. Combinator., 6 (1990), 293-296. doi: 10.1007/BF01787580. Google Scholar [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. Google Scholar [17] H. Suzuki, 2-designs over $GF(q)$, Graph. Combinator., 8 (1992), 381-389. doi: 10.1007/BF02351594. Google Scholar [18] S. Thomas, Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242. doi: 10.1007/BF00150939. Google Scholar [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. Google Scholar
Connections of parameters
$2\text{-}(9,k,\lambda;2)$ designs for $k\in\{3,4\}$
 $t\text{-}(n,k,\lambda;q)$ $G$ $|A_{t,k}^G|$ $\lambda$ $2\text{-}(9,3,\lambda;2)$ $N(3,2^3)$ $31\!\times\!529$ $21$, $22$, $42$, $43$, $63$ $N(8,2)\!\times\! 1$ $28\!\times\!408$ $7$, $12$, $19$, $24$, $31$, $36$, $43$, $48$, $55$, $60$ $M(3,2^3)$ $40\!\times\!460$ $49$ $2\text{-}(9,4,\lambda;2)$ $N(9,2)$ $11\!\times\!725$ $21$, $63$, $84$, $126$, $147$, $189$, $210$, $252$, $273$, $315$, $336$, $378$, $399$, $441$, $462$, $504$, $525$, $567$, $588$, $630$, $651$, $693$, $714$, $756$, $777$, $819$, $840$, $882$, $903$, $945$, $966$, $1008$, $1029$, $1071$, $1092$, $1134$, $1155$, $1197$, $1218$, $1260$, $1281$, $1323$
 $t\text{-}(n,k,\lambda;q)$ $G$ $|A_{t,k}^G|$ $\lambda$ $2\text{-}(9,3,\lambda;2)$ $N(3,2^3)$ $31\!\times\!529$ $21$, $22$, $42$, $43$, $63$ $N(8,2)\!\times\! 1$ $28\!\times\!408$ $7$, $12$, $19$, $24$, $31$, $36$, $43$, $48$, $55$, $60$ $M(3,2^3)$ $40\!\times\!460$ $49$ $2\text{-}(9,4,\lambda;2)$ $N(9,2)$ $11\!\times\!725$ $21$, $63$, $84$, $126$, $147$, $189$, $210$, $252$, $273$, $315$, $336$, $378$, $399$, $441$, $462$, $504$, $525$, $567$, $588$, $630$, $651$, $693$, $714$, $756$, $777$, $819$, $840$, $882$, $903$, $945$, $966$, $1008$, $1029$, $1071$, $1092$, $1134$, $1155$, $1197$, $1218$, $1260$, $1281$, $1323$
 [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] Sabyasachi Dey, Tapabrata Roy, Santanu Sarkar. Revisiting design principles of Salsa and ChaCha. Advances in Mathematics of Communications, 2019, 13 (4) : 689-704. doi: 10.3934/amc.2019041 [17] 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 [18] 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 [19] 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 [20] 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

2018 Impact Factor: 0.879