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. BraunA. 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. BraunM. KiermaierA. 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. BraunA. KohnertP. 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. KiermaierR. 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. MiyakawaA. 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. BraunA. 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. BraunM. KiermaierA. 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. BraunA. KohnertP. 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. KiermaierR. 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. MiyakawaA. 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

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