Search

In this article, we provide the first systematic analysis of bent functions $f$ on $\mathbb {F}_{2}^{n}$ in the Maiorana-McFarland class $\mathcal {M}$ regarding the origin and cardinality of their $\mathcal {M}$ -subspaces, i.e., vector subspaces such that for any two elements $a,b$ from this subspace, the second-order derivative $D_{a}D_{b}f$ is the zero function on $\mathbb {F}_{2}^{n}$ . By imposing restrictions on permutations $\pi $ of $\mathbb {F}_{2}^{n/2}$ , we specify the conditions so that Maiorana-McFarland bent functions $f(x,y)=x\cdot \pi (y) + h(y)$ admit a unique $\mathcal {M}$ -subspace of dimension $n/2$ . On the other hand, we show that permutations $\pi $ with linear structures give rise to Maiorana-McFarland bent functions that do not have this property. In this way, we contribute to the classification of Maiorana-McFarland bent functions, since the number of $\mathcal {M}$ -subspaces of a fixed dimension is invariant under equivalence. Additionally, we give several generic methods of specifying permutations $\pi $ so that $f\in \mathcal {M}$ admits a unique $\mathcal {M}$ -subspace. Most notably, using the knowledge about $\mathcal {M}$ -subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions on $\mathbb {F}_{2}^{n-2}$ , one can in a generic manner generate bent functions on $\mathbb {F}_{2}^{n}$ outside the completed Maiorana-McFarland class $\mathcal {M}^{\#}$ for any even $n\geq 8$ . Remarkably, with our construction methods, it is possible to obtain inequivalent bent functions on $\mathbb {F}_{2}^{8}$ not stemming from the two primary classes, the partial spread class $\mathcal {PS}$ and $\mathcal {M}$ . In this way, we contribute to a better understanding of the origin of bent functions in eight variables, since only a small fraction of about 276 bent functions stems from $\mathcal {PS}$ and $\mathcal {M}$ , whereas their total number on $\mathbb {F}_{2}^{8}$ is approximately 2106.

In this article, we propose several generic methods for constructing minimal linear codes over the prime field $ {\mathbb {F}}_{p}$ . The first construction uses the direct sum of an arbitrary function $f\colon {\mathbb {F}} _{p^{r}}\to {\mathbb {F}} _{p}$ and a bent function $g\colon {\mathbb {F}} _{p^{s}}\to {\mathbb {F}} _{p}$ 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 by Li and Mesnager. The second construction deals with a bent function $g\colon {\mathbb {F}} _{p^{m}}\to {\mathbb {F}} _{p}$ and a suitable subspace of derivatives of $g$ , i.e., functions of the form $g(y+a)-g(y)$ for some $a\in {\mathbb {F}} _{p^{m}}^{\star} $ . 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.