Some designs and codes invariant under the Tits group

Jamshid Moori; Amin Saeidi

doi:10.3934/amc.2017003

Article Contents

2017, Volume 11, Issue 1: 77-82. Doi: 10.3934/amc.2017003

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Some designs and codes invariant under the Tits group

Jamshid Moori and
Amin Saeidi

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

Received: June 2015

Revised: July 2015

Published: May 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 05B05, 05E20, 20D05.

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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

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

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