This paper presents enhanced reductions of the bounded-weight and exact-weight Syndrome Decoding Problem (SDP) to a system of quadratic equations. Over $\mathbb{F}_2$, we improve on a previous work and study the degree of regularity of the modeling of the exact weight SDP. Additionally, we introduce a novel technique that transforms SDP instances over $\mathbb{F}_q$ into systems of polynomial equations and thoroughly investigate the dimension of their varieties. Experimental results are provided to evaluate the complexity of solving SDP instances using our models through Gr\"obner bases techniques.

翻译：本文提出了将带权有界和精确权重的伴随式解码问题（SDP）归约至二次方程系统的增强方法。在$\mathbb{F}_2$域上，我们改进了先前的研究，并分析了精确权重SDP建模的正则度。此外，我们引入了一种创新技术，可将$\mathbb{F}_q$域上的SDP实例转化为多项式方程组，并深入研究了其簇的维数特性。通过Gröbner基技术，我们提供了实验数据以评估采用本模型求解SDP实例的计算复杂度。