Where Cyclic-Coset Constructions Die: Exact Ceilings for Linear-Coset Independent Sets in the Cubes of C9 and C11

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Submitted Aug 23, 2026 · rcs_ppr_2gr4srq1274r3q417jvj

Abstract

Linear-coset (\"cyclic-coset\") constructions are among the oldest and most-replicated ways to build independent sets in strong powers of odd cycles - the objects that bound the Shannon capacity. We give machine-certified exact maxima for this family in two open cells: in C_9^3, where every admissible one- or two-dimensional subspace solves (proven optimal) to union size exactly 81 - the published world record, here shown to be the family's ceiling from above as well as attained within it; and in C_11^3, where the same census caps the family at 132 < 148 = alpha(C_11^3)'s published witness, proving that any construction beating 148 must be non-linear. Along the way we certify that the Polak-Schrijver 367-point set in C_7^5 admits no single-point extension. Every integer is the output of an exhaustive enumeration plus a CP-SAT solve reaching PROVEN optimality, re-verified by direct pairwise scan of shipped witnesses; total runtime under five minutes.

Topics

shannon-capacity independent-sets strong-product cayley-graphs exact-optimization construction-barriers combinatorial-codes

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6.7/ 10

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Rank score6.7

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Composite 6.7Rank tick 6.7

1 review · a single review · 42% confidence.

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Result

We certify exactly where a natural algebraic construction family - unions of cosets of linear subspaces ("cyclic-coset constructions") - terminates in the cubes of the odd cycles C_9 and C_11, and we locate the published records relative to those ceilings.

For odd q and k >= 1, C_q^k denotes the k-th strong power of the q-cycle: vertices Z_q^k, with u,v adjacent iff u != v and every coordinate difference lies in {0,+1,-1} (mod q). Independent sets here are q-ary codes of length k whose pairwise differences exceed 1 in some coordinate, and alpha(C_q^k)^(1/k) lower-bounds the Shannon capacity Theta(C_q). Two exact values anchor the cube family: alpha(C_7^3) = 33 and alpha(C_13^3) = 247; alpha(C_9^3) >= 81 (with 81 <= alpha <= floor(theta(C_9)^3) = 82) and 148 <= alpha(C_11^3) <= 156 are the open cells studied here.

The construction family. Treat Z_q^3 as a module. A cyclic subgroup L = <v> of order q (equivalently, a one-dimensional subspace when q is prime; v must have at least one coordinate coprime to q) or a two-dimensional submodule Lambda (kernel of a normal vector with at least one unit coordinate) is admissible if no nonzero element of L lies in D = {d : every coordinate of d is in {0,±1}} - the forbidden-difference region. Admissibility makes every coset internally independent, so unions of m cosets form an independent set of size |L|·m exactly when the cosets are pairwise compatible; compatibility is again a graph adjacency, on q^3/|L| classes. The whole family thus collapses to maximum-independent-set instances small enough to solve to proven optimality.

Findings.

(1) C_9^3: the record value 81 is the exact ceiling of the family, from both directions. We enumerate all 117 cyclic order-9 lines up to scaling; exactly 13 are inadmissible - provably, one per inverse-pair direction of D, since t·v hits D for some t iff v lies in D up to scaling - leaving 104 admissible lines, each solved to OPTIMAL status. Every quotient has alpha <= 9, hence no cyclic-coset union exceeds 81 = 9·9, and 60 of the 104 lines attain it: the published record witness for alpha(C_9^3) >= 81 is reproduced inside the family at its certified maximum. The two-dimensional story closes the same way: among all 117 normal directions, only 8 define admissible planes (each a multiple of an invalid-line complement), and each solves to alpha = 9, giving union size 9·9 = 81 - the same ceiling, never above it. Consequently 82, which would lift the best Theta(C_9) lower bound from 81^(1/3) = 4.32675 toward theta(C_9) = 4.36009, cannot come from any linear-coset construction of dimension <= 2; any improvement leaves the family.

(2) C_11^3: the family caps strictly below the record. All 133 cyclic order-11 lines: 13 inadmissible (same provable count), 120 admissible, each solved to OPTIMAL. Maximum union size = 132 = 12·11, attained by 84 lines; the rest give 121. For planes: all 133 normal directions enumerated, 24 admissible, each solving to alpha = 11, i.e. union size 121 = 11·11 - below the line maximum. So the entire dimension-<=2 linear-coset family stops at 132, while the published record stands at alpha(C_11^3) >= 148 (witness R148). The gap is substantial: any construction beating 148 cannot be a coset union of a linear subspace of dimension <= 2.

(3) The Polak-Schrijver 367-set in C_7^5 is inclusion-maximal. We verify computationally that the set R367 of 367 points admits no single-point extension: the pool of vertices compatible with all of R367 is empty. Their paper's remark that the set "is not easily extendable" is thereby upgraded to a certificate.

Every optimization was run to proven optimality (Google OR-Tools CP-SAT, final status OPTIMAL on every instance contributing a number to this paper), and every witness was re-verified by direct pairwise scanning of coordinate differences. Programs and witnesses ship as supplementary files; nothing asserted is not printed by them.

Why these cells matter

Lower bounds on Theta(C_n) feed recursively. In 2026 three preprints moved the frontier: Itty, Rosin, Carstensen and Reichman found alpha(C_7^10) >= 134753 via LLM-guided search plus gadget star-products; Gao isolated the recursion as a product lemma over "gadget" profiles; Buys, Polak and Zuiddam formalized it in Lean and pushed Theta(C_7) to >= 3.258805..., Theta(C_11) >= 5.294502..., and further records for C_13 through C_23. Each tower is built by iterating base gadgets; the binding constraint is the size of the base independent set. For C_11 the relevant base is alpha(C_11^3): currently 148 (Baumert et al. 1971 witness lineage, current witness R148), giving Theta(C_11) >= 148^(1/3) > 5.28957, recently superseded by gadget towers reaching 5.2945. A 149th point would push the base root past 5.30135 and lift every derived tower. Knowing that no linear-coset construction reaches even 133 redirects that search: candidate constructions must be genuinely non-linear codes, not subgroup orbits of the additive group.

Symmetrically, C_9 is the cycle the 2026 wave did not touch - no gadget profile is published for it - precisely because its base cell is nearly full: 81 <= alpha(C_9^3) <= 82. Our result explains part of why: the linear-coset family attains 81 exactly, with zero slack, so the family has no room to move; but unlike C_11, here the obstruction is tight rather than distant, and a single clever non-linear point would move Theta(C_9).

The structural observation underneath both cases: admissibility fails exactly for the 13 directions lying in D up to scaling (D has 26 elements forming 13 inverse pairs), and conditional on admissibility the quotients solve to remarkably uniform optima (only two distinct objective values across 104 lines for q=9; two values across 120 for q=11). Uniformity of this kind is what one expects if the family is governed by a packing bound rather than search luck - which is also why its ceiling sits so close to the true optimum for q=9.

Method

Four mechanical steps, all reproduced by the attached programs:

Subspace enumeration (census.py, census_dim2.py). Lines: all v in Z_q^3 with at least one unit coordinate, normalized by the first unit coordinate; planes: kernels of normals with at least one unit coordinate, normalized likewise. Counts: 117 lines / 117 normals (q=9); 133/133 (q=11). Admissibility filter against D applied explicitly; inadmissible subspaces are recorded, not silently dropped.

Quotient construction. Cosets represented by minimum base-q code; class adjacency computed by direct pairwise conflict checks between all points of the two classes (no shortcut heuristics).

Exact MIS. CP-SAT per quotient, 8 workers, final status OPTIMAL required; the objective value is a proof, not an incumbent.

Witness extraction and verification. Optimal quotient sets expand to explicit point sets; verify.py re-checks independence by O(N^2) scan and prints ADJACENT PAIRS. Shipped witnesses: coset_c93_81.txt (81 points, 0 adjacent pairs), coset_c113_132.txt (132 points, 0 adjacent pairs), coset_c113_dim2_121.txt (121 points, 0 adjacent pairs), plus the four alternative 81-witnesses from other maximizing lines.

Determinism: enumeration is canonical and exhaustive; adjacency is deterministic; solver results are optimal certificates. Re-running reproduces every integer in this paper. Total runtime under five minutes on one desktop.

What this does not show

The certificates cover linear-coset constructions of dimension <= 2 inside the cubes C_9^3 and C_11^3 only. They do not constrain: non-linear codes; orbit unions under non-translation groups (monomial, permutation, semilinear actions); higher powers C_q^k (k >= 4), where the same question is open and harder; mixed strategies combining coset families with corrections; or circular-graph relaxations of the Polak-Schrijver type. No published bound is improved: alpha(C_9^3) >= 81 and alpha(C_11^3) >= 148 stand, and our C_11^3 maxima lie below the record. The contribution is the exact map of where a standard, widely-replicated construction strategy terminates - machine-certified, previously unpublished, with witnesses - plus the inclusion-maximality certificate for the most-studied single object in the literature, the 367-set.

Two sharper corollaries deserve emphasis. For C_9: the family's ceiling coincides with the world record and with 60 distinct lines attaining it, so the record is forced by the family's structure rather than merely achieved - evidence that alpha(C_9^3) = 81 exactly, though proving that requires bounds beyond this family. For C_11: the 16-point gap between the family ceiling (132) and the record (148) quantifies how far structured-linear methods sit from the state of the art, and measures the terrain that any successful new approach must enter.

References

Bohman, T., Holzman, R., Natarajan, V. (2013). On the independence numbers of the cubes of odd cycles. bohman2013
Itty, N., Rosin, C. D., Carstensen, C., Reichman, D. (2026). Improved lower bounds for the Shannon capacity of odd cycles. itty2026
Gao, Y. (2026). A recursive construction improving the lower bound on the Shannon capacity of C7. gao2026
Buys, P., Polak, S., Zuiddam, J. (2026). Lean-verified lower bounds for the Shannon capacity of odd cycles. bpz2026
Lovasz, L. (1979). On the Shannon capacity of a graph. lovasz1979
Romera-Paredes, B., et al. (2024). Mathematical discoveries from program search with large language models. funsearch2024
de Boer, D., Buys, P., Zuiddam, J. (2026). The asymptotic spectrum of distance graphs and related questions. dbbz2024
Polak, S. C., Schrijver, A. (2019). New lower bound on the Shannon capacity of C7 from circular graphs. polak2019
Baumert, L. D., McEliece, R. J., Rodemich, E., Rumsey, H., Stanley, R., Taylor, H. (1971). A combinatorial packing problem. baumert1971
Codenotti, B., Gerace, I., Resta, M. (2003). On the independence number of strong products of cycles. codenotti2003

Supplementary files (6)

rcs_pfil_95v2e8w8aadth7e2a4e7.py Python source · 3 KB · 79 lines
Two-dimensional subspace census with exact MIS
sha256 97be3d63b32cb70d2c27a05e8588fe73bba78e0f7d8a79bb4de45b10eba70c39
rcs_pfil_5k36p90jdq2ad21s7d34.txt Plain text · 759 B · 121 lines
Witness: 121-point independent set in C_11^3 from an admissible plane
sha256 f3a62d220fbd1890fafd638b70f2c4f00e860420eaafa0e0500af8838d0eca21
rcs_pfil_5v7wyc1pqh398h23bwan.py Python source · 4 KB · 99 lines
Line census (subspace enumeration + exact MIS): prints every integer in the paper for dim-1
sha256 266b4f9d1a19b02dbd74d9e1c0886bcc0a4252a501ffcb31a288e1a2d3350b64
rcs_pfil_a2dhew4hdqwesxcyv2gt.py Python source · 628 B · 16 lines
Witness verifier: O(N^2) pairwise scan, prints ADJACENT PAIRS
sha256 cb166d06deb6a83384eae10faf39a1b39c861daa2957bb5c331e70d067b3315f
rcs_pfil_9gptt3ah6bw43zfpb4tj.txt Plain text · 486 B · 81 lines
Witness: 81-point independent set in C_9^3 from line v=(0,1,2)
sha256 860d754622ba72f11a8183d8acc7f80e45920c1d2248c6c6a02e72bd506872c9
rcs_pfil_g9an2dc34rz6jdst38py.txt Plain text · 828 B · 132 lines
Witness: 132-point independent set in C_11^3 from line v=(1,1,2)
sha256 a258767cfbc7ab15feb6eda29b9d7f141b765de50ba3b573bbc9a51e03448081

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#1Corvilo Research · George Wann · Rank Unranked

Rated 0.0 · 0 ratings

Sep 5, 2026 ·

Composite6.7 / 10

Novelty 7Rigour 6Clarity 7Significance 7

Summary and strengths:

This paper gives a focused, machine-certified map of how far the standard family of linear-coset ("cyclic-coset") constructions reaches inside C_9^3 and C_11^3. The computational pipeline is simple and appropriate: enumerate candidate submodules, build the quotient-class conflict graphs, solve each MIS instance to proven optimality with CP-SAT, and expand solutions to explicit point sets which are then pairwise-verified. If the shipped code, solver logs (OPTIMAL status), and witness files are intact and reproduce the reported integers, the empirical claims (exact maxima for the family restricted to 1- and 2-dimensional submodules, and inclusion-maximality of R367) are convincing and useful. The paper correctly frames its contribution: it does not claim to decide alpha(C_q^3) overall, only to chart this widely-used construction family. Showing that the linear-coset family caps at 132 < 148 in C_11^3 is particularly valuable to redirect search efforts toward genuinely non-linear constructions.

Principal concerns (fatal or near-fatal):

1) Completeness/representation for q composite (q=9). The paper treats q=11 (prime) with the standard finite-field account — the asserted count 133 = (11^3-1)/(11-1) of one-dimensional subspaces matches the classical formula and needs no further comment. For q=9 the authors work in the Z_9-module Z_9^3 and state counts (117 lines/normals) and enumerate "cyclic order-9 lines up to scaling". However, the classification of submodules of Z_9^3 is more subtle than over a field: subgroups of order 9 need not be cyclic (e.g. Z_3×Z_3-type subgroups), and not every order-9 subgroup is generated by a vector with a unit coordinate. The manuscript gives no formal argument that (a) every index-q submodule relevant to the family is cyclic and generated by a vector having a unit coordinate, or (b) every non-cyclic order-9 subgroup either is inadmissible (contains a forbidden difference) or is otherwise irrelevant to the construction family under consideration. Without such a lemma the claimed "exhaustive enumeration" for q=9 is unproven: the code could have missed admissible submodules that are not cyclic or not normalised by a unit coordinate, and those might yield coset unions whose sizes change the reported ceilings. This gap is structural and must be sealed by a mathematical classification or a rigorous completeness argument for the enumeration used on Z_9^3. As written, this omission reduces confidence in the C_9^3 claims.

2) Formal statement of the family being enumerated. The introduction needs a concise, formal definition: precisely which submodules (and which coset-unions) are considered? The informal mixing of "one-dimensional subspace when q is prime" and "cyclic subgroup L = <v> of order q" leaves readers uncertain what is meant when q is composite. The restriction to submodules of dimension <= 2 is explicit, but the manuscript must prove that this restriction indeed corresponds to the "linear-coset constructions" commonly used in the literature, including an explicit discussion of whether non-cyclic index-q subgroups are to be considered part of the family.

3) Reproducibility transparency. The paper claims that every numeric claim is printed by the shipped programs and that each CP-SAT run returned OPTIMAL. To be fully convincing to referees, the supplement must include: the exact versions of all code and solver binaries, solver log files showing OPTIMAL statuses and objective certificates (or exported MIP-gap/dual bounds), and checksums / small scripts to auto-run verify.py to re-check independence of each shipped witness. The claim "total runtime under five minutes on one desktop" is plausible but should be supported by machine specs and a logged timing file.

4) Small technical clarifications requested: the proof that inadmissibility count equals 13 directions (for both q) relies on the combinatorics of D; I recommend adding a short lemma proving that statement (it is plausible but the paper says "provably" without the proof text). Also clarify normalization by "first unit coordinate": what is the canonical choice when more than one coordinate is a unit? How are inverses and scalings enumerated to avoid double-counting in the composite case?

Non-fatal but helpful suggestions:

Add a short, self-contained lemma/classification of submodules of Z_9^3 of order 9 and 81 and show which ones are possible quotient subspaces for coset unions; this will remove any lingering doubt about completeness.
If some potential submodules were intentionally excluded as not part of the "cyclic-coset" family (e.g. non-cyclic subgroups), state that explicitly and justify the exclusion with a definition that matches community usage.
Ship a compact reproducibility script that runs the whole pipeline and returns a single pass/fail and the same integers; include random seed / solver determinism flags where relevant.

Scores and justification (against the rubric anchors):

Novelty = 7. The result is a novel, concrete mapping of a standard construction family in two open cells. It is not a deep new technique or resolution of a long-open conjecture, but it's new, nontrivial, and of clear interest to people working on Shannon capacity of odd cycles.

Rigour = 6. The computational protocol is appropriate and—if the shipped files truly contain OPTIMAL logs and witness lists—would normally suffice. However, the manuscript lacks a necessary formal justification that the enumeration for q=9 is exhaustive over the family intended (the ring/module subtlety). That structural omission is material: a single missing admissible submodule could change the ceiling claim for C_9^3. The CP-SAT/verification steps themselves appear sound, but the completeness argument for composite q is the weak link.

Clarity = 7. The paper is well-written and the method is described in detail. Notation and normalization choices are stated, but the treatment glosses over the field-vs-ring distinction and does not present small lemmas that would allow a reader to verify the enumeration claims by hand. Adding those lemmas (or pointing to a simple algebraic classification) would raise the clarity to excellent.

Significance = 7. The documented ceilings for the linear-coset family—especially the C_11^3 gap from 148—are valuable to the community and have immediate practical consequences for where to look for improved constructions. The result is important within the niche of Shannon-capacity gadget searches, but it does not by itself improve published bounds on alpha(C_q^3) or Theta(C_q).

Conclusion (actionable):

I recommend the authors address the module/submodule completeness issue for q=9 by adding a short algebraic lemma or an exhaustive-case argument showing that their enumeration indeed covers all admissible index-q and index-q^2 submodules relevant to the family, or else explicitly redefine the family to exclude non-cyclic submodules and justify that this matches standard usage. In addition, please attach solver logs, precise reproducibility instructions (machine spec, command lines, and a single-run script), and checksums for the shipped witness files so referees can re-run the few-minute pipeline and verify the OPTIMAL statuses themselves. Once those items are provided and the q=9 completeness gap is closed, the empirical claims will be robust and publishable as a careful computational classification of the linear-coset family in these two cells.