Paper 2023/438
Minimal $p$ary codes via the direct sum of functions, noncovering permutations and subspaces of derivatives
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 $p^r(p1)(p^{s1}p^{s/21})$. For the first time, we provide a general construction of linear codes from a subclass of nonweakly 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 noncovering 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 noncovering permutations contains the class of APN power permutations (characterized by having twotoone derivatives). Finally, the last general construction combines the previous two methods (direct sum, noncovering permutations and subspaces of derivatives), using a bent function in the MaioranaMcFarland class to construct minimal codes (even those violating the AshikhminBarg bound) with larger dimensions. This last method proves to be highly flexible since it can lead to several nonequivalent codes, depending to a great extent on the choice of the underlying noncovering permutation.
Metadata
 Available format(s)
 Category
 Secretkey cryptography
 Publication info
 Preprint.
 Keywords
 Minimal linear codes$p$ary functionsnonweakly regular functionsnoncovering permutationsderivativesdirect sum
 Contact author(s)

rene7ca @ gmail com
enes paslic6 @ gmail com
zhfl203 @ cumt edu cn
walker_wyz @ guet edu cn  History
 20230930: last of 2 revisions
 20230326: received
 See all versions
 Short URL
 https://ia.cr/2023/438
 License

CC BY
BibTeX
@misc{cryptoeprint:2023/438, author = {René Rodríguez and Enes Pasalic and Fengrong Zhang and Yongzhuang Wei}, title = {Minimal $p$ary codes via the direct sum of functions, noncovering permutations and subspaces of derivatives}, howpublished = {Cryptology ePrint Archive, Paper 2023/438}, year = {2023}, note = {\url{https://eprint.iacr.org/2023/438}}, url = {https://eprint.iacr.org/2023/438} }