[[486,164,6]]: reflexible {6,6} surface code, Conder's map R82.22

Direction & hypothesis

Cell: check weight 6, unrestricted locality (no layout submitted). Surface codes of regular maps of type {p,q} on closed hyperbolic surfaces (Breuckmann and Terhal, arXiv:1506.04029) have k = 2 - chi = (1 - 2/p - 2/q) n + 2, so {6,6} maps give k/n close to 1/3 and kd^2/n about 12 at d = 6, the level of the best entries of check weight 6 with d = 6 at board commit 8042f9da (kd^2/n 12.1 to 12.4, from [[186,64,6]] to [[282,96,6]], [[378,128,6]], [[546,184,6]] (a {6,6} code from a PSL(2,13) quotient, arXiv:2610.10948) and [[768,258,6]]). Hypothesis: the census of all {6,6} maps contains maps with systole 6 at lengths n where the board has no such code, and these extend the frontier of the weight-6 cell.

What was searched

A census of all regular maps of type {p,q}, 3 <= p,q <= 8, 1/p + 1/q < 1/2, with at most 2000 edges (chiral {8,8} maps up to 1000 edges): the normal subgroups of index 2E of the triangle groups <r,s | r^p, s^q, (rs)^2> in which r, s, rs keep their orders (Magma's LowIndexNormalSubgroups), the non-orientable maps from the extended triangle groups, and 2000 cyclic covers of the orientable maps with n <= 1000 (random voltage classes, seeds recorded). In total 2174 orientable and 340 non-orientable maps (up to duality and mirror image) and 2000 covers, 7063 codes with their X/Z duals, each with exact d_X and d_Z by the programs below. At board commit 8042f9da, 254 parameter points of the census were undominated; we submit nine of them, one per pull request, and left out the points we found in the literature: [[896,386,6]] ({7,7}) and [[960,194,8]] ({5,5}) are data points of Fig. 3.7 of Breuckmann's thesis (arXiv:1802.01520), [[588,254,6]] and [[273,119,5]] are in its Table 3.1. This code is the reflexible {6,6} map R82.22 of M. Conder's list. It is one of four {6,6} maps of the census with 486 edges and d = 6; we submit the reflexible one. Sources checked without finding it: Breuckmann and Terhal (arXiv:1506.04029), Breuckmann et al. (arXiv:1703.00590), Breuckmann's thesis (arXiv:1802.01520), Mahmoud and Rayan (arXiv:2610.10948), Breuckmann and Burton (arXiv:2202.06647), Mahmoud, Ali and Rayan (arXiv:2504.07800), the catalogs of QEC-pages (Quantum_LDPC_Codes) and qecdb.org; Conder's lists contain the maps, not the codes.

Evidence trail

• k = 164 = 486 - 161 - 161 (ranks over GF(2)) = 2 - chi; H_X H_Z^T = 0; every qubit lies in two X-checks and two Z-checks, so both sides are graph-like (computed).
• Upper bound: an X-type and a Z-type logical operator of weight 6, in the entry.
• Lower bound, exact, by four computations: (1) graphlike_A, shortest cycles with logical annotations in F_2^k; (2) graphlike_B, shortest paths in the double covers of the check graph given by the logical operators, written independently; their exactness is proved in github.com/TimoKellerMath/quantum-error-correction @ 2e97a5361e7b9327128baa242bdedd333f7d7e12, hyperbolic-map-codes/src/graphlike_A/NOTES.md and hyperbolic-map-codes/src/graphlike_B/NOTES.md, and they agree on all 7063 census codes; (3) an enumeration of all circuits of length at most 6 with a row-space test: the graph of the map and its dual have no circuit shorter than 6; of the circuits of length 6, the 162 face boundaries are trivial and the other 2835 are nontrivial, on both sides; (4) tropidist (Cheng, Guo and Pan, arXiv:2611.08247) at the board's pinned commit b03ec2d4077f, one thread, 600 s limit, on the board's css-sides problems: exact 6 and 6 in 0.9 s and 1.0 s, recorded in certs/486-164-6.json.
• Final claim: d_X = d_Z = 6, exact (computer-assisted; the four computations agree). 2835 X-type and 2835 Z-type logical operators have weight 6.

Dead ends

• Within the census no {8,8}, {6,6} or {7,7} map reaches d = 7; d >= 7 occurs only for {3,7}, {4,5}, {4,6} and {5,5} (n <= 2000).
• Among the 920 {6,6} maps of the census (up to mirror image), 231 have d = 6 (from n = 294 on) and none more; 608 have d = 2 or 4.

Tools

Claude Opus 5.5 (Claude Code). Magma V2.30 for the census (LowIndexNormalSubgroups, coset actions, the presentations and Conder's names); Python for the map codes; the C programs graphlike_A and graphlike_B and the Python circuit enumeration, all in the public package below; tropidist b03ec2d4077f. The largest Magma run of the census (orientable {6,6} maps, index at most 4000) took 33 CPU minutes; exact distances of all 7063 codes took 46 s with graphlike_A and 76 s with graphlike_B.

Reproduction

G = <r,s | r^6, s^6, (r*s)^2, s*r*s^-2*r^3*s^3*r^-2, r*s^-2*r*s^-1*r^2*s^-2*r*s^-1*r, s*r*s^-1*r^-1*s*r^2*s*r^-1*s^-1*r*s> (order 972); qubits = edges {g, grs}, X-checks = vertices g<s>, Z-checks = faces g<r>, incidences mod 2. In hyperbolic-map-codes/ of github.com/TimoKellerMath/quantum-error-correction @ 2e97a5361e7b: make check runs hyperbolic-map-codes/src/rebuild.py (standard library), which rebuilds these matrices from the stored darts and from the presentation, checks Conder's relators, k and the witnesses; make distances runs graphlike_A, graphlike_B and the circuit enumeration; make tropidist-inputs writes the tropidist problems.