Welcome to the resource topic for 2023/438
Title:
Minimal p-ary codes from non-covering permutations
Authors: René Rodríguez, Enes Pasalic, Fengrong Zhang, Yongzhuang Wei
Abstract:In this article, we propose generalizations to the non-binary scenario of the methods employed in [44] for constructing minimal linear codes. Specifically, we provide three constructions of minimal codes over \mathbb{F}_p. The first construction uses the method of direct sum of an arbitrary function f:\mathbb{F}_{p^r}\to \mathbb{F}_{p} and a bent function g:\mathbb{F}_{p^s}\to \mathbb{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. The second construction deals with a bent function g:\mathbb{F}_{p^m}\to \mathbb{F}_p and a subspace of suitable derivatives U of g, i.e., functions of the form g(y+a)-g(y) for some a\in \mathbb{F}_{p^m}^*. We also provide a generalization of the recently introduced concept of non-covering permutations [44] and prove important properties of this class of permutations. The most notable observation is that the class of non-covering permutations contains the class of 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) to construct minimal codes with a larger dimension. This method proves to be quite flexible since it can lead to several non-equivalent codes, depending exclusively on the choice of the underlying non-covering permutation.
ePrint: https://eprint.iacr.org/2023/438
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