用于码界的蜂窝框架
The Honeycomb Framework for Code Bounds
AI总结：
该研究提出蜂窝层级框架，得到新的 $R_2(\delta)$ 渐近上界，通过优化指数得到更优的 $\kappa_{\mathrm{best}}$，其层级可拓展，能改进现有通用界，为更紧的界提供途径。
AI中文摘要：
我们提出了蜂窝层级（honeycomb hierarchy）这一表示论框架，它为 $R_2(\delta)$ 提供了新的渐近上界。该框架的第一层是与类型 $S^{(n-k,k)}$ 相关的两行超八面体表示图。保留所有两行不可约表示、所有坐标盒转移通道，结合移动投影定理，可得到显式四参数指数 $\kappa_{\mathrm{HC}}$。早期的全立方指数 $\kappa_H$ 是该优化的边界限制，而完全优化的第二个 MRRW 指数 $M_2$ 是精确对称切片。此前最优曲线是组合的 $\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$，它使用常重量分支 $\kappa_{\mathrm{CW}}$。仅用蜂窝界替换全立方分支，得到 $\kappa_{\mathrm{best}}=\min\{\kappa_{\mathrm{CW}}, \kappa_{\mathrm{HC}}\}$。我们证明，在 $0<\delta<1/2$ 范围内，$R_2(\delta)\le \kappa_{\mathrm{best}}(\delta) \le \kappa_{\mathrm{bin}}(\delta) \le R_{\mathrm{2MQC}}(\delta)<M_2(\delta)$，且 $\kappa_{\mathrm{best}}(\delta) \le \min\{\kappa_{\mathrm{CW}}(\delta), \kappa_{\mathrm{bal}}(\delta)\} <R_{\mathrm{2MQC}}(\delta)$，$\kappa_H(\delta)=R_{\mathrm{MQC}}(\delta)$。该层级还有两个拓展方向：增加表示深度，将蜂巢上的标量转移替换为矩阵值转移；增加锚点深度，将其定位于稳定集层级。所得界在两个方向上均单调，最终可恢复 $A_2(n,d)$。互补的 Horn 通道层级产生矩阵优化，其 $2\times2$ 层级对应 $\kappa_{\mathrm{HC}}$，$3\times3$ 层级对应更强的界。即使在低层级，这些界也可用于改进此前最强的通用界，而蜂窝框架为获得更紧的界提供了途径。
英文摘要：
We introduce the honeycomb hierarchy, a representation-theoretic framework that gives new asymptotic upper bounds on $R_2(δ)$. Its first level is the two-row hyperoctahedral representation graph associated with type $S^{(n-k,k)}$. Retaining every two-row irreducible and every coordinate box-transfer channel, together with a moving-projection theorem, yields an explicit four-parameter exponent $κ_{\mathrm{HC}}$. The earlier whole-cube exponent $κ_H$ is a boundary restriction of this optimization, whereas the fully optimized second MRRW exponent $M_2$ is an exact symmetric slice. The prior best curve is the combined $κ_{\mathrm{bin}}=\min\{κ_{\mathrm{CW}},κ_H\}$, which uses a constant-weight branch $κ_{\mathrm{CW}}$. Replacing only the whole-cube branch by the honeycomb bound gives $κ_{\mathrm{best}}=\min\{κ_{\mathrm{CW}}, κ_{\mathrm{HC}}\}$. We prove, on $0<δ<1/2$, \[ R_2(δ)\le κ_{\mathrm{best}}(δ) \le κ_{\mathrm{bin}}(δ) \le R_{\mathrm{2MQC}}(δ)<M_2(δ),\\[-1mm] κ_{\mathrm{best}}(δ) \le \min\{κ_{\mathrm{CW}}(δ), κ_{\mathrm{bal}}(δ)\} <R_{\mathrm{2MQC}}(δ), \qquad κ_H(δ)=R_{\mathrm{MQC}}(δ). \] The hierarchy has two further directions. Increasing the representation depth replaces scalar by matrix-valued transfers on the hive. Increasing the anchor depth localizes it in a stable-set hierarchy. The resulting bounds are monotone in both directions and eventually recover $A_2(n,d)$. A complementary Horn--channel hierarchy gives matrix optimizations whose $2\times2$ level is $κ_{\mathrm{HC}}$ and whose $3\times3$ level is a stronger bound. Already at low levels, they can be used to improve the strongest previous general bounds, while the honeycomb framework provides a route towards tighter bounds.
