Linear codes over finite fields parameterized by functions have proven to be a powerful tool in coding theory, yielding optimal and few-weight codes with significant applications in secret sharing, authentication codes, and association schemes. In 2023, Xu et al. introduced a construction framework for 3-dimensional linear codes parameterized by two functions, which has demonstrated considerable success in generating infinite families of optimal linear codes. Motivated by this approach, we propose a construction that extends the framework to three functions, thereby enhancing the flexibility of the parameters. Additionally, we introduce a vectorial setting by allowing vector-valued functions, expanding the construction space and the set of achievable structural properties. We analyze both scalar and vectorial frameworks, employing Bent and s-Plateaued functions, including Almost Bent, to define the code generators. By exploiting the properties of the Walsh transform, we determine the explicit parameters and weight distributions of these codes and their punctured versions. A key result of this study is that the constructed codes have few weights, and their duals are distance and dimensionally optimal with respect to both the Sphere Packing and Griesmer bounds. Furthermore, we establish a theoretical connection between our vectorial approach and the classical first generic construction of linear codes, providing sufficient conditions for the resulting codes to be minimal and self-orthogonal. Finally, we investigate applications to quantum coding theory within the Calderbank-Shor-Steane framework.

翻译：由函数参数化的有限域上线性码已被证明是编码理论中的有力工具，可产生在秘密共享、认证码和关联方案中具有重要应用的最优码和少重量码。2023年，Xu等人提出了一个由两个函数参数化的三维线性码构造框架，该框架在生成无限族最优线性码方面已展现出显著成效。受此方法启发，我们提出了一种将框架扩展至三个函数的构造，从而增强了参数的灵活性。此外，我们通过引入向量值函数建立了一个向量化设置，扩展了构造空间和可实现的结构性质集合。我们分析了标量与向量化两种框架，采用Bent函数和s-平台函数（包括Almost Bent函数）来定义码生成器。通过利用Walsh变换的性质，我们确定了这些码及其删减版本的显式参数与重量分布。本研究的一个关键结论是：所构造的码具有少重量特性，且其对偶码在球填充界和Griesmer界下均达到距离和维数最优。进一步地，我们在向量化方法与经典的线性码第一类通用构造之间建立了理论联系，为所得码成为极小码和自正交码提供了充分条件。最后，我们探讨了其在Calderbank-Shor-Steane框架下量子编码理论中的应用。