Minimal p -Ary Codes via the Direct Sum of Functions, Non-Covering Permutations and Subspaces of Derivatives
René Rodríguez-Aldama, Enes Pašalić, Fengrong Zhang, Yongzhuang Wei · IEEE Transactions on Information Theory · 2023
In this article, we propose several generic methods for constructing minimal linear codes over the prime field Fp. The first construction uses the direct sum of an arbitrary functionf: Fpr→ Fpand a bent functiong: Fps→ Fpto induce minimal codes with parameters [pr+s- 1,r+s+ 1] and minimum distance larger thanpr(p- 1)(ps-1-ps/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 functiong: Fpm→ Fpand a suitable subspace of derivatives ofg, i.e., functions of the formg(y+a) -g(y) for somea∈ F*pm. 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.