Given the importance of hyperbolic quantum color codes and Euclidean quantum color codes, this paper considers the study of the former codes on compact surfaces with genus \(g \ge 2\) from the mathematical point of view. Identifying the normal subgroup in the decomposition of the full symmetry group of the \(\{p,3\}\) tessellation is relevant because it provides the algebraic structure for identifying and constructing a class of linear shrunk hyperbolic quantum color codes. Under this assumption, the normal subgroup’s presentation, the whole process’s kernel, is derived from the Reidemeister–Schreier method. As a result, we present a class of regular normal hyperbolic quantum color codes derived from the \(\{6j,3\}\) tessellation with encoding rate going asymptotically to 1. The regular tessellation \(\{6j,3\}\) includes the two types of tessellations: (1) the densest tessellation \(\{12i-6,3\}\) when \(j=2i-1\) and (2) the tessellation \(\{12i,3\}\) when \(j=2i\), for \(i \in \mathbb {N}\). An analysis of the minimum distance achieved by this class of regular normal hyperbolic quantum color codes is performed.

鉴于双曲量子颜色代码和欧几里得量子颜色代码的重要性，本文从数学角度考虑对属\(g \ge 2\)紧致表面上的双曲量子颜色代码的研究。识别\(\{p,3\}\)曲面细分的完全对称群分解中的正规子群是相关的，因为它提供了用于识别和构造一类线性收缩双曲量子颜色代码的代数结构。在此假设下，正规子群的表示，即整个过程的核心，是从Reidemeister-Schreier方法推导出来的。因此，我们提出了一类从\(\{6j,3\}\)曲面细分导出的一类常规正态双曲量子颜色代码，其编码率渐近为 1。常规曲面细分\(\{6j,3\} \)包括两种类型的曲面细分：(1) 当 \(j=2i-1\) 时最密集的曲面细分\(\{12i-6,3\}\)和 (2) 当\(j=2i-1\)时的曲面细分\(\{12i, 3\}\)当\(j=2i\)时，对于\(i \in \mathbb {N}\)。对此类常规正双曲量子颜色代码实现的最小距离进行了分析。