Paper deep dive
Improved lower bounds for the Shannon capacity of odd cycles
Nathaniel Itty, Christopher D. Rosin, Chase Carstensen, Daniel Reichman
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
Last extracted: 7/25/2026, 1:15:38 AM
Summary
This paper presents improved lower bounds for the Shannon capacity of odd cycles C7, C11, and C13 by constructing larger independent sets in their strong graph powers. Specifically, the authors found independent sets of size 134,753 in C7^10, 21,909 in C11^6, and 62,530 in C13^6. These constructions were discovered through iterative interactions with a Large Language Model (LLM), demonstrating the utility of AI in combinatorial optimization. The new bounds are Theta(C7) > 3.258020, Theta(C11) > 5.289773, and Theta(C13) > 6.300109.
Entities (12)
Relation Signals (9)
C7 → haslowerbound → 3.258020
confidence 98% · Θ(C7)≥134753^{1/10}>3.258020
C11 → haslowerbound → 5.289773
confidence 98% · Θ(C11)≥21909^{1/6}>5.289773
C13 → haslowerbound → 6.300109
confidence 98% · Θ(C13)≥62530^{1/6}>6.300109
C7^10 → contains → 134753
confidence 95% · We construct independent sets of size 134753 in C7^{10}
C13^6 → contains → 62530
confidence 95% · 62530 in C13^{6}
C11^6 → contains → 21909
confidence 95% · 21909 in C11^{6}
Shannon capacity → isdefinedby → independent set
confidence 95% · The Shannon capacity Θ(G) of a graph G quantifies the maximum rate... It is lower bounded by α(G^d)^{1/d}... where α(G^d) is the independence number
Large Language Model → usedfordiscovery → independent set
confidence 90% · The constructions were discovered through iterative interactions with a Large Language Model (LLM)
ChatGPT-5.6 Sol Pro → usedby → Nathaniel Itty
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:The Shannon capacity $\Theta(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by $\alpha(G^d)^{1/d}$ for any $d$, where $\alpha(G^d)$ is the independence number of the $d$-th strong power of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, and $62530$ in $C_{13}^{6}$, improving the best known lower bounds for the Shannon capacity of these graphs to $\Theta(C_7)\geq 134753^{1/10}>3.258020$, $\Theta(C_{11})\geq 21909^{1/6}>5.289773$, and $\Theta(C_{13})\geq 62530^{1/6}>6.300109$. We also improve the best known lower bounds on the independence numbers of several individual strong powers of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions.
Tags
Links
- Source: https://arxiv.org/abs/2607.21517v1
- Canonical: https://arxiv.org/abs/2607.21517v1
Trouble viewing inline? Open PDF directly →
Full Text
36,451 characters extracted from source content.
Expand or collapse full text
Improved lower bounds for the Shannon capacity of odd cycles Nathaniel Itty Email: nathanielitty@gmail.com Worcester Polytechnic Institute Christopher D. Rosin Email: christopher.rosin@gmail.com Constructive Codes Chase Carstensen Worcester Polytechnic Institute Daniel Reichman Worcester Polytechnic Institute Abstract The Shannon capacity Θ(G) (G) of a graph G quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by α(Gd)1/dα(G^d)^1/d for any d, where α(Gd)α(G^d) is the independence number of the d-th strong power of G. We construct independent sets of size 134753134753 in C710C_7^10, 2190921909 in C116C_11^6, and 6253062530 in C136C_13^6, improving the best known lower bounds for the Shannon capacity of these graphs to Θ(C7)≥1347531/10>3.258020 (C_7)≥ 134753^1/10>3.258020, Θ(C11)≥219091/6>5.289773 (C_11)≥ 21909^1/6>5.289773, and Θ(C13)≥625301/6>6.300109 (C_13)≥ 62530^1/6>6.300109. We also improve the best known lower bounds on the independence numbers of several individual strong powers of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions. 1 Introduction Let G=(V,E)G=(V,E) be an undirected graph. The strong d-product of G, denoted by GdG^d has vertex-set VdV^d. Two distinct vertices (u1,…ud)(u_1,… u_d) and (v1,…vd)(v_1,… v_d) in VdV^d are connected by an edge in GdG^d if and only if for every i,ui=vii,u_i=v_i or (ui,vi)∈E(u_i,v_i)∈ E. Recall that for a graph H,H, α(H)α(H) is the size of the largest independent set in H. The Shannon Capacity of G, denoted Θ(G) (G) is defined as Θ(G)=supd∈ℕα(Gd)1/d. (G)= _d α(G^d)^1/d. It is well known that for any two positive integers d1,d2d_1,d_2, α(Gd1+d2)≥α(Gd1)α(Gd2)α(G^d_1+d_2)≥α(G^d_1)α(G^d_2), which implies that Θ(G)=limd→∞α(Gd)d. (G)= _d→∞ [d]α(G^d). The Shannon capacity was introduced by Shannon [15] in 1956 to quantify the communication capacity of a noisy channel. It has been extensively studied: for an excellent recent survey we refer the reader to [9]. For the even cycle with p-vertices, it is straightforward to prove that Θ(Cp)=p/2 (C_p)=p/2 (e.g., [12]). The celebrated work of Lovász [10] introduced the ϑ -function and used it to prove that for the 55-cycle C5C_5, Θ(C5)=5. (C_5)= 5. For odd cycles C2r+1C_2r+1 with r>2r>2 the value of Θ(C2r+1) (C_2r+1) is not known. Even for the simplest case, determining the value Θ(C7) (C_7) is considered a notoriously difficult problem that is still open despite extensive effort [12, 16, 11, 7]. The best known lower bound of Θ(C7) (C_7) was found [12] by clever construction of an independent set of size 367367 in C75C_7^5. The construction in [12] combines combinatorial ideas along with combinatorial search using an optimization solver (Gurobi). This immediately implies that Θ(C7)≥(367)1/5>3.2578. (C_7)≥(367)^1/5>3.2578. Here we provide modest improvements on the best known lower bound of C7C_7 and C11C_11, and a somewhat larger improvement for C13C_13: Theorem 1. We have that Θ(C7)>3.258020 (C_7)>3.258020, Θ(C11)>5.289773 (C_11)>5.289773 and Θ(C13)>6.300109 (C_13)>6.300109. Proof. This follows from the existence of an independent set of size 134753134753 in C710C_7^10 as well as the existence of an independent set of size 2190921909 in C116C_11^6. Finally, we found an independent set in C136C_13^6 of size 62530 which implies the lower bound on Θ(C13) (C_13). The independent sets can be found in the GitHub repository. A summary of the resulting lower bounds and comparison to previous lower bounds can be found in Table 1. ∎ Table 1: Our improvements to the lower bounds on the Shannon Capacity for C7C_7, C11C_11 and C13C_13. Graph Previous lower bound New lower bound C7C_7 3671/5>3.257865367^1/5>3.257865 [12] 1347531/10>3.258020134753^1/10>3.258020 C11C_11 1481/3>5.289572148^1/3>5.289572 [2] 219091/6>5.28977321909^1/6>5.289773 C13C_13 2471/3>6.274305247^1/3>6.274305 [2, 3] 625301/6>6.30010962530^1/6>6.300109 We also improve several known lower bounds on the independence numbers of products of odd cycles that do not lead to improved lower bounds on the Shannon capacity. For example, we prove that α(C153)≥383α(C_15^3)≥ 383 improving upon the previous bound in [6]. We report these in Appendix B as they could be instrumental for future improvements. Statement on AI use Theorem 1 was obtained using multiple prompts to ChatGPT-5.6 Sol Pro. The correctness of the constructions (feasibility of the independent sets constructed) was verified by the authors. The paper was written by the authors (LLMs were used for minor edits to improve grammar and style). Perhaps surprisingly, ChatGPT-5.6 Sol Pro produced independent sets that were not found by the search heuristics manually implemented by the authors, including simulated annealing. Even local search algorithms that were constructed with generative AI (CPro1 [14]) failed to reach the improved lower bounds reported in Theorem 1 despite more than 3 months of repeated attempts. We believe that this points to the importance of mathematical knowledge in improving lower bounds for the Shannon capacity of odd cycles. Interestingly several state of the art lower bounds for C7C_7 and C15C_15 [12, 6] rely on a combination of heuristic search coupled with expert knowledge. 2 Finding independent sets in strong products of odd cycles The idea of improving the lower bounds for the Shannon capacity of G by finding successively larger independent sets in suitable strong products of G is a natural and common approach that we follow as well. However, this approach is not without difficulties as it requires finding a quantity that is NP-hard to compute even approximately [8] in general, for graphs that reach thousands or more vertices for a modest value of the exponent d. The exponential growth rate of the size of the strong product makes it difficult to compute the size of α(Gd)α(G^d). For example, α(C74)α(C_7^4) is currently not known. It is only known that α(C74)≥108α(C_7^4)≥ 108 as it contains an independent set of this size [16]. It is a major open problem whether deciding if Θ(G) (G) is larger than a given threshold is a decidable problem [1]. Even understanding the independence number of the 33rd product of odd cycles is challenging. While the exact value of α(C73)α(C_7^3) is known, only partial results are known about α(C2r+13)α(C^3_2r+1) for r>3r>3 and devising an exact formula for this quantity is currently open [4]. The recent use of Large Language Models (LLMs) for finding mathematical constructions has been applied to the Shannon capacity of odd cycles. In [13] LLMs were used to rediscover α(C75)≥367α(C_7^5)≥ 367, state-of-the-art lower bounds for α(C9d)α(C_9^d) for d=3,..,7d=3,..,7 and improve the best lower bound on α(C114)α(C_11^4) by finding an independent set of size 754 in C114.C_11^4. These results were obtained by a simple greedy algorithm discovered using the FunSearch framework for finding mathematical constructions leveraging algorithms created with a combination of evolutionary algorithms and LLMs. This paradigm was used recently [17] to improve the best known lower bound for C155C_15^5 by constructing an independent set of size 19946. 2.1 Our approach Searches were conducted through the standard ChatGPT web interface using ChatGPT-5.6 Sol Pro. For each instance, we specified the cycle length, product dimension, best construction known to us, and target cardinality required for an improvement. The model generated search programs, executed them within its environment, and returned resulting independent sets in a specified format. The prompt can be found in Appendix A. 3 Constructions We represent the vertices of the cycle CkC_k by ℤk=0,1,…,k−1Z_k=\0,1,…,k-1\, and vertices in the strong n-product by vectors in ℤknZ^n_k. Two distinct vertices have an edge if and only if their circular distance is at most 1 in every coordinate of their vectors. 3.1 7-Cycle C7C_7 For the 7-cycle C7C_7, the best current bound comes from the strong 55-product, for which the largest known independent set has size 367 [12], giving Θ(C7)≥3671/5≈3.257866 (C_7)≥ 367^1/5≈ 3.257866. Let R be the explicit size-367 set of 5-vectors as given in the Appendix of the original publication [12]. Note that R×R× R would give a set of 367×367=134689367× 367=134689 10-vectors that constitute an independent set in the strong 1010-product. We are instead going to delete several vectors from R to yield B, and then take B×B× B augmented with additional vectors derived from R, to reach a larger independent set. Define rjr_j and qjq_j according to the table below. j rjr_j qjq_j 0 (1,3,4,4,6) (2,3,5,4,6) 1 (3,4,0,3,5) (2,4,6,3,5) 2 (5,3,1,3,4) (5,3,2,3,5) 3 (4,4,6,1,6) (5,4,6,0,6) 4 (6,0,6,4,5) (6,1,6,5,5) 5 (0,3,5,6,5) (6,3,5,0,5) 6 (6,4,3,4,0) (6,4,2,4,6) 7 (6,4,5,3,2) (6,5,5,3,1) Let B be a subset of 359 vectors from R, defined as B=R∖rj:0≤j<8B=R \r_j:0≤ j<8\. Taking arithmetic modulo 7, define: X0=(2−w1,w3,w0,(2−w2),w4):w∈RX_0=\(2-w_1,w_3,w_0,(2-w_2),w_4):w∈ R\ Let X be X0X_0 with the vector (2,4,6,3,5) replaced by (1,5,6,3,5). Define J0=0,5,6J_0=\0,5,6\ and J1=1,2,3,4,7J_1=\1,2,3,4,7\. Let: PH=rj:j∈J0∪qj:j∈J1P_H=\r_j:j∈ J_0\∪\q_j:j∈ J_1\ PV=qj:j∈J0∪rj:j∈J1P_V=\q_j:j∈ J_0\∪\r_j:j∈ J_1\ Let x∼yx y indicate that the vertices corresponding to x and y have an edge; the vectors differ by at most 1 (taken modulo 7) in each coordinate. Define: A=x∈X|∃y∈PV:x∼yA=\x∈ X|∃ y∈ P_V:x y\ D=x∈X|∃y∈PH:x∼yD=\x∈ X|∃ y∈ P_H:x y\ The set A has 20 vectors and D has 26. Define functions hj(x)h_j(x) and vj(x)v_j(x) for x∈Xx∈ X as follows: If j∈J0j∈ J_0 and x∈Ax∈ A, or j∈J1j∈ J_1 and x∈Dx∈ D, then let hj(x)=qjh_j(x)=q_j. Otherwise let hj(x)=rjh_j(x)=r_j. If j∈J0j∈ J_0 and x∈Dx∈ D, or j∈J1j∈ J_1 and x∈Ax∈ A, then let vj(x)=qjv_j(x)=q_j. Otherwise let vj(x)=rjv_j(x)=r_j. Now define this set of 10-vectors: I=(B×B)∪(hj(x),x):x∈X,0≤j<8∪(x,vj(x)):x∈X,0≤j<8I=(B× B)∪\(h_j(x),x):x∈ X,0≤ j<8\∪\(x,v_j(x)):x∈ X,0≤ j<8\ I has 359×359+8×367+8×367=134753359× 359+8× 367+8× 367=134753 vectors. We verified that the vertices given by these vectors constitute an independent set in the strong 1010-product of C7C_7; for any two distinct x,y∈Ix,y∈ I it is not the case that x∼yx y. This raises the lower bound on the Shannon capacity for C7C_7, giving Θ(C7)≥1347531/10>3.258020 (C_7)≥ 134753^1/10>3.258020. 3.2 11-Cycle C11C_11 For the 11-cycle C11C_11, the best current bound comes from the strong 3-product, for which the largest known independent set has size 148 [2], giving Θ(C11)≥1481/3≈5.289572 (C_11)≥ 148^1/3≈ 5.289572. We start from this independent set. Let R be the size-148 independent set of 3-vectors described in [2] (see Table 4). Define the following sets of 3-vectors (with arithmetic taken modulo 11): DL D_L =(0,0,2),(3,0,2),(1,3,2),(3,2,2), =\(0,0,2),(3,0,2),(1,3,2),(3,2,2)\, DR D_R =(0,0,2),(3,0,2), =\(0,0,2),(3,0,2)\, BL B_L =R∖DL, =R D_L, BR B_R =R∖DR, =R D_R, X X =(x0+1,x1+10,x2+10):x∈R, =\(x_0+1,x_1+10,x_2+10):x∈ R\, Y Y =(x0+1,x1+1,x2+10):x∈R, =\(x_0+1,x_1+1,x_2+10):x∈ R\, FA F_A =(1,10,1),(10,0,2), =\(1,10,1),(10,0,2)\, FA′ F _A =(0,1,2),(1,3,2),(2,1,2), =\(0,1,2),(1,3,2),(2,1,2)\, FB F_B =(2,0,4),(4,10,1),(4,10,3), =\(2,0,4),(4,10,1),(4,10,3), (4,1,1),(4,1,3),(0,0,4), (4,1,1),(4,1,3),(0,0,4)\, FB′ F _B =(1,0,2),(1,2,2),(3,0,2),(3,2,2), =\(1,0,2),(1,2,2),(3,0,2),(3,2,2)\, FC F_C =(2,2,1),(2,2,3),(0,2,1),(0,2,3), =\(2,2,1),(2,2,3),(0,2,1),(0,2,3)\, FC′ F _C =(1,0,3),(1,3,2),(3,0,2),(3,2,2), =\(1,0,3),(1,3,2),(3,0,2),(3,2,2)\, GA G_A =(1,1,1), =\(1,1,1)\, GA′ G _A =(1,1,2), =\(1,1,2)\, GB G_B =(1,10,2),(2,2,4),(2,0,4),(3,10,2),(4,1,1), =\(1,10,2),(2,2,4),(2,0,4),(3,10,2),(4,1,1), (4,1,3),(4,3,1),(4,3,3),(0,2,4),(0,0,4), (4,1,3),(4,3,1),(4,3,3),(0,2,4),(0,0,4)\, GB′ G _B =(0,0,2),(2,0,2), =\(0,0,2),(2,0,2)\, GC G_C =(2,4,1),(2,4,3),(10,2,2),(10,0,2), =\(2,4,1),(2,4,3),(10,2,2),(10,0,2), (0,4,1),(0,4,3), (0,4,1),(0,4,3)\, GC′ G _C =(1,0,3),(3,0,2). =\(1,0,3),(3,0,2)\. Define set-valued functions H(x)H(x) and V(y)V(y) for x∈Xx∈ X and y∈Yy∈ Y as follows: If x∈FAx∈ F_A then H(x)=FA′H(x)=F _A. If x∈FBx∈ F_B then H(x)=FB′H(x)=F _B. If x∈FCx∈ F_C then H(x)=FC′H(x)=F _C. For all other values of x, H(x)=DLH(x)=D_L. If y∈GAy∈ G_A then V(y)=GA′V(y)=G _A. If y∈GBy∈ G_B then V(y)=GB′V(y)=G _B. If y∈GCy∈ G_C then V(y)=GC′V(y)=G _C. For all other values of y, V(y)=DRV(y)=D_R. Now define this set of 66-vectors: I=BL×BR∪(h,x):x∈X,h∈H(x)∪(y,v):y∈Y,v∈V(y)I=B_L× B_R∪\(h,x):x∈ X,h∈ H(x)\∪\(y,v):y∈ Y,v∈ V(y)\ I has: |I| |I| =144⋅146+(2⋅3+6⋅4+4⋅4+(148−12)⋅4) =144· 146+ (2· 3+6· 4+4· 4+(148-12)· 4 ) +(1⋅1+10⋅2+6⋅2+(148−17)⋅2) + (1· 1+10· 2+6· 2+(148-17)· 2 ) =21909. =21909. vectors. We verified that the vertices given by these vectors constitute an independent set in the strong 66-product of C11C_11; for any two distinct x,y∈Ix,y∈ I it is not the case that x∼yx y. This raises the lower bound on the Shannon capacity for C11C_11, giving Θ(C11)≥219091/6>5.289773 (C_11)≥ 21909^1/6>5.289773. 3.3 13-Cycle C13C_13 For the 13-cycle C13C_13, the best current bound comes from the strong 33-product, in which the maximum independent set size was found to be exactly 247 [3], giving Θ(C13)≥2471/3≈6.274305 (C_13)≥ 247^1/3≈ 6.274305. Let A=(100011120100911001710000010).A= pmatrix1&0&0&0&11&12\\ 0&1&0&0&9&11\\ 0&0&1&7&1&0\\ 0&0&0&0&1&0\\ pmatrix. Define a graph GAG_A with 13413^4 vertices corresponding to ℤ134Z^4_13. Two distinct vertices s and t are adjacent if ∃d∈−1,0,16:s−t≡Admod13∃ d∈\-1,0,1\^6:s-t≡ Ad 13. A randomized local search was used to find a large independent set in GAG_A. Let S be the set of 370 4-vectors in this independent set (see Table 3). Let I=x∈ℤ136:Ax∈SI=\x ^6_13:Ax∈ S\ with arithmetic modulo 13. The resulting set I has 62530 6-vectors. We verified that each pair of vectors of I differs by more than 1 (modulo 1313) in at least one coordinate. Therefore, the vertices corresponding to I constitute an independent set in the strong 66-product of cycle C13C_13. This gives Θ(C13)≥625301/6>6.300109 (C_13)≥ 62530^1/6>6.300109. Data and Code Availability Independent-set constructions, generated programs, prompts, and supporting documentation are available at https://github.com/nathanielitty/lower-bounds-for-shannon-capacity. References [1] N. Alon and E. Lubetzky (2006) The Shannon capacity of a graph and the independence numbers of its powers. IEEE Transactions on Information Theory 52 (5), p. 2172–2176. Cited by: §2. [2] L. D. Baumert, R. J. McEliece, E. Rodemich, H. Rumsey, R. Stanley, and H. Taylor (1971) A combinatorial packing problem. Computers in algebra and number theory 4, p. 97–108. Cited by: Table 1, Table 1, §3.2, §3.2. [3] T. Bohman, R. Holzman, and V. Natarajan (2013) On the independence numbers of the cubes of odd cycles. The Electronic Journal of Combinatorics, p. P10. Cited by: Table 1, §3.3. [4] T. Bohman (2003) A limit theorem for the Shannon capacities of odd cycles I. Proceedings of the American Mathematical Society 131 (11), p. 3559–3569. Cited by: §2. [5] B. Codenotti, I. Gerace, G. Resta, et al. (2003) Some remarks on the Shannon capacity of odd cycles. Ars Combinatoria 66, p. 243–258. Cited by: Table 2. [6] D. de Boer, P. Buys, and J. Zuiddam (2024) The asymptotic spectrum distance, graph limits, and the Shannon capacity. arXiv preprint arXiv:2404.16763. Cited by: §1, §1. [7] V. Guruswami and A. Riazanov (2021) Linear Shannon capacity of Cayley graphs. In 2021 IEEE International Symposium on Information Theory (ISIT), p. 988–992. Cited by: §1. [8] J. Håstad (1999) Clique is hard to approximate within n1−εn^1- . Acta Mathematica 182 (1), p. 105–142. Cited by: §2. [9] N. Lavi and I. Sason (2025) Advances in the Shannon capacity of graphs. arXiv preprint arXiv:2509.24600. Cited by: §1. [10] L. Lovász (1979) On the Shannon capacity of a graph. IEEE Transactions on Information theory 25 (1), p. 1–7. Cited by: §1. [11] K. A. Mathew and P. R. Östergård (2017) New lower bounds for the Shannon capacity of odd cycles. Designs, Codes and Cryptography 84 (1), p. 13–22. Cited by: Table 2, §1. [12] S. C. Polak and A. Schrijver (2019) New lower bound on the Shannon capacity of C7 from circular graphs. Information Processing Letters 143, p. 37–40. Cited by: Table 2, Appendix B, §1, Table 1, §1, §3.1, §3.1. [13] B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog, M. P. Kumar, E. Dupont, F. J. Ruiz, J. S. Ellenberg, P. Wang, O. Fawzi, et al. (2024) Mathematical discoveries from program search with large language models. Nature 625 (7995), p. 468–475. Cited by: Table 2, §2. [14] C. D. Rosin (2025) Using reasoning models to generate search heuristics that solve open instances of combinatorial design problems. arXiv preprint arXiv:2505.23881. Cited by: §1. [15] C. Shannon (1956) The zero error capacity of a noisy channel. IRE Transactions on Information Theory 2 (3), p. 8–19. Cited by: §1. [16] A. Vesel and J. Žerovnik (2002) Improved lower bound on the Shannon capacity of C7. Information processing letters 81 (5), p. 277–282. Cited by: §1, §2. [17] Y. Zhai, Z. Wei, R. Li, K. Pan, S. Liu, L. Zhang, J. Ji, W. Zhang, Y. Zhang, and Y. Zhang (2025) X-Evolve: solution space evolution powered by large language models. arXiv preprint arXiv:2508.07932. Cited by: §2. Appendix A ChatGPT Interactions Each search centered on a detailed initial prompt specifying the construction problem, the required output format, the existing construction to be improved, and the target cardinality. After the model produced an improved construction, we used brief follow-up instructions asking it to continue beyond the newly obtained result. The following initial prompt was used to search for an independent set in C710C_7^10 of size greater than 134689134689. The prompts for C11C_11 and C13C_13 followed the same format, adapted to the corresponding cycle length, product dimension, existing construction, and target cardinality. We are working on the following problem: A "Cycle Code" C(n,k,m) is set of m length-n codewords over the alphabet 0,1,...,k-1, such that any pair of distinct codewords differ by more than 1 (taken circularly) in at least one of the n positions. That is, for every two codewords x and y of length n, there is a position i such that min((x_i - y_i)%k, (y_i - x_i)%k)> 1. For our purposes, k<=15, n<=10, and m< 200000. Output C(n,k,m) with one codeword per line, with each codeword as n space-separated integers, each integer in 0,1,...,k-1. This is essentially the Shannon capacity problem for cycles. For k=7, the best known bound on Shannon capacity is from the existence of C(5,7,367). We have explored this extensively and we do not believe the 367 can be increased. We have found some small improvements for k=7 with n in 6,7,8,9, but these are nowhere close to improving the bound on Shannon capacity. We believe the best opportunity for improving the Shannon capacity bound for k=7 will come from n=10, where the best known result is m=367*367=134689 by simple composition of C(5,7,367). Find C(10,7,m) with m> 134689. The initial search produced a construction CC(10,7,134690)C(10,7,134690). We continued the same conversation with the instruction That’s great. Now push further to m>134690. This follow-up produced CC(10,7,134693)C(10,7,134693). Subsequent similar iterations in the same chat interaction yielded 134698134698, 134714134714, 134738134738, and 134751134751, and then the final construction CC(10,7,134753)C(10,7,134753) used in Theorem 1. Analogous initial and follow-up interactions were used to discover the other constructions. Appendix B Additional Lower Bounds on Independence Numbers In addition to the constructions that improve the known lower bounds on Θ(C7) (C_7), Θ(C11) (C_11), and Θ(C13) (C_13), our searches produced several improved lower bounds on the independence numbers of individual strong powers of odd cycles. These constructions, summarized in Table 2, improve the size of the largest independent set, but do not improve the bound on the Shannon capacity of the underlying cycle. Table 2: Improvements to lower bounds on independence numbers. Graph Previous bound Current bound C114C_11^4 754754 [13] 766766 C134C_13^4 15341534 [11] 15351535 C76C_7^6 11011101 [12] 11201120 C153C_15^3 382382 [5] 383383 The entry 11011101 for C76C_7^6 is the elementary product construction obtained from an independent set of size 367367 in C75C_7^5 [12]. Appendix C Independent Set Constructions C.1 Auxiliary Set S for the C136C_13^6 Construction 7 3 0 0 0 7 11 1 0 7 9 3 12 11 7 5 8 11 6 7 0 7 4 9 2 3 2 11 8 11 0 0 1 11 11 1 2 7 9 3 1 3 8 5 8 3 7 7 11 7 4 9 1 7 2 11 10 11 0 0 12 11 11 1 8 7 10 3 12 3 8 5 10 3 7 7 5 7 5 9 8 3 3 11 0 3 1 0 5 11 12 1 1 7 11 3 7 3 9 5 1 11 7 7 0 11 5 9 5 7 3 11 2 3 1 0 4 3 0 2 12 7 11 3 3 7 9 5 12 11 7 7 11 11 5 9 0 7 4 11 6 3 2 0 8 11 0 2 9 11 11 3 5 7 9 5 1 3 8 7 4 11 6 9 11 7 4 11 11 7 2 0 10 3 1 2 0 11 12 3 11 7 10 5 3 3 8 7 8 3 7 9 4 7 5 11 2 7 3 0 12 11 1 2 2 11 12 3 2 7 11 5 7 3 9 7 10 11 7 9 1 11 5 11 4 7 3 0 1 3 2 2 4 3 0 4 4 7 11 5 1 7 9 7 12 11 7 9 3 11 5 11 10 7 4 0 3 3 2 2 6 3 0 4 9 11 11 5 3 7 9 7 1 3 8 9 7 11 6 11 1 7 5 0 2 7 2 2 12 11 0 4 2 11 12 5 7 7 10 7 3 3 8 9 5 3 7 11 3 7 5 0 6 7 3 2 10 3 1 4 4 11 12 5 0 7 11 7 7 3 9 9 7 3 7 11 0 11 5 0 8 7 3 2 3 3 2 4 6 3 0 6 2 7 11 7 3 7 9 9 0 11 7 11 4 11 6 0 1 7 4 2 5 3 2 4 8 3 0 6 11 11 11 7 5 7 9 9 0 3 8 11 6 11 6 0 7 11 4 2 9 3 3 4 8 11 0 6 4 11 12 7 9 7 10 9 4 3 9 11 9 3 7 0 5 7 5 2 4 7 3 4 12 3 1 6 6 11 12 7 3 7 11 9 6 3 9 11 11 3 7 0 11 11 5 2 6 7 3 4 5 3 2 6 4 3 0 8 7 11 11 9 3 7 9 11 12 11 7 0 4 11 6 2 10 7 4 4 7 3 2 6 10 11 0 8 9 11 11 9 7 7 10 11 4 3 8 0 6 11 6 2 11 11 4 4 0 7 2 6 8 3 1 8 2 11 12 9 9 7 10 11 8 3 9 0 6 3 7 2 3 7 5 4 6 7 3 6 10 3 1 8 7 3 0 10 2 7 11 11 10 3 9 0 8 3 7 2 2 11 5 4 10 7 4 6 3 3 2 8 8 11 0 10 10 11 11 11 0 7 9 0 10 11 7 2 8 11 6 4 12 7 4 6 1 7 2 8 10 11 0 10 12 11 11 11 6 7 10 0 12 3 8 2 10 11 6 4 5 7 5 6 7 3 3 8 0 3 1 10 5 11 12 11 8 7 10 0 5 3 9 2 8 3 7 4 0 11 5 6 7 7 3 8 11 3 1 10 4 3 0 12 12 7 11 0 7 3 9 2 1 11 7 4 11 11 5 6 0 7 4 8 6 3 2 10 7 11 0 12 9 11 11 0 4 7 9 2 1 3 8 4 6 11 6 6 11 7 4 8 12 7 2 10 9 11 0 12 11 11 11 0 10 7 10 2 12 3 8 4 10 3 7 6 9 11 4 8 10 3 3 10 10 3 1 12 2 11 12 0 12 7 10 2 7 3 9 4 10 11 7 6 0 11 5 8 4 7 3 10 12 3 1 12 3 2 0 1 3 7 11 2 0 7 9 4 12 11 7 6 2 11 5 8 10 7 4 10 3 3 2 12 0 11 0 1 9 11 11 2 2 7 9 4 1 3 8 6 6 11 6 8 12 7 4 10 1 7 2 12 11 11 0 1 0 11 12 2 8 7 10 4 3 3 8 6 8 11 6 8 3 7 5 10 3 7 2 12 10 3 1 1 2 11 12 2 1 7 11 4 9 3 9 6 6 3 7 8 0 11 5 10 9 3 3 12 1 3 2 1 6 3 0 3 12 7 11 4 2 7 9 6 1 11 7 8 2 11 5 10 7 7 3 12 3 3 2 1 8 3 0 3 0 11 11 4 4 7 9 6 12 11 7 8 5 3 6 10 0 7 4 12 12 7 2 1 8 11 0 3 4 11 12 4 8 7 10 6 1 3 8 8 6 11 6 10 2 7 4 12 3 7 3 1 1 3 1 3 6 11 12 4 1 7 11 6 12 3 8 8 9 3 7 10 6 7 5 12 5 7 3 1 12 11 1 3 4 3 0 5 3 7 11 6 5 3 9 8 1 11 7 10 12 11 5 12 11 7 4 1 5 3 2 3 6 3 0 5 9 11 11 6 3 7 9 8 12 11 7 10 3 11 6 12 10 11 4 1 7 3 2 3 8 11 0 5 2 11 12 6 5 7 9 8 2 3 8 10 5 11 6 12 2 7 5 1 4 7 3 3 10 3 1 5 4 11 12 6 9 7 10 8 4 3 8 10 6 3 7 12 4 7 5 1 6 7 3 3 3 3 2 5 6 3 0 7 3 7 11 8 8 3 9 10 8 3 7 12 3 11 5 1 10 7 4 3 5 3 2 5 10 11 0 7 5 7 11 8 1 7 9 10 11 11 7 12 7 11 6 1 7 11 4 3 1 7 2 5 12 3 1 7 11 11 11 8 6 7 10 10 1 3 8 12 9 11 6 1 3 7 5 3 9 3 3 5 5 3 2 7 4 11 12 8 8 7 10 10 5 3 9 12 6 3 7 1 11 11 5 3 7 7 3 5 12 7 2 7 6 3 0 9 1 7 11 10 7 3 9 12 8 3 7 1 4 11 6 3 9 7 3 5 9 3 3 7 6 11 0 9 9 11 11 10 5 7 9 12 2 11 7 1 6 11 6 3 0 7 4 5 5 7 3 7 8 11 0 9 11 11 11 10 9 7 10 12 12 3 8 1 10 3 7 3 6 7 5 5 9 7 4 7 10 3 1 9 4 11 12 10 11 7 10 12 5 3 9 1 12 3 7 3 0 11 5 5 11 7 4 7 12 3 1 9 3 3 0 11 4 7 11 12 7 3 9 1 10 11 7 3 11 11 5 5 9 11 4 7 5 3 2 9 9 11 0 11 8 11 11 12 1 7 9 1 3 3 8 3 6 11 6 5 4 7 5 7 1 7 2 9 11 11 0 11 10 11 11 12 7 7 10 1 9 3 9 3 8 3 7 5 0 11 5 7 9 3 3 9 9 3 1 11 1 11 12 12 9 7 10 1 11 3 9 3 10 11 7 5 2 11 5 7 7 7 3 9 11 3 1 11 Table 3: The set S⊆ℤ134S _13^4 consisting of 370370 vectors. C.2 Base Construction in C113C_11^3 0 0 0 1 6 7 3 2 2 4 8 3 6 4 9 7 9 4 9 5 10 0 0 2 1 7 0 3 2 4 4 9 7 6 5 2 7 10 8 9 6 5 0 1 7 1 8 6 3 3 8 4 10 0 6 5 4 8 0 5 9 7 9 0 2 0 1 9 10 3 4 1 5 0 8 6 6 8 8 1 10 9 8 2 0 3 6 1 10 5 3 4 3 5 1 2 6 7 1 8 2 5 9 8 4 0 4 10 2 0 0 3 5 7 5 1 4 6 7 3 8 3 9 9 9 8 0 5 5 2 1 7 3 6 0 5 2 8 6 8 7 8 4 2 9 10 1 0 6 9 2 2 0 3 7 6 5 3 1 6 9 0 8 4 4 9 10 3 0 7 2 2 3 6 3 8 10 5 3 3 6 10 6 8 5 8 10 0 9 0 7 4 2 4 10 3 9 5 5 4 7 7 0 1 8 6 1 10 1 5 0 8 8 2 5 5 3 10 9 5 5 0 7 0 3 8 6 3 10 2 9 0 9 1 2 6 9 4 0 6 5 6 6 7 1 8 8 7 7 10 3 2 0 9 3 2 7 2 4 1 0 5 7 10 7 2 1 8 8 0 10 3 4 0 10 7 2 7 4 4 2 6 5 8 5 7 2 3 8 9 6 10 4 8 1 0 9 2 8 8 4 3 10 5 9 9 7 3 7 8 10 10 10 5 1 1 1 5 2 9 1 4 4 5 5 10 2 7 4 0 9 0 7 10 5 3 1 2 9 2 9 3 4 5 9 5 10 4 7 5 6 9 1 1 10 6 7 1 3 2 2 10 7 4 6 2 6 0 10 7 6 10 9 1 3 10 7 0 1 3 4 3 0 2 4 6 4 6 1 6 7 7 5 9 2 7 10 8 6 1 4 8 3 0 4 4 7 8 6 2 10 7 8 9 9 3 0 10 9 10 1 5 1 3 1 9 4 8 1 6 3 5 7 9 2 9 4 6 10 10 5 1 5 3 Table 4: The 148 vectors in ℤ113Z_11^3 that form the basis for the C11C_11 construction. C.3 C114C_11^4 0 0 2 1 0 0 3 6 0 0 3 8 0 0 7 0 0 0 7 9 0 1 0 1 0 1 0 5 0 1 5 6 0 1 5 8 0 1 5 10 0 1 9 0 0 2 2 1 0 2 3 6 0 2 3 8 0 2 3 10 0 2 7 0 0 3 0 1 0 3 1 6 0 3 1 8 0 3 1 10 0 3 5 0 0 3 9 1 0 3 10 6 0 3 10 8 0 3 10 10 0 4 3 0 0 4 7 1 0 4 8 6 0 4 8 8 0 4 8 10 0 5 1 0 0 5 5 1 0 5 6 8 0 5 6 10 0 5 10 0 0 6 3 1 0 6 4 8 0 6 4 10 0 6 8 0 0 7 2 6 0 7 2 8 0 7 2 10 0 7 6 0 0 8 0 6 0 8 0 8 0 8 0 10 0 8 4 0 0 8 8 1 0 8 9 6 0 8 9 8 0 8 9 10 0 9 2 0 0 9 6 1 0 9 7 6 0 9 7 8 0 9 7 10 0 10 0 0 0 10 4 1 0 10 5 6 0 10 5 8 0 10 5 10 0 10 9 0 1 0 1 3 1 0 3 10 1 0 5 4 1 0 6 2 1 0 8 7 1 0 9 5 1 1 1 7 1 1 1 9 1 1 3 3 1 1 4 1 1 1 7 4 1 1 8 2 1 1 10 3 1 1 10 7 1 1 10 9 1 2 1 3 1 2 5 4 1 2 6 2 1 2 8 7 1 2 8 9 1 3 3 4 1 3 4 2 1 3 6 7 1 3 6 9 1 3 8 3 1 3 10 3 1 4 1 4 1 4 2 2 1 4 4 7 1 4 4 9 1 4 5 5 1 4 6 3 1 5 0 2 1 5 2 7 1 5 2 9 1 5 3 5 1 5 4 3 1 5 8 4 1 5 9 2 1 5 10 4 1 6 0 7 1 6 0 9 1 6 2 3 1 6 6 4 1 6 7 2 1 6 9 7 1 6 9 9 1 7 0 3 1 7 1 1 1 7 4 4 1 7 5 2 1 7 7 7 1 7 7 9 1 7 9 3 1 8 2 4 1 8 3 2 1 8 5 7 1 8 5 9 1 8 7 3 1 8 10 1 1 9 0 4 1 9 1 2 1 9 3 7 1 9 3 9 1 9 5 3 1 9 9 4 1 10 1 7 1 10 1 9 1 10 3 3 1 10 7 4 1 10 8 2 1 10 10 2 1 10 10 8 2 0 0 5 2 0 4 6 2 0 4 8 2 0 8 0 2 0 8 9 2 1 1 0 2 1 2 5 2 1 6 6 2 1 6 8 2 1 6 10 2 1 10 0 2 2 0 5 2 2 4 6 2 2 4 8 2 2 4 10 2 2 8 0 2 2 9 5 2 3 0 10 2 3 2 6 2 3 2 8 2 3 2 10 2 3 6 0 2 3 7 5 2 3 10 1 2 4 0 6 2 4 0 8 2 4 4 0 2 4 9 6 2 4 9 8 2 4 9 10 2 5 0 0 2 5 2 0 2 5 7 6 2 5 7 8 2 5 7 10 2 6 1 5 2 6 5 6 2 6 5 8 2 6 5 10 2 6 9 0 2 7 3 6 2 7 3 8 2 7 3 10 2 7 7 0 2 7 8 5 2 7 10 5 2 8 1 6 2 8 1 8 2 8 1 10 2 8 5 0 2 8 6 5 2 8 10 8 2 8 10 10 2 9 3 0 2 9 4 5 2 9 8 6 2 9 8 8 2 9 8 10 2 9 10 6 2 10 1 0 2 10 2 5 2 10 6 6 2 10 6 8 2 10 6 10 2 10 10 0 3 0 0 7 3 0 2 3 3 0 3 1 3 0 4 10 3 0 6 4 3 0 7 2 3 0 9 6 3 1 0 2 3 1 0 9 3 1 2 7 3 1 2 9 3 1 4 3 3 1 5 1 3 1 9 2 3 2 0 7 3 2 2 3 3 2 3 1 3 2 7 2 3 2 9 7 3 2 9 9 3 3 0 3 3 3 1 1 3 3 4 4 3 3 5 2 3 3 7 7 3 3 7 9 3 3 9 3 3 4 2 4 3 4 3 2 3 4 5 7 3 4 5 9 3 4 7 3 3 4 8 1 3 5 1 2 3 5 3 7 3 5 3 9 3 5 5 3 3 5 6 1 3 5 9 4 3 5 10 2 3 6 1 7 3 6 1 9 3 6 3 3 3 6 4 1 3 6 8 2 3 6 10 7 3 6 10 9 3 7 1 3 3 7 2 1 3 7 6 2 3 7 8 7 3 7 8 9 3 7 10 3 3 8 0 1 3 8 4 2 3 8 6 7 3 8 6 9 3 8 8 3 3 8 9 1 3 9 2 2 3 9 4 7 3 9 4 9 3 9 6 3 3 9 7 1 3 9 10 4 3 10 0 2 3 10 0 9 3 10 2 7 3 10 2 9 3 10 4 3 3 10 5 1 3 10 8 4 3 10 9 2 4 0 1 5 4 0 5 6 4 0 5 8 4 0 8 10 4 0 9 8 4 0 10 4 4 1 1 0 4 1 3 5 4 1 6 10 4 1 7 6 4 1 7 8 4 1 8 4 4 1 10 0 4 2 1 5 4 2 4 10 4 2 5 6 4 2 5 8 4 2 6 4 4 2 8 0 4 2 10 5 4 3 2 10 4 3 3 6 4 3 3 8 4 3 6 0 4 3 8 5 4 4 0 10 4 4 1 6 4 4 1 8 4 4 4 0 4 4 6 5 4 4 9 10 4 4 10 6 4 4 10 8 4 5 0 4 4 5 2 0 4 5 4 5 4 5 7 10 4 5 8 6 4 5 8 8 4 6 0 0 4 6 2 5 4 6 5 10 4 6 6 6 4 6 6 8 4 6 7 4 4 6 9 0 4 7 0 5 4 7 3 10 4 7 4 6 4 7 4 8 4 7 5 4 4 7 7 0 4 7 9 5 4 8 0 8 4 8 1 10 4 8 2 6 4 8 2 8 4 8 3 4 4 8 5 0 4 8 7 5 4 8 10 10 4 9 0 6 4 9 1 4 4 9 3 0 4 9 5 5 4 9 8 10 4 9 9 6 4 9 9 8 4 10 1 0 4 10 3 5 4 10 6 10 4 10 7 6 4 10 7 8 4 10 10 0 5 0 0 9 5 0 1 7 5 0 3 3 5 0 4 1 5 0 4 10 5 0 8 1 5 0 10 6 5 1 1 2 5 1 2 9 5 1 3 7 5 1 6 1 5 1 10 2 5 2 0 9 5 2 1 7 5 2 3 3 5 2 4 1 5 2 8 2 5 2 9 9 5 2 10 7 5 3 0 1 5 3 1 3 5 3 2 1 5 3 6 2 5 3 7 9 5 3 8 7 5 3 10 3 5 4 4 2 5 4 5 9 5 4 6 7 5 4 8 3 5 4 9 1 5 5 0 2 5 5 2 2 5 5 3 9 5 5 4 7 5 5 7 1 5 6 0 7 5 6 1 9 5 6 2 7 5 6 5 1 5 6 9 2 5 6 10 9 5 7 0 3 5 7 3 1 5 7 7 2 5 7 8 9 5 7 9 7 5 8 1 1 5 8 5 2 5 8 6 9 5 8 7 7 5 8 9 3 5 8 10 1 5 9 3 2 5 9 4 9 5 9 5 7 5 9 7 3 5 9 8 1 5 10 1 2 5 10 2 9 5 10 3 7 5 10 5 3 5 10 6 1 5 10 10 2 6 0 0 0 6 0 0 4 6 0 2 5 6 0 5 7 6 0 6 5 6 0 7 3 6 0 9 8 6 0 9 10 6 1 2 0 6 1 4 5 6 1 5 3 6 1 7 8 6 1 7 10 6 1 8 6 6 1 9 4 6 2 0 5 6 2 2 5 6 2 5 8 6 2 5 10 6 2 6 6 6 2 7 4 6 2 9 0 6 3 3 8 6 3 3 10 6 3 4 6 6 3 5 4 6 3 7 0 6 3 9 5 6 4 0 6 6 4 1 8 6 4 1 10 6 4 2 6 6 4 3 4 6 4 5 0 6 4 7 5 6 4 10 8 6 4 10 10 6 5 1 4 6 5 3 0 6 5 5 5 6 5 6 3 6 5 8 8 6 5 8 10 6 5 9 6 6 5 10 4 6 6 1 0 6 6 3 5 6 6 4 3 6 6 6 8 6 6 6 10 6 6 7 6 6 6 8 4 6 6 10 0 6 7 1 5 6 7 2 3 6 7 4 8 6 7 4 10 6 7 5 6 6 7 6 4 6 7 8 0 6 7 10 5 6 8 2 8 6 8 2 10 6 8 3 6 6 8 4 4 6 8 6 0 6 8 8 5 6 9 0 4 6 9 0 8 6 9 0 10 6 9 1 6 6 9 2 4 6 9 4 0 6 9 6 5 6 9 9 8 6 9 9 10 6 9 10 6 6 10 2 0 6 10 4 5 6 10 7 8 6 10 7 10 6 10 8 6 6 10 9 4 7 0 1 7 7 0 1 9 7 0 3 2 7 0 4 0 7 0 5 9 7 0 8 1 7 0 10 6 7 1 1 2 7 1 3 7 7 1 3 9 7 1 6 1 7 1 10 2 7 2 0 0 7 2 1 7 7 2 1 9 7 2 3 3 7 2 4 1 7 2 8 2 7 2 10 7 7 2 10 9 7 3 1 3 7 3 2 1 7 3 6 2 7 3 8 7 7 3 8 9 7 3 10 3 7 4 0 1 7 4 4 2 7 4 6 7 7 4 6 9 7 4 8 3 7 4 9 1 7 5 2 2 7 5 4 7 7 5 4 9 7 5 7 1 7 6 0 2 7 6 2 7 7 6 2 9 7 6 5 1 7 6 9 2 7 7 0 7 7 7 0 9 7 7 3 1 7 7 7 2 7 7 9 7 7 7 9 9 7 8 1 1 7 8 5 2 7 8 7 7 7 8 7 9 7 8 9 3 7 8 10 1 7 9 3 2 7 9 5 7 7 9 5 9 7 9 7 3 7 9 8 1 7 10 1 2 7 10 3 7 7 10 3 9 7 10 5 3 7 10 6 1 7 10 10 2 8 0 2 0 8 0 2 4 8 0 6 5 8 0 6 7 8 0 7 3 8 0 10 8 8 0 10 10 8 1 0 4 8 1 4 5 8 1 5 3 8 1 8 6 8 1 8 8 8 1 8 10 8 1 9 4 8 2 2 5 8 2 6 6 8 2 6 8 8 2 6 10 8 2 7 4 8 3 0 5 8 3 4 6 8 3 4 8 8 3 4 10 8 3 5 4 8 3 9 5 8 4 2 6 8 4 2 8 8 4 2 10 8 4 3 4 8 4 7 5 8 5 0 6 8 5 0 8 8 5 0 10 8 5 1 4 8 5 5 5 8 5 6 3 8 5 9 6 8 5 9 8 8 5 9 10 8 5 10 4 8 6 3 5 8 6 4 3 8 6 7 6 8 6 7 8 8 6 7 10 8 6 8 4 8 7 1 5 8 7 2 3 8 7 5 6 8 7 5 8 8 7 5 10 8 7 6 4 8 7 10 5 8 8 0 3 8 8 3 6 8 8 3 8 8 8 3 10 8 8 4 4 8 8 8 5 8 9 1 6 8 9 1 8 8 9 1 10 8 9 2 4 8 9 6 5 8 9 10 6 8 9 10 8 8 9 10 10 8 10 0 4 8 10 4 5 8 10 8 6 8 10 8 8 8 10 8 10 8 10 9 4 9 0 0 1 9 0 2 2 9 0 2 6 9 0 2 8 9 0 6 0 9 0 6 9 9 1 0 6 9 1 4 0 9 1 4 7 9 1 4 9 9 1 9 1 9 2 2 0 9 2 2 7 9 2 2 9 9 2 3 2 9 2 7 1 9 3 0 0 9 3 0 7 9 3 0 9 9 3 1 2 9 3 5 1 9 3 9 0 9 3 9 7 9 3 9 9 9 3 10 2 9 4 3 1 9 4 7 0 9 4 7 7 9 4 7 9 9 4 8 2 9 5 1 1 9 5 5 1 9 5 5 8 9 5 5 10 9 5 10 1 9 6 3 1 9 6 3 8 9 6 3 10 9 6 8 1 9 7 1 0 9 7 1 7 9 7 1 9 9 7 6 1 9 7 10 0 9 7 10 7 9 7 10 9 9 8 4 1 9 8 8 0 9 8 8 2 9 8 8 7 9 8 8 9 9 9 0 1 9 9 2 1 9 9 6 0 9 9 6 2 9 9 6 7 9 9 6 9 9 10 4 0 9 10 4 2 9 10 4 7 9 10 4 9 9 10 8 1 10 0 2 4 10 0 2 10 10 0 7 3 10 0 7 5 10 0 7 7 10 1 0 3 10 1 0 8 10 1 0 10 10 1 5 2 10 1 5 4 10 1 9 3 10 1 9 5 10 1 9 7 10 1 9 9 10 2 3 4 10 2 7 3 10 2 7 5 10 2 7 7 10 2 7 9 10 3 1 4 10 3 5 3 10 3 5 5 10 3 5 7 10 3 5 9 10 3 10 4 10 4 3 3 10 4 3 5 10 4 3 7 10 4 3 9 10 4 8 4 10 5 1 3 10 5 1 5 10 5 1 7 10 5 1 9 10 5 6 3 10 5 6 5 10 5 10 3 10 5 10 7 10 5 10 9 10 6 4 3 10 6 4 5 10 6 8 3 10 6 8 5 10 6 8 7 10 6 8 9 10 6 10 5 10 7 1 2 10 7 2 4 10 7 6 3 10 7 6 5 10 7 6 7 10 7 6 9 10 7 10 2 10 8 0 4 10 8 4 3 10 8 4 5 10 8 4 7 10 8 4 9 10 8 9 4 10 9 2 3 10 9 2 5 10 9 2 7 10 9 2 9 10 9 6 4 10 10 0 3 10 10 0 5 10 10 0 7 10 10 0 9 10 10 4 4 10 10 9 3 10 10 9 5 10 10 9 7 10 10 9 9 Table 5: Independent set of size 766766 in C114C_11^4. C.4 C153C_15^3 8 0 0 3 1 0 14 1 0 7 2 0 2 3 0 13 3 0 4 4 0 10 5 0 12 5 0 3 6 0 7 6 0 13 7 0 2 8 0 6 8 0 12 9 0 1 10 0 5 10 0 9 11 0 11 11 0 0 12 0 5 12 0 9 13 0 11 13 0 0 14 0 4 14 0 6 0 1 1 1 1 10 1 1 12 1 1 5 2 1 0 3 1 9 3 1 11 3 1 6 4 1 14 5 1 1 6 1 5 6 1 9 7 1 11 7 1 0 8 1 4 8 1 8 9 1 10 9 1 3 10 1 14 10 1 7 11 1 3 12 1 13 12 1 7 13 1 2 14 1 13 14 1 4 0 2 8 1 2 14 1 2 3 2 2 13 3 2 2 4 2 4 4 2 8 5 2 10 5 2 12 5 2 3 6 2 7 7 2 2 8 2 13 8 2 6 9 2 1 10 2 12 10 2 5 11 2 1 12 2 9 12 2 11 12 2 5 13 2 0 14 2 9 14 2 11 14 2 2 0 3 6 1 3 10 1 3 12 1 3 1 2 3 7 3 3 9 3 3 11 3 3 0 4 3 6 5 3 1 6 3 14 6 3 5 7 3 0 8 3 9 8 3 11 8 3 4 9 3 8 10 3 10 10 3 14 10 3 3 11 3 7 12 3 13 12 3 3 13 3 7 14 3 13 14 3 0 0 4 4 1 4 8 1 4 14 2 4 3 3 4 5 3 4 13 4 4 4 5 4 8 6 4 10 6 4 12 6 4 3 7 4 7 8 4 13 8 4 2 9 4 6 10 4 12 10 4 1 11 4 5 12 4 9 12 4 11 12 4 1 13 4 5 14 4 9 14 4 11 14 4 13 0 5 2 1 5 6 1 5 10 2 5 12 2 5 1 3 5 7 4 5 9 4 5 11 4 5 0 5 5 2 5 5 6 6 5 1 7 5 5 8 5 9 8 5 11 8 5 0 9 5 4 10 5 8 10 5 10 10 5 14 11 5 3 12 5 7 12 5 14 13 5 3 14 5 7 14 5 0 0 6 9 0 6 11 0 6 4 1 6 8 2 6 14 2 6 3 3 6 5 3 6 13 4 6 4 5 6 8 6 6 10 6 6 12 6 6 3 7 6 14 7 6 7 8 6 2 9 6 13 9 6 6 10 6 1 11 6 12 11 6 5 12 6 1 13 6 10 13 6 12 13 6 5 14 6 7 0 7 13 0 7 2 1 7 10 2 7 12 2 7 1 3 7 7 4 7 9 4 7 11 4 7 0 5 7 2 5 7 6 6 7 1 7 7 5 8 7 0 9 7 9 9 7 11 9 7 4 10 7 8 11 7 10 11 7 14 11 7 3 12 7 8 13 7 14 13 7 3 14 7 5 0 8 9 0 8 11 0 8 0 1 8 4 2 8 6 2 8 8 2 8 14 3 8 5 4 8 13 5 8 4 6 8 8 7 8 10 7 8 12 7 8 14 7 8 3 8 8 7 9 8 13 9 8 2 10 8 6 11 8 12 11 8 1 12 8 6 13 8 10 13 8 12 13 8 1 14 8 3 0 9 7 0 9 13 1 9 2 2 9 10 3 9 12 3 9 1 4 9 3 4 9 7 5 9 9 5 9 11 5 9 2 6 9 6 7 9 1 8 9 5 9 9 9 9 9 11 9 9 0 10 9 4 11 9 8 11 9 10 11 9 14 12 9 4 13 9 8 13 9 14 14 9 1 0 10 5 0 10 9 1 10 11 1 10 0 2 10 4 2 10 6 3 10 8 3 10 14 4 10 5 5 10 0 6 10 13 6 10 4 7 10 8 7 10 10 7 10 14 8 10 3 9 10 7 9 10 13 10 10 2 11 10 6 11 10 12 12 10 2 13 10 6 13 10 10 14 10 12 14 10 3 0 11 7 1 11 13 1 11 2 2 11 10 3 11 12 3 11 1 4 11 3 4 11 7 5 11 9 5 11 11 5 11 2 6 11 6 7 11 1 8 11 12 8 11 5 9 11 0 10 11 9 10 11 11 10 11 4 11 11 0 12 11 8 12 11 10 12 11 4 13 11 8 14 11 14 14 11 1 0 12 5 1 12 9 1 12 11 1 12 0 2 12 6 3 12 8 3 12 14 4 12 5 5 12 0 6 12 13 6 12 4 7 12 8 8 12 10 8 12 14 8 12 3 9 12 7 10 12 13 10 12 2 11 12 6 12 12 13 12 12 2 13 12 6 14 12 10 14 12 12 14 12 14 0 13 3 1 13 7 1 13 13 2 13 2 3 13 4 3 13 10 4 13 12 4 13 3 5 13 7 6 13 9 6 13 11 6 13 2 7 13 6 8 13 12 8 13 1 9 13 5 10 13 9 10 13 11 10 13 0 11 13 4 12 13 9 12 13 11 12 13 0 13 13 4 14 13 8 14 13 10 0 14 12 0 14 1 1 14 5 1 14 9 2 14 11 2 14 0 3 14 6 4 14 8 4 14 1 5 14 14 5 14 5 6 14 0 7 14 4 8 14 8 8 14 10 8 14 14 9 14 3 10 14 7 10 14 13 11 14 2 12 14 7 12 14 13 13 14 2 14 14 6 14 14 Table 6: Independent set of size 383383 in C153C_15^3.