We study square-base Calderbank--Shor--Steane (CSS) hypergraph-product codes as a finite-length class for regular high-girth quantum low-density parity-check (LDPC) design. For base matrices of small column weight, we give checkable conditions for regularity, rank deficiency, and short-cycle exclusion, and we present explicit column-weight-three and column-weight-four examples with Tanner girth 6 and 8. We also analyze circulant permutation matrix (CPM) lifts of this class. Using the standard voltage-sum criterion, we identify orthogonality-forced Tanner 8-cycles and show that CPM lifting cannot raise the Tanner girth beyond 8 when these cycles are present. As a representative finite-length instance, a randomized CPM lift of the girth-8 base construction gives a $[[28800,62]]$ girth-8 $(3,6)$-regular CSS-LDPC code. Under degeneracy-aware belief-propagation decoding with optional ordered-statistics-decoding-lite post-processing, this code produced zero decoding failures in $2.993\times 10^8$ independent trials at depolarizing probability $p=0.1402$; the Wilson 95% upper confidence bound is $1.28\times 10^{-8}$.

翻译：我们研究基于平方基的Calderbank-Shor-Steane（CSS）型超图乘积码，将其作为一类有限长正则高围长量子低密度奇偶校验（LDPC）码的设计方案。针对小列重基矩阵，给出了正则性、秩亏损及短环排除的可检验条件，并给出了列重为3和列重为4的显式示例，其Tanner围长分别为6和8。同时分析了该类码的循环置换矩阵（CPM）提升方法。利用标准电压和准则，识别出正交强迫型Tanner 8环，并证明当此类环存在时，CPM提升无法将Tanner围长提升至8以上。作为代表性有限长实例，对围长-8基构造进行随机化CPM提升，得到一个围长为8、$(3,6)$-正则的$[[28800,62]]$ CSS-LDPC码。在退化感知置信传播译码（辅以可选轻量级有序统计译码后处理）下，该码在去极化概率$p=0.1402$的$2.993\times 10^8$次独立实验中实现了零译码失败；Wilson 95%置信上限为$1.28\times 10^{-8}$。