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

[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] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
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 |
[17] |
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 |
[18] |
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 |
[19] |
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 |
[20] |
Josep M. Miret, Jordi Pujolàs, Anna Rio. Explicit 2-power torsion of genus 2 curves over finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 155-168. doi: 10.3934/amc.2010.4.155 |
2017 Impact Factor: 0.564
Tools
Metrics
Other articles
by authors
[Back to Top]