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2023, Volume 17, Issue 1: 207-226. Doi: 10.3934/amc.2022016
This issue Previous Article Classification of $ \mathbf{(3 \!\mod 5)} $ arcs in $ \mathbf{ \operatorname{PG}(3,5)} $ Next Article Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes

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

    * Corresponding author

Dedicated to the Memory of Kay Magaard

Received:  August 2021
Revised:  February 2022
Early access:  March 2022
Published:  December 2022

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

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    Abstract

  • 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.

    Mathematics Subject Classification: Primary: 05B05, 94B05; Secondary: 20D45.

    Citation:
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  • Figure 1.  Partial submodule lattice for the $ 351 $-dimensional representation over $ \mathbb{F}_2 $

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    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
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    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
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    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] $ $ - $
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