Minimal p-Ary Codes via the Direct Sum of Functions, Non-Covering Permutations and Subspaces of Derivatives

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.