doi: 10.3934/amc.2022016
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On some codes from rank 3 primitive actions of the simple Chevalley group $ G_2(q) $

Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0028, South Africa

* Corresponding author

Dedicated to the Memory of Kay Magaard

Received  August 2021 Revised  February 2022 Early access March 2022

Fund Project: The second author acknowledges research support by the National Research Foundation of South Africa (Grant Number 120846)

Let $ G_2(q) $ be a Chevalley group of type $ G_2 $ over a finite field $ \mathbb{F}_q $. Considering the $ G_2(q) $-primitive action of rank $ 3 $ on the set of $ \frac{q^3(q^3-1)}{2} $ hyperplanes of type $ O_{6}^{-}(q) $ in the $ 7 $-dimensional orthogonal space $ {{\rm{PG}}}(7, q) $, we study the designs, codes, and some related geometric structures. We obtained the main parameters of the codes, the full automorphism groups of these structures, and geometric descriptions of the classes of minimum weight codewords.

Citation: Tung Le, Bernardo G. Rodrigues. On some codes from rank 3 primitive actions of the simple Chevalley group $ G_2(q) $. Advances in Mathematics of Communications, doi: 10.3934/amc.2022016
References:
[1]

E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, Vol. 103. Cambridge: Cambridge University Press, 1992. doi: 10.1017/CBO9781316529836.

[2]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.

[3]

S. BraićA. GolemacJ. Mandić and T. Vučičić, Primitive symmetric designs with up to $2500$ points, J. Combin Designs, 19 (2011), 463-474.  doi: 10.1002/jcd.20291.

[4]

T. Breuer, Decomposition Matrices, Available at:, The Modular Atlas homepage. http://www.math.rwth-aachen.de/MOC/decomposition. 1999.

[5]

A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-1939-6.

[6]

A. E. Brouwer and C. A. van Eijl, On the $p$-rank of the adjacency matrices of strongly regular graphs, J. Algebraic Combin., 1 (1992), 329-346.  doi: 10.1023/A:1022438616684.

[7]

A. E. Brouwer and H. van Maldeghem, Strongly Regular Graphs, Encyclopedia of Mathematics and its Applications, 182. Cambridge University Press, Cambridge, 2022. https://homepages.cwi.nl/~aeb/math/srg/rk3/srgw.pdf. doi: 10.1017/9781009057226.

[8]

F. Buekenhout and H. Van Maldeghen, A characterization of some rank 2 incidence geometries by their automorphism group, Mitt. Math. Sem. Giessen, 218 (1994), 1-70. 

[9]

A. R. Calderbank and J. -M. Goethals, Three weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. 

[10]

P. J. Cameron, Permutation Groups, London Mathematical Society Student Texts, 45, Cambridge: Cambridge University Press, 1999. doi: 10.1017/CBO9780511623677.

[11]

P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links, London Mathematical Society Student Texts, 22, Cambridge: Cambridge University Press, 1991. doi: 10.1017/CBO9780511623714.

[12]

N. ChigiraM. Harada and M. Kitazume, Permutation groups and binary self-orthogonal codes, J. Algebra, 309 (2007), 610-621.  doi: 10.1016/j.jalgebra.2006.06.001.

[13] J. H. ConwayR. T. CurtisS. P. NortonR. A. Parker and R. A. Wilson, An Atlas of Finite Groups, Oxford University Press, Oxford, 1985. 
[14]

A. Cossidente and O. H. King, On the geometry of the exceptional group $G_2(q), $ $q$ even, Des. Codes Cryptogr., 47 (2008), 145-157.  doi: 10.1007/s10623-007-9107-0.

[15]

D. CrnkovićV. Mikulić and B. G. Rodrigues, Some strongly regular graphs and self-orthogonal codes from the unitary group ${U}_4(3)$, Glas. Mat. Ser. Ⅲ, 45 (2010), 307-323.  doi: 10.3336/gm.45.2.02.

[16]

H. J. CouttsM. Quick and C. M. Roney-Dougal, The primitive permutation groups of degree less than $4096$, Comm. Algebra, 39 (2011), 3526-3546.  doi: 10.1080/00927872.2010.515521.

[17]

P. Dankelmann, J. D. Key and B. G. Rodrigues, A characterization of graphs by codes from their incidence matrices, Electron. J. of Combin., 20 (2013), Paper 18, 22 pp. doi: 10.37236/2770.

[18]

P. DankelmannJ. D. Key and B. G. Rodrigues, Codes from incidence matrices of graphs, Des. Codes Cryptogr., 68 (2013), 373-393.  doi: 10.1007/s10623-011-9594-x.

[19]

U. Dempwolff, Primitive rank-3 groups on symmetric designs, Des. Codes Crypt., 22 (2001), 191-207.  doi: 10.1023/A:1008373207617.

[20]

R. H. Dye, Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra, 59 (1979), 202-221.  doi: 10.1016/0021-8693(79)90157-1.

[21]

W. FishJ. D. Key and E. Mwambene, Binary codes from reexive uniform subset graphs on 3-sets, Adv. Math. Commun., 9 (2015), 211-232.  doi: 10.3934/amc.2015.9.211.

[22]

W. FishJ. D. Key and E. Mwambene, Ternary codes from some reflexive uniform subset graphs, Appl. Algebra Engrg. Comm. Comput., 25 (2014), 363-382.  doi: 10.1007/s00200-014-0233-4.

[23]

A. Günther and G. Nebe, Automorphisms of doubly even self-dual codes, Bull. London Math. Soc., 41 (2009), 769-778.  doi: 10.1112/blms/bdp026.

[24]

C. Jansen, K. Lux, R. Parker and R. Wilson, An Atlas of Brauer Characters, LMS Monographs New Series 11. Oxford: Oxford Science Publications, Clarendon Press, 1995.

[25]

D. Jungnickel and V. D. Tonchev, Exponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankin bound, Des. Codes and Cryptogr., 1 (1991), 247-253.  doi: 10.1007/BF00123764.

[26]

J. D. Key and J. Moori, Codes, designs and graphs from the Janko groups ${J}_1$ and ${J}_2$, J. Combin. Math and Combin. Comput., 40 (2002), 143-159. 

[27]

W. Knapp and H.-J. Schaeffer, On the codes related to the Higman-Sims graph, Electron. J. of Combin., 22 (2015), Paper 1.19, 58 pp.

[28]

T. Le and B. G. Rodrigues, Magma computations for $G_2(q)$ of rank 3, Available at:, https://bgrodrigues.weebly.com/uploads/1/2/8/4/12846738/g2qrank3.txt

[29]

M. W. LiebeckC. E. Praeger and J. Saxl, On the $2$-closures of finite permutation groups, J. Lond. Math. Soc., 37 (1987), 241-252.  doi: 10.1112/jlms/s2-37.2.241.

[30]

K. Lux and H. Pahlings, Representations of Groups: A Computational Approach, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010.

[31]

A. A. Makhnev, $GQ(4, 2)$-extensions, the strongly regular case, Math. Notes, 68, 97–102.

[32]

J. Moori, Finite groups, designs and codes, Information security, coding theory and related combinatorics, 202–230, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS, Amsterdam, 2011.

[33]

J. Moori and B. G. Rodrigues, Some designs and codes invariant under the simple group Co2, J. of Algebra, 316 (2007), 649-661.  doi: 10.1016/j.jalgebra.2007.02.004.

[34]

J. Moori and B. G. Rodrigues, Some designs and binary codes preserved by the simple group Ru of Rudvalis, J. Algebra, 372 (2012), 702-710.  doi: 10.1016/j.jalgebra.2012.09.032.

[35]

B. G. Rodrigues, Linear codes with complementary duals related to the complement of the Higman-Sims graph, Bull. Iranian Math. Soc., 43 (2017), 2183-2204. 

[36]

V. D. Tonchev, Codes, in: Handbook of Combinatorial Designs, 2nd ed. (C. J. Colbourn and J. H. Dinitz, Eds.), Chapman and Hall/CRC, Boca Raton, 2007,667–702.

[37]

R. A. Wilson, The Finite Simple Groups, London: Springer-Verlag London Ltd., 2009. Graduate Texts in Mathematics, Vol. 251. doi: 10.1007/978-1-84800-988-2.

[38]

R. A. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. A. Parker, S. P. Norton, S. Nickerson, S. Linton, J. Bray and R. Abbott, Atlas of Finite Group Representations - version 3, http://brauer.maths.qmul.ac.uk/Atlas/v3/exc/.

show all references

References:
[1]

E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, Vol. 103. Cambridge: Cambridge University Press, 1992. doi: 10.1017/CBO9781316529836.

[2]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.

[3]

S. BraićA. GolemacJ. Mandić and T. Vučičić, Primitive symmetric designs with up to $2500$ points, J. Combin Designs, 19 (2011), 463-474.  doi: 10.1002/jcd.20291.

[4]

T. Breuer, Decomposition Matrices, Available at:, The Modular Atlas homepage. http://www.math.rwth-aachen.de/MOC/decomposition. 1999.

[5]

A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-1939-6.

[6]

A. E. Brouwer and C. A. van Eijl, On the $p$-rank of the adjacency matrices of strongly regular graphs, J. Algebraic Combin., 1 (1992), 329-346.  doi: 10.1023/A:1022438616684.

[7]

A. E. Brouwer and H. van Maldeghem, Strongly Regular Graphs, Encyclopedia of Mathematics and its Applications, 182. Cambridge University Press, Cambridge, 2022. https://homepages.cwi.nl/~aeb/math/srg/rk3/srgw.pdf. doi: 10.1017/9781009057226.

[8]

F. Buekenhout and H. Van Maldeghen, A characterization of some rank 2 incidence geometries by their automorphism group, Mitt. Math. Sem. Giessen, 218 (1994), 1-70. 

[9]

A. R. Calderbank and J. -M. Goethals, Three weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. 

[10]

P. J. Cameron, Permutation Groups, London Mathematical Society Student Texts, 45, Cambridge: Cambridge University Press, 1999. doi: 10.1017/CBO9780511623677.

[11]

P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links, London Mathematical Society Student Texts, 22, Cambridge: Cambridge University Press, 1991. doi: 10.1017/CBO9780511623714.

[12]

N. ChigiraM. Harada and M. Kitazume, Permutation groups and binary self-orthogonal codes, J. Algebra, 309 (2007), 610-621.  doi: 10.1016/j.jalgebra.2006.06.001.

[13] J. H. ConwayR. T. CurtisS. P. NortonR. A. Parker and R. A. Wilson, An Atlas of Finite Groups, Oxford University Press, Oxford, 1985. 
[14]

A. Cossidente and O. H. King, On the geometry of the exceptional group $G_2(q), $ $q$ even, Des. Codes Cryptogr., 47 (2008), 145-157.  doi: 10.1007/s10623-007-9107-0.

[15]

D. CrnkovićV. Mikulić and B. G. Rodrigues, Some strongly regular graphs and self-orthogonal codes from the unitary group ${U}_4(3)$, Glas. Mat. Ser. Ⅲ, 45 (2010), 307-323.  doi: 10.3336/gm.45.2.02.

[16]

H. J. CouttsM. Quick and C. M. Roney-Dougal, The primitive permutation groups of degree less than $4096$, Comm. Algebra, 39 (2011), 3526-3546.  doi: 10.1080/00927872.2010.515521.

[17]

P. Dankelmann, J. D. Key and B. G. Rodrigues, A characterization of graphs by codes from their incidence matrices, Electron. J. of Combin., 20 (2013), Paper 18, 22 pp. doi: 10.37236/2770.

[18]

P. DankelmannJ. D. Key and B. G. Rodrigues, Codes from incidence matrices of graphs, Des. Codes Cryptogr., 68 (2013), 373-393.  doi: 10.1007/s10623-011-9594-x.

[19]

U. Dempwolff, Primitive rank-3 groups on symmetric designs, Des. Codes Crypt., 22 (2001), 191-207.  doi: 10.1023/A:1008373207617.

[20]

R. H. Dye, Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra, 59 (1979), 202-221.  doi: 10.1016/0021-8693(79)90157-1.

[21]

W. FishJ. D. Key and E. Mwambene, Binary codes from reexive uniform subset graphs on 3-sets, Adv. Math. Commun., 9 (2015), 211-232.  doi: 10.3934/amc.2015.9.211.

[22]

W. FishJ. D. Key and E. Mwambene, Ternary codes from some reflexive uniform subset graphs, Appl. Algebra Engrg. Comm. Comput., 25 (2014), 363-382.  doi: 10.1007/s00200-014-0233-4.

[23]

A. Günther and G. Nebe, Automorphisms of doubly even self-dual codes, Bull. London Math. Soc., 41 (2009), 769-778.  doi: 10.1112/blms/bdp026.

[24]

C. Jansen, K. Lux, R. Parker and R. Wilson, An Atlas of Brauer Characters, LMS Monographs New Series 11. Oxford: Oxford Science Publications, Clarendon Press, 1995.

[25]

D. Jungnickel and V. D. Tonchev, Exponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankin bound, Des. Codes and Cryptogr., 1 (1991), 247-253.  doi: 10.1007/BF00123764.

[26]

J. D. Key and J. Moori, Codes, designs and graphs from the Janko groups ${J}_1$ and ${J}_2$, J. Combin. Math and Combin. Comput., 40 (2002), 143-159. 

[27]

W. Knapp and H.-J. Schaeffer, On the codes related to the Higman-Sims graph, Electron. J. of Combin., 22 (2015), Paper 1.19, 58 pp.

[28]

T. Le and B. G. Rodrigues, Magma computations for $G_2(q)$ of rank 3, Available at:, https://bgrodrigues.weebly.com/uploads/1/2/8/4/12846738/g2qrank3.txt

[29]

M. W. LiebeckC. E. Praeger and J. Saxl, On the $2$-closures of finite permutation groups, J. Lond. Math. Soc., 37 (1987), 241-252.  doi: 10.1112/jlms/s2-37.2.241.

[30]

K. Lux and H. Pahlings, Representations of Groups: A Computational Approach, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010.

[31]

A. A. Makhnev, $GQ(4, 2)$-extensions, the strongly regular case, Math. Notes, 68, 97–102.

[32]

J. Moori, Finite groups, designs and codes, Information security, coding theory and related combinatorics, 202–230, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS, Amsterdam, 2011.

[33]

J. Moori and B. G. Rodrigues, Some designs and codes invariant under the simple group Co2, J. of Algebra, 316 (2007), 649-661.  doi: 10.1016/j.jalgebra.2007.02.004.

[34]

J. Moori and B. G. Rodrigues, Some designs and binary codes preserved by the simple group Ru of Rudvalis, J. Algebra, 372 (2012), 702-710.  doi: 10.1016/j.jalgebra.2012.09.032.

[35]

B. G. Rodrigues, Linear codes with complementary duals related to the complement of the Higman-Sims graph, Bull. Iranian Math. Soc., 43 (2017), 2183-2204. 

[36]

V. D. Tonchev, Codes, in: Handbook of Combinatorial Designs, 2nd ed. (C. J. Colbourn and J. H. Dinitz, Eds.), Chapman and Hall/CRC, Boca Raton, 2007,667–702.

[37]

R. A. Wilson, The Finite Simple Groups, London: Springer-Verlag London Ltd., 2009. Graduate Texts in Mathematics, Vol. 251. doi: 10.1007/978-1-84800-988-2.

[38]

R. A. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. A. Parker, S. P. Norton, S. Nickerson, S. Linton, J. Bray and R. Abbott, Atlas of Finite Group Representations - version 3, http://brauer.maths.qmul.ac.uk/Atlas/v3/exc/.

Figure 1.  Partial submodule lattice for the $ 351 $-dimensional representation over $ \mathbb{F}_2 $
Table 1.  The weight distribution of $ C_3(\Gamma) $
$ i $ $A_i $ $ i $ $ A_i $
0 1 243 1899969548750
126 702 252 376258697100
144 132678 261 22893588900
162 264810 270 1272627720
180 15877134 279 107557632
189 125095689 288 3027024
198 2147437656 306 88452
207 26912233530 315 88452
216 395941284648 324 21840
225 1844882687232 351 756
234 3055067224272
$ i $ $A_i $ $ i $ $ A_i $
0 1 243 1899969548750
126 702 252 376258697100
144 132678 261 22893588900
162 264810 270 1272627720
180 15877134 279 107557632
189 125095689 288 3027024
198 2147437656 306 88452
207 26912233530 315 88452
216 395941284648 324 21840
225 1844882687232 351 756
234 3055067224272
Table 2.  Incidence matrix of the poset of submodules of $ { \mathbb{F}_2}^{351 \times 1 } $
dim 0 1 78 79 92 93 168 168 169 169 182 182 183 183 258 259 272 273 350 351
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 . 1 . 1 . 1 . . 1 1 . . 1 1 . 1 . 1 . 1
78 . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
79 . . . 1 . 1 . . 1 1 . . 1 1 . 1 . 1 . 1
92 . . . . 1 1 . . . . 1 1 1 1 . . 1 1 1 1
93 . . . . . 1 . . . . . . 1 1 . . . 1 . 1
168 . . . . . . 1 . 1 . 1 . 1 . 1 1 1 1 1 1
168 . . . . . . . 1 . 1 . 1 . 1 1 1 1 1 1 1
169 . . . . . . . . 1 . . . 1 . . 1 . 1 . 1
169 . . . . . . . . . 1 . . . 1 . 1 . 1 . 1
182 . . . . . . . . . . 1 . 1 . . . 1 1 1 1
182 . . . . . . . . . . . 1 . 1 . . 1 1 1 1
183 . . . . . . . . . . . . 1 . . . . 1 . 1
183 . . . . . . . . . . . . . 1 . . . 1 . 1
258 . . . . . . . . . . . . . . 1 1 1 1 1 1
259 . . . . . . . . . . . . . . . 1 . 1 . 1
272 . . . . . . . . . . . . . . . . 1 1 1 1
273 . . . . . . . . . . . . . . . . . 1 . 1
350 . . . . . . . . . . . . . . . . . . 1 1
351 . . . . . . . . . . . . . . . . . . . 1
dim 0 1 78 79 92 93 168 168 169 169 182 182 183 183 258 259 272 273 350 351
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 . 1 . 1 . 1 . . 1 1 . . 1 1 . 1 . 1 . 1
78 . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
79 . . . 1 . 1 . . 1 1 . . 1 1 . 1 . 1 . 1
92 . . . . 1 1 . . . . 1 1 1 1 . . 1 1 1 1
93 . . . . . 1 . . . . . . 1 1 . . . 1 . 1
168 . . . . . . 1 . 1 . 1 . 1 . 1 1 1 1 1 1
168 . . . . . . . 1 . 1 . 1 . 1 1 1 1 1 1 1
169 . . . . . . . . 1 . . . 1 . . 1 . 1 . 1
169 . . . . . . . . . 1 . . . 1 . 1 . 1 . 1
182 . . . . . . . . . . 1 . 1 . . . 1 1 1 1
182 . . . . . . . . . . . 1 . 1 . . 1 1 1 1
183 . . . . . . . . . . . . 1 . . . . 1 . 1
183 . . . . . . . . . . . . . 1 . . . 1 . 1
258 . . . . . . . . . . . . . . 1 1 1 1 1 1
259 . . . . . . . . . . . . . . . 1 . 1 . 1
272 . . . . . . . . . . . . . . . . 1 1 1 1
273 . . . . . . . . . . . . . . . . . 1 . 1
350 . . . . . . . . . . . . . . . . . . 1 1
351 . . . . . . . . . . . . . . . . . . . 1
Table 3.  $p$-ary codes from the rank 3 actions of $G_2(3)$ and $G_2(4)$
$ p $ $ G_2(3) $ code $ { \mbox{Aut}} $ $ G_2(4) $ code $ { \mbox{Aut}} $
$ 2 $ $ [351,78,48] $ $ O_7(3){:}2 $ $ [2016,14,976] $ $ {{\rm{PSp}}}_{6}(4) $
$ 2 $ $ [351,79,48] $ $ O_7(3){:}2 $ $ [2016,2002,4] $ $ {{\rm{PSp}}}_{6}(4) $
$ 2 $ $ [2016,13,992] $ $ {{\rm{PSp}}}_{12}(2) $
$ 2 $ $ [2016,2003,4] $ $ {{\rm{PSp}}}_{12}(2) $
$ 3 $ $ [351,27,126] $ $ O_7(3){:}2 $ $ [2016,651,d],\,d \geq 975 $ $ - $
$ 3 $ $ [351,324,6] $ $ O_7(3){:}2 $ $ [2016,651,d],\,d \geq 1041 $ $ - $
$ 3 $ $ [351,28,108] $ $ O_7(3){:}2 $
$ 5 $ $ [2016,650,d] $ $ - $
$ 5 $ $ [2016,651,d] $ $ - $
$ p $ $ G_2(3) $ code $ { \mbox{Aut}} $ $ G_2(4) $ code $ { \mbox{Aut}} $
$ 2 $ $ [351,78,48] $ $ O_7(3){:}2 $ $ [2016,14,976] $ $ {{\rm{PSp}}}_{6}(4) $
$ 2 $ $ [351,79,48] $ $ O_7(3){:}2 $ $ [2016,2002,4] $ $ {{\rm{PSp}}}_{6}(4) $
$ 2 $ $ [2016,13,992] $ $ {{\rm{PSp}}}_{12}(2) $
$ 2 $ $ [2016,2003,4] $ $ {{\rm{PSp}}}_{12}(2) $
$ 3 $ $ [351,27,126] $ $ O_7(3){:}2 $ $ [2016,651,d],\,d \geq 975 $ $ - $
$ 3 $ $ [351,324,6] $ $ O_7(3){:}2 $ $ [2016,651,d],\,d \geq 1041 $ $ - $
$ 3 $ $ [351,28,108] $ $ O_7(3){:}2 $
$ 5 $ $ [2016,650,d] $ $ - $
$ 5 $ $ [2016,651,d] $ $ - $
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