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We contribute to the knowledge of linear codes from special polynomials andfunctions, which have been studied intensively in the past few years. Suchcodes have several applications in secret sharing, authentication codes,association schemes and strongly regular graphs.  This is the first work in which we study the dual codes in the framework ofthe two generic constructions; in particular, we propose a Gram-Schmidt(complexity of $\mathcal{O}(n^3)$) process to compute them explicitly. Theoriginality of this contribution is in the study of the existence or not ofdefining sets $D'$, which can be used as ingredients to construct the dual code$\mathcal{C}'$ for a given code $\mathcal{C}$ in the context of the secondgeneric construction. We also determine a necessary condition expressed byemploying the Walsh transform for a codeword of $\mathcal{C}$ to belong in thedual. This achievement was done in general and when the involved functions areweakly regularly bent. We shall give a novel description of the Hull code inthe framework of the two generic constructions. Our primary interest isconstructing linear codes of fixed Hull dimension and determining the (Hamming)weight of the codewords in their duals.

Dual and Hull code in the first two generic constructions and relationship with the Walsh transform of cryptographic functions