We contribute to the knowledge of linear codes from special polynomials and functions, which have been studied intensively in the past few years. Such codes 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 of the two generic constructions; in particular, we propose a Gram-Schmidt (complexity of $\mathcal{O}(n^3)$) process to compute them explicitly. The originality of this contribution is in the study of the existence or not of defining 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 second generic construction. We also determine a necessary condition expressed by employing the Walsh transform for a codeword of $\mathcal{C}$ to belong in the dual. This achievement was done in general and when the involved functions are weakly regularly bent. We shall give a novel description of the Hull code in the framework of the two generic constructions. Our primary interest is constructing linear codes of fixed Hull dimension and determining the (Hamming) weight of the codewords in their duals.

翻译：本文深化了对由特殊多项式与函数构造的线性码的研究，这类码在秘密共享、认证码、结合方案及强正则图等领域具有重要应用。这是首次在两种通用构造框架下研究对偶码的工作；特别地，我们提出了一种复杂度为$\mathcal{O}(n^3)$的Gram-Schmidt过程来显式计算这些对偶码。本贡献的创新性在于研究定义集$D'$的存在性条件——在第二种通用构造中，该集合可作为构建给定码$\mathcal{C}$的对偶码$\mathcal{C}'$的要素。我们还确定了$\mathcal{C}$中码字属于对偶码的一个必要条件，该条件通过Walsh变换表述，并在一般情形及所涉函数为弱正则弯曲函数时均成立。本文将在两种通用构造框架下给出壳码的新描述，重点关注固定壳维数的线性码构造及其对偶码中码字的汉明重量确定问题。