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Dual transform and projective self-dual codes

  • *Corresponding author: Stefka Bouyuklieva

    *Corresponding author: Stefka Bouyuklieva

Both authors are supported by [the Bulgarian National Science Fund under Contract No KP-06-H62/2/13.12.2022]

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  • We present and analyze the relation between different types of linear codes through the (projective) dual transform. The considered codes are represented by a generator matrix or a characteristic vector, and the dual transform is defined in these terms. This allows us to extend its use to study codes with special properties, which we call self-polar, and their relation to Boolean functions. The self-polar codes are a special class of projective self-dual codes and are closely connected with self-dual and anti-self-dual bent functions. We also give some computational results for self-dual bent functions in eight variables.

    Mathematics Subject Classification: Primary: 94B05, 05B25, 06E30.

    Citation:

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  • Table 1.  Orders of the automorphism groups of the codes $ C(f) $ for the constructed self-dual bent functions $ f $

    $ |Aut(C)| $ $ \sharp $ codes $ |Aut(C)| $ $ \sharp $ codes $ |Aut(C)| $ $ \sharp $ codes $ |Aut(C)| $ $ \sharp $ codes
    1032192 1 98304 2 144 2 96 12
    589824 1 24576 2 768 6 48 13
    786432 1 49152 4 1536 8 24 29
    344064 1 3840 1 240 1 72 3
    393216 2 1344 2 384 5 12 39
    294912 1 3072 7 192 2 30 1
    196608 2 6144 3 42 1 9 1
    86016 1 9216 1 18 1 6 50
    51840 1 1920 2 36 4 3 23
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