The Honeycomb Framework for Code Bounds

We introduce the honeycomb hierarchy, a representation-theoretic framework that gives new asymptotic upper bounds on $R_2(\delta )$. 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 ${\kappa }_{\mathrm{H}\mathrm{C}}$. The earlier whole-cube exponent ${\kappa }_H$ 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 ${\kappa }_{\mathrm{b}\mathrm{i}\mathrm{n}}=\min \{{\kappa }_{\mathrm{C}\mathrm{W}},{\kappa }_H\}$, which uses a constant-weight branch ${\kappa }_{\mathrm{C}\mathrm{W}}$. Replacing only the whole-cube branch by the honeycomb bound gives ${\kappa }_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}=\min \{{\kappa }_{\mathrm{C}\mathrm{W}},{\kappa }_{\mathrm{H}\mathrm{C}}\}$. We prove, on $0<\delta <1\mathrm{/}2$, $$R_2(\delta )\le {\kappa }_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}(\delta )\le {\kappa }_{\mathrm{b}\mathrm{i}\mathrm{n}}(\delta )\le R_{2\mathrm{M}\mathrm{Q}\mathrm{C}}(\delta )<M_2(\delta ),{\kappa }_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}(\delta )\le \min \{{\kappa }_{\mathrm{C}\mathrm{W}}(\delta ),{\kappa }_{\mathrm{b}\mathrm{a}\mathrm{l}}(\delta )\}<R_{2\mathrm{M}\mathrm{Q}\mathrm{C}}(\delta ),{\kappa }_H(\delta )=R_{\mathrm{M}\mathrm{Q}\mathrm{C}}(\delta )\mathrm{.}$$ 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 $2\times 2$ level is ${\kappa }_{\mathrm{H}\mathrm{C}}$ and whose $3\times 3$ 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.