Weighted projective spaces are natural generalizations of projective spaces with a rich structure. Projective Reed-Muller codes are error-correcting codes that played an important role in reliably transmitting information on digital communication channels. In this case study, we explore the power of commutative and homological algebraic techniques to study weighted projective Reed-Muller (WPRM) codes on weighted projective spaces of the form $\mathbb{P}(1,1,a)$. We compute minimal free resolutions and thereby obtain Hilbert series for the vanishing ideal of the $\mathbb{F}_q$-rational points, and compute main parameters for these codes.
翻译:加权投影空间是具有丰富结构的投影空间的自然一般化。投影 Reed-Muller 代码是更正错误的代码,在可靠传输数字通信频道信息方面发挥了重要作用。在本案例研究中,我们探索了移动和同系代数技术在研究关于以$\mathbb{P}(1,1,a)$为表的加权投影空间的加权投影Reed-Muller(WPRM)代码方面的力量。我们计算了最低自由分辨率,从而获取了用于 $\mathb{F ⁇ q$-立标点的消失理想的Hilbert系列,并计算了这些代码的主要参数。