Paper 2023/438
Minimal $p$-ary codes from non-covering permutations
Abstract
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) together with a bent function in the Maiorana-McFarland class to construct minimal codes (even those violating the Ashikhmin-Barg bound) with a larger dimension. This last method proves to be quite flexible since it can lead to several non-equivalent codes, depending to a great extent on the choice of the underlying non-covering permutation.
Metadata
- Available format(s)
-
PDF
- Category
- Secret-key cryptography
- Publication info
- Preprint.
- Keywords
- Minimal linear codes$p$-ary functionsnon-covering permutationsderivativesdirect sum
- Contact author(s)
-
rene7ca @ gmail com
enes paslic6 @ gmail com
zhfl203 @ cumt edu cn
walker_wyz @ guet edu cn - History
- 2023-04-07: revised
- 2023-03-26: received
- See all versions
- Short URL
- https://ia.cr/2023/438
- License
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CC BY
BibTeX
@misc{cryptoeprint:2023/438, author = {René Rodríguez and Enes Pasalic and Fengrong Zhang and Yongzhuang Wei}, title = {Minimal $p$-ary codes from non-covering permutations}, howpublished = {Cryptology ePrint Archive, Paper 2023/438}, year = {2023}, note = {\url{https://eprint.iacr.org/2023/438}}, url = {https://eprint.iacr.org/2023/438} }