February  2017, 11(1): 77-82. doi: 10.3934/amc.2017003

Some designs and codes invariant under the Tits group

School of Mathematical Sciences North-West University (Mafikeng), Mmabatho, South Africa

Received  June 2015 Revised  July 2015 Published  February 2017

Fund Project: The first author acknowledges support of NRF and NWU (Mafikeng). The second author acknowledges support of NWU (Mafikeng) postdoctoral fellowship

In this paper, we construct some designs and associated binary codes from a primitive permutation representation of degree 1755 of the sporadic simple Tits group $^2F_4 \left(2 \right)'$. In particular, we construct a binary code $[1755, 26,1024]_2$ on which $^2F_4 \left(2 \right)'$ acts irreducibly. This is the smallest non-trivial irreducible $GF(2)$-module for our group.

Citation: Jamshid Moori, Amin Saeidi. Some designs and codes invariant under the Tits group. Advances in Mathematics of Communications, 2017, 11 (1) : 77-82. doi: 10.3934/amc.2017003
References:
[1] Jr. E. F. Assmus and J. D. Key, Designs and their Codes, Cambridge Univ. Press, Cambridge, 1993. doi: 10.1017/CBO9781316529836. Google Scholar
[2]

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

[3]

A. R. Calderbank and D. B. Wales, A global code invariant under the Higman-Sims group, Algebra J., 75 (1982), 233-260. doi: 10.1016/0021-8693(82)90073-4. Google Scholar

[4] J. H. ConwayR. T. CurtisS. P. NortonR. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford Univ. Press, Oxford, 1985. Google Scholar
[5]

The GAP Group, GAP -Groups, Algorithms, and Programming, Version 4.7.6, 2014, available at http://www.gap-system.orgGoogle Scholar

[6]

W. H. HaemersC. ParkerV. Pless and V. D. Tonchev, A design and a code invariant under the simple group Co3, J. Combin. Theory Ser. A, 62 (1993), 225-233. doi: 10.1016/0097-3165(93)90045-A. Google Scholar

[7]

W. C. Huffman, Codes and groups, in Handbook of Coding Theory, Elsevier, 1998,1345-1440. Google Scholar

[8]

C. Jansen, K. Lux, R. Parker and R Wilson, An Atlas of Brauer Characters, Oxford Sci. Publ. , Oxford, 1995. Google Scholar

[9]

J. D. Key and J. Moori, Designs, codes and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002), 143-159. Google Scholar

[10]

J. D. KeyJ. Moori and B. G. Rodrigues, On some designs and codes from primitive representations of some finite simple groups, J. Combin. Math. Combin. Comput., 45 (2003), 3-19. Google Scholar

[11]

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

[12]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly even code invariant under McL : 2, J. Combin. Theory Ser. A, 110 (2005), 53-69. doi: 10.1016/j.jcta.2004.10.001. Google Scholar

[13]

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

[14]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly-even code invariant under the McL, Ars Combin., 91 (2009), 321-332. Google Scholar

[15]

J. Moori and B. G. Rodrigues, On some designs and codes invariant under the Higman-Sims group, Util. Math., 86 (2011), 225-239. Google Scholar

[16]

B. G. Rodrigues, Codes of Designs and Graphs from Finite Simple Groups, Ph. D thesis, Univ. Natal, 2002.Google Scholar

[17]

J. Tits, Algebraic and abstract simple groups, Ann. Math., 80 (1964), 313-329. doi: 10.2307/1970394. Google Scholar

show all references

References:
[1] Jr. E. F. Assmus and J. D. Key, Designs and their Codes, Cambridge Univ. Press, Cambridge, 1993. doi: 10.1017/CBO9781316529836. Google Scholar
[2]

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

[3]

A. R. Calderbank and D. B. Wales, A global code invariant under the Higman-Sims group, Algebra J., 75 (1982), 233-260. doi: 10.1016/0021-8693(82)90073-4. Google Scholar

[4] J. H. ConwayR. T. CurtisS. P. NortonR. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford Univ. Press, Oxford, 1985. Google Scholar
[5]

The GAP Group, GAP -Groups, Algorithms, and Programming, Version 4.7.6, 2014, available at http://www.gap-system.orgGoogle Scholar

[6]

W. H. HaemersC. ParkerV. Pless and V. D. Tonchev, A design and a code invariant under the simple group Co3, J. Combin. Theory Ser. A, 62 (1993), 225-233. doi: 10.1016/0097-3165(93)90045-A. Google Scholar

[7]

W. C. Huffman, Codes and groups, in Handbook of Coding Theory, Elsevier, 1998,1345-1440. Google Scholar

[8]

C. Jansen, K. Lux, R. Parker and R Wilson, An Atlas of Brauer Characters, Oxford Sci. Publ. , Oxford, 1995. Google Scholar

[9]

J. D. Key and J. Moori, Designs, codes and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002), 143-159. Google Scholar

[10]

J. D. KeyJ. Moori and B. G. Rodrigues, On some designs and codes from primitive representations of some finite simple groups, J. Combin. Math. Combin. Comput., 45 (2003), 3-19. Google Scholar

[11]

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

[12]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly even code invariant under McL : 2, J. Combin. Theory Ser. A, 110 (2005), 53-69. doi: 10.1016/j.jcta.2004.10.001. Google Scholar

[13]

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

[14]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly-even code invariant under the McL, Ars Combin., 91 (2009), 321-332. Google Scholar

[15]

J. Moori and B. G. Rodrigues, On some designs and codes invariant under the Higman-Sims group, Util. Math., 86 (2011), 225-239. Google Scholar

[16]

B. G. Rodrigues, Codes of Designs and Graphs from Finite Simple Groups, Ph. D thesis, Univ. Natal, 2002.Google Scholar

[17]

J. Tits, Algebraic and abstract simple groups, Ann. Math., 80 (1964), 313-329. doi: 10.2307/1970394. Google Scholar

Table 1.  Maximal subgroups of ${}^2F_4 \left(2 \right)^\prime$
No. Max. sub. Deg.
1 $L_3(3){:}2$ 1600
2 $L_3(3){:}2$ 1600
3 $2.[2^8].5.4$ 1755
4 $L_2(25)$ 2304
5 $2^2.[2^8].S_3$ 2925
6 $A_6.2^2$ 12480
7 $A_6.2^2$ 12480
8 $5^2{:}4A_4$ 14976
No. Max. sub. Deg.
1 $L_3(3){:}2$ 1600
2 $L_3(3){:}2$ 1600
3 $2.[2^8].5.4$ 1755
4 $L_2(25)$ 2304
5 $2^2.[2^8].S_3$ 2925
6 $A_6.2^2$ 12480
7 $A_6.2^2$ 12480
8 $5^2{:}4A_4$ 14976
Table 2.  The stabilizers of codewords of C
Weight Number of words Structure of the stabilizer
768 $11700$ $((((((4 \times 2) {:} 4) {:} 3) {:} 2) {:} 2){:}2){:}(2 \times 2)$
768 $93600$ $((((4 \times 2) {:} 4) {:}3) {:}2) {:} 2$
800 $44928$ $((5 {:} 4) \times (5 {:} 4)) {:} 2$
832 $56160$ $2 \times (((2 \times 2 \times 2 \times 2) {:} 5) {:} 4)$
832 $280800$ $2 \times ((4 \times 2 \times 2) {:} 4)$
832 $69120$ $13{:}4$
832 $898560$ $2\times (5 {:}4)$
832 $1797120$ $D_{20}$
864 $24960$ $(A_6. 2) {:} 2$
864 $449280$ $4 \times (5{:}4)$
864 $748800$ $2 \times S_4$
864 $1123200$ $(2 \times D_8){:}2$
864 $4492800$ $4 \times (5{:}4)$
864 $4492800$ $D_8$
864 $5990400$ $S_3$
864 $5990400$ $S_3$
864 $8985600$ $2 \times 2$
896 $140400$ $(4 \times 2 \times 2). (8 \times 2)$
896 $280800$ $((2 \times 2 \times Q_8) {:} 2) {:} 2$
896 $2246400$ $(4\times 2){:}2$
896 $4492800$ $2\times4$
896 $4492800$ $2\times4$
896 $8985600$ $2\times2$
896 $8985600$ $4$
928 $1123200$ $(4 \times4) {:}2$
960 $187200$ $((2 \times 2 \times 2). (2 \times 2 \times 2)) {:} 3$
1024 $1755$ $2.[2^9].5.4$
Weight Number of words Structure of the stabilizer
768 $11700$ $((((((4 \times 2) {:} 4) {:} 3) {:} 2) {:} 2){:}2){:}(2 \times 2)$
768 $93600$ $((((4 \times 2) {:} 4) {:}3) {:}2) {:} 2$
800 $44928$ $((5 {:} 4) \times (5 {:} 4)) {:} 2$
832 $56160$ $2 \times (((2 \times 2 \times 2 \times 2) {:} 5) {:} 4)$
832 $280800$ $2 \times ((4 \times 2 \times 2) {:} 4)$
832 $69120$ $13{:}4$
832 $898560$ $2\times (5 {:}4)$
832 $1797120$ $D_{20}$
864 $24960$ $(A_6. 2) {:} 2$
864 $449280$ $4 \times (5{:}4)$
864 $748800$ $2 \times S_4$
864 $1123200$ $(2 \times D_8){:}2$
864 $4492800$ $4 \times (5{:}4)$
864 $4492800$ $D_8$
864 $5990400$ $S_3$
864 $5990400$ $S_3$
864 $8985600$ $2 \times 2$
896 $140400$ $(4 \times 2 \times 2). (8 \times 2)$
896 $280800$ $((2 \times 2 \times Q_8) {:} 2) {:} 2$
896 $2246400$ $(4\times 2){:}2$
896 $4492800$ $2\times4$
896 $4492800$ $2\times4$
896 $8985600$ $2\times2$
896 $8985600$ $4$
928 $1123200$ $(4 \times4) {:}2$
960 $187200$ $((2 \times 2 \times 2). (2 \times 2 \times 2)) {:} 3$
1024 $1755$ $2.[2^9].5.4$
[1]

Peter Vandendriessche. LDPC codes associated with linear representations of geometries. Advances in Mathematics of Communications, 2010, 4 (3) : 405-417. doi: 10.3934/amc.2010.4.405

[2]

Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036

[3]

Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505

[4]

Ivica Martinjak, Mario-Osvin Pavčević. Symmetric designs possessing tactical decompositions. Advances in Mathematics of Communications, 2011, 5 (2) : 199-208. doi: 10.3934/amc.2011.5.199

[5]

Chun-Gil Park. Stability of a linear functional equation in Banach modules. Conference Publications, 2003, 2003 (Special) : 694-700. doi: 10.3934/proc.2003.2003.694

[6]

Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157

[7]

Anuradha Sharma, Saroj Rani. Trace description and Hamming weights of irreducible constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 123-141. doi: 10.3934/amc.2018008

[8]

Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83

[9]

Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055

[10]

Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161

[11]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977

[12]

Pankaj Kumar, Monika Sangwan, Suresh Kumar Arora. The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$. Advances in Mathematics of Communications, 2015, 9 (3) : 277-289. doi: 10.3934/amc.2015.9.277

[13]

Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385

[14]

Cristina García Pillado, Santos González, Victor Markov, Consuelo Martínez, Alexandr Nechaev. New examples of non-abelian group codes. Advances in Mathematics of Communications, 2016, 10 (1) : 1-10. doi: 10.3934/amc.2016.10.1

[15]

Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Advances in Mathematics of Communications, 2010, 4 (1) : 69-81. doi: 10.3934/amc.2010.4.69

[16]

David Clark, Vladimir D. Tonchev. A new class of majority-logic decodable codes derived from polarity designs. Advances in Mathematics of Communications, 2013, 7 (2) : 175-186. doi: 10.3934/amc.2013.7.175

[17]

Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191

[18]

Antonio Cafure, Guillermo Matera, Melina Privitelli. Singularities of symmetric hypersurfaces and Reed-Solomon codes. Advances in Mathematics of Communications, 2012, 6 (1) : 69-94. doi: 10.3934/amc.2012.6.69

[19]

Diego Napp, Carmen Perea, Raquel Pinto. Input-state-output representations and constructions of finite support 2D convolutional codes. Advances in Mathematics of Communications, 2010, 4 (4) : 533-545. doi: 10.3934/amc.2010.4.533

[20]

Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (8)
  • HTML views (1)
  • Cited by (1)

Other articles
by authors

[Back to Top]