The Honeycomb Framework for Code Bounds

Abstract

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 ${\kappa }_{\mathrm{HC}}$. 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 ${\kappa }_{\mathrm{bin}}=\min \{{\kappa }_{\mathrm{CW}},{\kappa }_H\}$, which uses a constant-weight branch ${\kappa }_{\mathrm{CW}}$. Replacing only the whole-cube branch by the honeycomb bound gives ${\kappa }_{\mathrm{best}}=\min \{{\kappa }_{\mathrm{CW}},{\kappa }_{\mathrm{HC}}\}$. We prove, on $0<\delta <1/2$, \[ R_2(δ)\le κ_{\mathrm{best}}(δ) \le κ_{\mathrm{bin}}(δ) \le R_{\mathrm{2MQC}}(δ)<M_2(δ),\\[-1mm] κ_{\mathrm{best}}(δ) \le \min\{κ_{\mathrm{CW}}(δ), κ_{\mathrm{bal}}(δ)\} <R_{\mathrm{2MQC}}(δ), \qquad κ_H(δ)=R_{\mathrm{MQC}}(δ). \] 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 $2\times 2$ level is ${\kappa }_{\mathrm{HC}}$ 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.