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.
[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.

[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.

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

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

[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.

[7]

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

[8]

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

[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. 

[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. 

[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,

[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.

[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.

[14]

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

[15]

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

[16]

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

[17]

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

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.
[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.

[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.

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

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

[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.

[7]

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

[8]

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

[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. 

[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. 

[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,

[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.

[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.

[14]

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

[15]

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

[16]

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

[17]

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

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 $1755$ $2.[2^9].5.4$
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