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 F-p . The first construction uses the direct sum of an arbitrary function f: F-p(r )-> F-p and a bent function g: F-p(s )-> 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: F-p(m) -> F-p and a suitable subspace of derivatives of g , i.e., functions of the form g(y + a) - g(y) for some a is an element of 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.