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$ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $

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  • In this paper we construct $ 2 $-PD-sets of $ 16 $ elements for codes from the Desarguesian projective planes $ \mathrm{PG}(2,q) $, where $ q = 2^h $ and $ 5\leq h \leq 9 $. We also construct $ 3 $-PD-sets of $ 75 $ elements for the code from the Desarguesian projective plane $ \mathrm{PG}(2,q) $, where $ q = 2^9 $. These $ 2 $-PD-sets and $ 3 $-PD-sets can be used for partial permutation decoding of codes obtained from the Desarguesian projective planes.

    Mathematics Subject Classification: Primary: 51E20, 94B05.

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  • Table 1.  Codes of $ \mathrm{PG}(2,q) $: lower bounds on sizes of PD-sets and $ 2 $-PD-sets

    $ q $ Code $ t $ $ r $ $ b $ $ b_2 $
    $ 32 $ [1057,244, 33] 16 813 180 3
    $ 64 $ [4161,730, 65] 32 3431 1623 3
    $ 128 $ [16513, 2188,129] 64 14325 40696 3
    $ 256 $ [65793, 6562,257] 128 59231 3965945 3
    $ 512 $ [262657, 19684,513] 256 242973 3625171287 3
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