Paper 2023/438

Minimal $p$-ary codes via the direct sum of functions, non-covering permutations and subspaces of derivatives

René Rodríguez, University of Primorska
Enes Pasalic, University of Primorska, Guilin University of Electronic Technology
Fengrong Zhang, Xidian University
Yongzhuang Wei, Guilin University of Electronic Technology

In this article, we propose several generic methods for constructing minimal linear codes over the field $\mathbb{F}_p$. 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, which partially answers an open problem posed in [22]. 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 sound generalization of the recently introduced concept of non-covering permutations [45]. 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 contains the class of APN power permutations (characterized by having two-to-one derivatives). Finally, the last general 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.

Available format(s)
Secret-key cryptography
Publication info
Minimal linear codes$p$-ary functionsnon-weakly regular functionsnon-covering permutationsderivativesdirect sum
Contact author(s)
rene7ca @ gmail com
enes paslic6 @ gmail com
zhfl203 @ cumt edu cn
walker_wyz @ guet edu cn
2023-09-30: last of 2 revisions
2023-03-26: received
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      author = {René Rodríguez and Enes Pasalic and Fengrong Zhang and Yongzhuang Wei},
      title = {Minimal $p$-ary codes via the direct sum of functions, non-covering permutations and subspaces of derivatives},
      howpublished = {Cryptology ePrint Archive, Paper 2023/438},
      year = {2023},
      note = {\url{}},
      url = {}
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