I introduce a method to generate families of CSS codes with interesting code parameters. The object of study is Coxeter groups, both finite and infinite (reducible or not), and a geometrically motivated partial order of Coxeter group elements named after Bruhat. The Bruhat order is known to provide a link to algebraic topology -- it doubles as a face poset capturing the inclusion relations of the $p$-dimensional cells of a regular CW~complex and that is what makes it interesting for QEC code design. Assisted by the Bruhat face poset interval structure unique to Coxeter groups I show that the corresponding chain complexes can be turned into multitudes of CSS codes. Depending on the approach, I obtain CSS codes (and their families) with controlled stabilizer weights, for example $[6006, 924, \{{\leq14},{\leq7}\}]$ (stabilizer weights~14 and 9) and $[22880,3432,\{{\leq8},{\leq16}\}]$ (weights 16 and 10), and CSS codes with highly irregular stabilizer weight distributions such as $[571,199,\{5,5\}]$. For the latter, I develop a weight-reduction method to deal with rare heavy stabilizers. Finally, I show how to extract four-term (length three) chain complexes that can be interpreted as CSS codes with a metacheck.

翻译：本文提出了一种生成具有有趣码参数的CSS码族的方法。研究对象是Coxeter群（包括有限与无限、可约与不可约情形），以及一种几何动机驱动的Coxeter群元素偏序——Bruhat序。已知Bruhat序与代数拓扑存在联系：它同时作为面偏序集，刻画了正则CW复形中$p$维胞腔的包含关系，这一特性使其在量子纠错码设计中具有重要意义。借助Coxeter群特有的Bruhat面偏序区间结构，本文证明了对应的链复形可转化为多类CSS码。根据具体构造方法，可获得具有受控稳定子权重的CSS码（及其码族），例如$[6006, 924, \{{\leq14},{\leq7}\}]$（稳定子权重为14和9）与$[22880,3432,\{{\leq8},{\leq16}\}]$（权重为16和10），以及具有高度非均匀稳定子权重分布的CSS码如$[571,199,\{5,5\}]$。针对后者，本文开发了权重约简方法以处理罕见的重型稳定子。最后，展示了如何提取可解释为带元校验的CSS码的四项（长度为三）链复形。