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  • Minimal p-ary codes via the direct sum of functions, non-covering permutations and subspaces of derivatives [Elektronski vir]
    Rodríguez, René ...
    In this article, we propose several generic methods for constructing minimal linear codes over the prime field F p . The first construction uses the direct sum of an arbitrary function f : F pr → F p ... and a bent function g : F ps → F p to induce minimal codes with parameters [ p r+s - 1, r + s + 1] and minimum distance larger than p r ( p - 1)( p s-1 - p s/2-1 ). For the first time, we provide a general construction of linear codes from a subclass of non-weakly regular plateaued functions, which partially answers an open problem posed by Li and Mesnager. The second construction deals with a bent function g : F pm → F p and a suitable subspace of derivatives of g , i.e., functions of the form g ( y + a ) - g ( y ) for some a ∈ F* pm . We also provide a sound generalization of the recently introduced concept of non-covering permutations. Some important structural properties of this class of permutations are derived in this context. The most remarkable observation is that the class of non-covering permutations includes all APN power permutations (characterized by having two-to-one derivatives). Finally, the last construction combines the previous two methods (direct sum, non-covering permutations and subspaces of derivatives), using a bent function in the Maiorana-McFarland class, to construct minimal codes (even those violating the Ashikhmin-Barg bound) with larger dimensions. This last method proves to be highly flexible since it can lead to several non-equivalent codes, depending to a great extent on the choice of the underlying non-covering permutation.
    Source: IEEE transactions on information theory. - ISSN 1557-9654 (2023, str. 1-19)
    Type of material - e-article ; adult, serious
    Publish date - 2023
    Language - english
    COBISS.SI-ID - 174595331

    Link(s):

    https://ieeexplore.ieee.org/document/10336830
    https://doi.org/10.1109/TIT.2023.3338577
    DOI

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