从向量板状函数构造线性码及其子域码及其在量子CSS码中的应用
Constructions of linear codes from vectorial plateaued functions and their subfield codes with applications to quantum CSS codes
AI总结：
本文提出通过向量板状函数构造线性码及其子域码，以生成具有少重性和最优距离的码，并应用于量子CSS码。
摘要：
有限域上由函数参数化的线性码已被证明是编码理论中的强大工具，能够产生最优和少重的线性码，并在秘密共享、认证码和关联方案中具有重要应用。2023年，徐等人提出了一种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表现出显著成效。受此方法启发，我们提出了一种扩展框架，将其扩展到三个函数，从而增强了参数的灵活性。此外，我们引入了向量设置，允许向量值函数，扩大了构造空间和可实现的结构属性集。我们分析了标量和向量框架，利用Bent和s-Plateaued函数，包括Almost Bent，来定义码生成器。通过利用沃尔什变换的性质，我们确定了这些码及其删减版本的显式参数和重量分布。本研究的关键结果是所构造的码具有少重性，且其对偶码在球面包装和格里斯默界方面在距离和维度上都是最优的。此外，我们建立了我们的向量方法与经典第一通用线性码构造之间的理论联系，提供了所得到的码为最小和自正交的充分条件。最后，我们研究了在Calderbank-Shor-Steane框架下的量子编码理论应用。
英文摘要：
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.
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