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On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $

This work has been partially supported by the Spanish grant TIN2016-77918-P (AEI/FEDER, UE)

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  • We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is $ C_{2t}\times C_2 $. We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by $ 3 $ if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order $ 16 $.

    Mathematics Subject Classification: Primary: 5B, 5E, 94B.

    Citation:

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  • Table 2.  Allowable values of the rank $ r $, and dimension of the kernel $ k $ for nonlinear $ \operatorname{HFP}(\cdot,\cdot,\cdot) $-codes of length $ 4t $

    $ \operatorname{HFP}(\cdot,\cdot,\cdot) $ $ t $ $ r $ $ k $
    $ (4t_ \mathbf{u},2) $ even $ \leq 2t $ $ 1 $
    $ (2t,2,2_ \mathbf{u}) $ even square $ \leq 2t $ $ 1,2,3 $
    $ (2t,4_ \mathbf{u}) $ even $ \leq 2t $ $ 1,2,3 $
    $ (t,Q_ \mathbf{u}) $ odd $ 4t-1 $ $ 1 $
     | Show Table
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    Table 1.  Rank and dimension of the kernel of Hadamard full propelinear codes with associated group $ C_{2t} \times C_2 $. Symbol x means that the non-existence was checked with Magma by exhaustive search, symbol $\checkmark$ means that the non-existence was proved analytically, and "-" means that the code does not have $ C_{2t}\times C_2 $ as associated group. When the values for the rank and the dimension of the kernel appears in a box it means that they are the only values for that box

    t $ (4t_ \mathbf{u},2) $ $ (2t,2,2_ \mathbf{u}) $ $ (2t,4_ \mathbf{u}) $ $ (t,Q_ \mathbf{u}) $
    $ r $ $ k $ $ r $ $ k $ $ r $ $ k $ $ r $ $ k $
    1 3 3 3 3 3 3 x x
    2 4 4 $\checkmark$ $\checkmark$ 4 4 - -
    3 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 11 1
    4 x x 5 5 7 2 - -
    6 3
    5 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 19 1
    6 x x $\checkmark$ $\checkmark$ x x - -
    7 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 27 1
    8 x x $\checkmark$ $\checkmark$ 11 2 - -
    13 1
    9 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 35 1
    10 x x $\checkmark$ $\checkmark$ x x - -
     | Show Table
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  • [1] V. ÁlvarezF. Gudiel and M. B. Güemes, On $\mathbb{Z}_t\times \mathbb{Z}_2^2$-cocyclic Hadamard matrices, J. Combin. Des., 23 (2015), 352-368.  doi: 10.1002/jcd.21406.
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