# American Institute of Mathematical Sciences

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:

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##### References:
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
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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