The Honeycomb Framework for Code Bounds

Abstract: We introduce the honeycomb hierarchy, a representation-theoretic framework that gives new asymptotic upper bounds on $R_2(δ)$. Its first level is the two-row hyperoctahedral representation graph associated with type $S<sup>{(n-k,k)}$. Retaining every two-row irreducible and every coordinate box-transfer channel, together with a moving-projection theorem, yields an explicit four-parameter exponent $κ<em>{\mathrm{HC}}$. 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 $κ</em>{\mathrm{bin}}=\min{κ<em>{\mathrm{CW}},κ_H}$, which uses a constant-weight branch $κ</em>{\mathrm{CW}}$. Replacing only the whole-cube branch by the honeycomb bound gives $$κ<em>{\mathrm{best}}=\min{κ</em>{\mathrm{CW}}, κ<em>{\mathrm{HC}}}$$. 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 κH(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.

• The paper develops a complete two-row hyperoctahedral representation graph and a four-parameter honeycomb exponent that strictly improves earlier moving-projection bounds and 2MQC for every nontrivial relative distance.
• The framework recovers the optimized second MRRW exponent on an exact symmetric slice while proving stronger analytic bounds through boundary, one-sided, and entropy-balanced perturbations.
• The paper builds Horn–channel and anchored moment hierarchies that generalize to higher representation depth and alphabet size, with finite convergence guaranteed but stronger higher-level numerical gains not yet fully certified.

Background and problem

The rate–distance function $R_2(\delta)$, the supremal asymptotic rate of binary codes with relative minimum distance $\delta$, remains bounded above and below by quantities that differ by an exponential factor in block length. The strongest general upper bounds have long come from Delsarte's association-scheme linear program in its McEliece–Rodemich–Rumsey–Welch form [MRRW77], with $M_1(\delta)$ and the fully optimized second exponent $M_2(\delta)$ defined by minimizing a one-parameter objective $F_\delta(\tau)$. Recently, two independent developments reset the benchmark. First, a moving-projection method applied to representation graphs on the Hamming cube produced a whole-cube exponent $\kappa_H$ and a constant-weight exponent $\kappa_{\mathrm{CW}}$, whose pointwise minimum $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$ strictly improves $M_2$. Second, Alrabiah and Guruswami introduced classical–quantum channel converses: an unmasked mixed-qubit-channel exponent MQC and a masked variant 2MQC that strictly improves $M_2$.

This paper, "The Honeycomb Framework for Code Bounds" by Gay, Granha Jeronimo, and Liu, enlarges the representation graph underlying $\delta$0 from a one-dimensional boundary family to the complete set of two-row hyperoctahedral irreducibles compatible with a moving stabilizer type $\delta$1. It derives a four-parameter asymptotic exponent $\delta$2, proves strict improvements over both $\delta$3 and 2MQC, embeds everything into a finitely convergent two-axis hierarchy (representation depth and anchor depth), and constructs a complementary Horn–channel hierarchy whose second level reproduces $\delta$4 exactly and whose third level is unconditional.

A profile-optimized moving-projection theorem

The finite engine is an abstract group-theoretic theorem. For a transitive finite-group action with stabilizer $\delta$5 and an irreducible $\delta$6-representation $\delta$7, the authors retain a set $\delta$8 of inequivalent irreducibles $\delta$9 satisfying multiplicity one, define directed contraction probabilities $M_1(\delta)$0 from coordinate intertwiners, and impose exact dimension balance $M_1(\delta)$1. A key departure from prior Perron-vector formulations is that an arbitrary positive profile $M_1(\delta)$2 is retained: a directed Collatz–Wielandt certificate $M_1(\delta)$3 controls the spectral term, while a blockwise trace argument yields an effective dimension $M_1(\delta)$4 rather than the total ambient dimension. If $M_1(\delta)$5, every code satisfies

$M_1(\delta)$6

The proof is a positive-definite kernel decomposition: a contractive combination $M_1(\delta)$7 of box-transfer intertwiners satisfies an exact identity $M_1(\delta)$8, splitting $M_1(\delta)$9 into three nonnegative kernels summed over $M_2(\delta)$0. The authors note explicitly that profile optimization can strictly beat the Perron-effective bound; equality holds only for complete normalized components, so genuine gains arise from proper truncations or profile trading of spectral slack against effective dimension.

The two-row hyperoctahedral graph

The Hamming cube is realized as $M_2(\delta)$1 with $M_2(\delta)$2. A central structural result classifies all bipartitions $M_2(\delta)$3 whose restriction to $M_2(\delta)$4 contains the fixed Specht module $M_2(\delta)$5: both components must have at most two rows, and the Littlewood–Richardson coefficient equals one exactly when the doubled spins satisfy the triangle inequality $M_2(\delta)$6. Thus the two-row vertex set is forced by compatibility, not chosen ad hoc; it fills a three-dimensional region parameterized by $M_2(\delta)$7 at fixed $M_2(\delta)$8.

Tensoring by the natural signed representation transfers exactly one box between components (multiplicity-free Pieri rule). The contraction coefficient along each of the four forward channels factors into a color probability, a Young branching ratio, and a half-spin Wigner $M_2(\delta)$9 Racah overlap; the closed formulas give squared coefficients $F_\delta(\tau)$0 and $F_\delta(\tau)$1 and verify exact dimension balance. At the boundary $F_\delta(\tau)$2, these reproduce the whole-cube matrix of the previous construction, confirming that the earlier graph is precisely a boundary slice. The full graph is normalized at every vertex ($F_\delta(\tau)$3).

Assembling any connected truncation gives a finite-length algorithmic bound on $F_\delta(\tau)$4 valid for arbitrary codes—unlike higher-order LP hierarchies for linear codes, which collapse to Delsarte's program without linearity.

The first honeycomb exponent

Passing to the continuum via Følner boxes in the compatibility lattice, the four channel weights converge uniformly to expressions involving $F_\delta(\tau)$5, giving a spectral symbol

$F_\delta(\tau)$6

and entropy potential $F_\delta(\tau)$7, where $F_\delta(\tau)$8 is the color fraction, $F_\delta(\tau)$9 are normalized second-row fractions, and $\kappa_H$0. The first honeycomb bound is $\kappa_H$1. The paper concedes a scoping point here: "complete" refers to the finite two-row family under nondegenerate bulk scalings; additional equal-row boundary layers (e.g., $\kappa_H$2 with bounded row difference) are not captured by the quadruple $\kappa_H$3 and could only strengthen the bound, but are omitted.

Three identifications structure the comparison theory:

• Boundary recovery: setting $\kappa_H$4 recovers $\kappa_H$5 and $\kappa_H$6 exactly.
• One-sided face: perturbing a whole-cube minimizer by opening a single second row produces a spectral gain of order $\kappa_H$7 against an entropy cost of order $\kappa_H$8, proving $\kappa_H$9 strictly for every $\kappa_{\mathrm{CW}}$0.
• Entropy-balanced branch: distributing total angle proportionally to partition sizes, $\kappa_{\mathrm{CW}}$1, $\kappa_{\mathrm{CW}}$2, gives $\kappa_{\mathrm{CW}}$3 with an explicit expansion $\kappa_{\mathrm{CW}}$4; a second-variation argument shows this ray is uniquely second-order optimal among fixed-ratio perturbations.
• MRRW embedding: the symmetric slice $\kappa_{\mathrm{CW}}$5 has spectral boundary and objective equal to $\kappa_{\mathrm{CW}}$6, so its infimum is exactly the fully optimized second MRRW exponent $\kappa_{\mathrm{CW}}$7. This is a notable claim: the honeycomb variational problem contains the entire fifty-year-old second LP optimization as an exact symmetric slice.

The combined statement keeps the constant-weight branch unchanged and replaces only the whole-cube component:

$\kappa_{\mathrm{CW}}$8

with the stronger pointwise strict inequality $\kappa_{\mathrm{CW}}$9 throughout $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$0, and the exact identity $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$1. All comparisons are analytic; no numerical experiment enters any theorem.

The Horn–channel hierarchy

A complementary construction realizes the same Horn geometry through explicit quantum channels. For positive semidefinite $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$2 with $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$3, the states built from $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$4 and its sign-flip conjugate form a binary-input output-symmetric channel whose pretty-good-measurement bit error is $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$5 and whose uniform-prior Holevo information is $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$6. Applying the Alrabiah–Guruswami criterion gives nested exponents $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$7 indexed by matrix size $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$8. The scalar level is $$\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$$9; the $M_2$0 level equals $M_2$1 exactly, via a Bloch-vector identification in which the feasible spectra trace the Horn triangle and the overlap parameter matches $M_2$2; the $M_2$3 level is an unconditional further improvement, with numerical gains up to roughly $M_2$4 bits per coordinate around $M_2$5. A rank-opening proposition shows that whenever a minimizer is positive definite, adding one dimension strictly decreases the optimum, explaining the observed monotone descent. Importantly, no equality between the channel hierarchy and the higher-row recoupling symbol is asserted beyond level one.

Finite completeness via anchored moment relaxations

Two axes generalize the framework. Increasing representation depth allows more rows, making Littlewood–Richardson multiplicities nontrivial—their fibers are Knutson–Tao hives—and scalar transfers become matrices acting on hive fibers. A frame-profile theorem handles arbitrary multiplicity: a Parseval frame $M_2$6 together with column-contractive coefficients $M_2$7 satisfying harmonic equations $M_2$8 yields the same shape of cardinality bound, recovering the scalar theorem as the rank-one case.

Increasing anchor depth localizes two quadratic certificates—a spectral polynomial $M_2$9 and a block-trace polynomial $M_2$0, both nonnegative on code indicators—as localizing matrices inside the stable-set moment hierarchy on the forbidden-distance graph. Each localizer entry conditions the honeycomb inequality on the presence of up to $M_2$1 anchor words in the code. The resulting bounds $M_2$2 are monotone in both axes, sound, and finitely convergent: $M_2$3 once $M_2$4. The proof uses inclusion–exclusion idempotents $M_2$5 of degree at most $M_2$6 in the stable-set quotient algebra, decomposing any moment functional as a mixture over codes. This convergence statement is deliberately weak in one respect: it does not claim that any fixed low row or anchor level determines the asymptotic rate $M_2$7. The extension to $M_2$8-ary alphabets proceeds through $M_2$9 with no essential change.

Limitations and open questions

Several qualifications are stated plainly in the paper. The asymptotic analysis covers nondegenerate bulk scalings and their ordinary boundary limits; bounded-row-difference boundary layers remain unexplored and could yield further improvements. The finite convergence of the anchored hierarchy is at anchor order $\delta$00—exponential in general—and whether a bounded level already yields a sharp asymptotic exponent is left open. Whether the full one-point representation family (all stabilizer types) is itself complete is also unknown; completeness is obtained only through joint-moment localization. The rank-three Horn–channel curve is supported by floating-point optimization (Sobol multistarts, SLSQP polishing, real orthogonal subfamily) that the authors explicitly do not certify: no interval arithmetic or global branch-and-bound proof of the displayed decimal decreases is provided, and the reported gains compare independently optimized infima. Certifying the global numerical curves is identified as the concrete next step, alongside a comprehensive analysis beyond the present proof-of-concept scope.

Conclusion

The paper replaces a boundary-restricted representation graph with the complete compatible two-row family, yielding an explicit four-parameter exponent that dominates the previous combined moving-projection bound, strictly improves 2MQC at every nontrivial distance, and contains the optimized second MRRW problem as an exact symmetric slice. The Horn–channel construction makes higher-row optimization unconditional and agrees with the first honeycomb level exactly, while the frame-profile and anchored moment machinery places the whole program in a monotone, finitely convergent hierarchy extending to all alphabet sizes. The gap between certified analytic results and numerically observed gains at levels above the first remains the principal open issue raised by the work.