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A Note on the Upper-Bound Columns

This note records the upper-bound mechanisms displayed in the live table on the main page. Throughout, the reported code parameter \(\left[\left[n,k,\le u\right]\right]\) is obtained by taking the minimum among the currently available structural and simulation-based upper bounds for the same row. The latent, \(m\)-block, and decoder-fail columns are the three core columns used throughout the manuscript, while the fiber-quotient, CRT, and direct-CSS columns report additional exploratory methods that we now track in parallel. The accompanying manuscript is available as arXiv:2604.15307.

1. Latent Upper Bound

For the \((J,L)=(3,12)\) family treated here, the latent upper bound is derived from explicit latent witnesses obtained from the kernels of \(\Psi_3\) and \(\Psi_3^{\mathsf T}\). A nonzero kernel vector \(\mathbf{u}\) or \(\mathbf{v}\) is converted into a latent logical candidate such as \(\mathbf{u}^{\mathsf T}\widetilde H_X\) or \(\mathbf{v}^{\mathsf T}\widetilde H_Z\), and the Hamming weight of the resulting support yields an upper bound on the latent \(X\)- or \(Z\)-distance.

In the parameter files this information is stored in latent_x_upper, latent_z_upper, and latent_min_upper. The live table reports the minimum of the \(X\)- and \(Z\)-side latent witnesses.

2. \(m\)-Block Upper Bound

The \(m\)-block upper bound is a non-latent structural upper bound derived from block-constant compression. Given a divisor \(m\mid P\), the APM construction is compressed to a quotient problem of length \(P/m\). A low-weight quotient witness is then lifted back to the original blocklength. In the non-latent setting, the lifted vector must remain outside the opposite row space, which makes it a genuine non-stabilizer logical operator.

The corresponding parameter-file fields are nonlatent_x_upper, nonlatent_z_upper, nonlatent_min_upper, together with the selected block sizes nonlatent_x_m and nonlatent_z_m. The live table shows the minimum non-latent structural upper bound in the \(m\)-block UB column.

3. Fiber-Quotient Upper Bound

The fiber-quotient upper bound refines the \(m\)-block idea without falling back to a full direct CSS search. Write \(P=mQ\). The ordinary \(m\)-block lift is constant on each fiber \(\{t+kQ:0\le k\lt m\}\). In the fiber-quotient search we instead fix a proper nonempty pattern \(F\subsetneq\{0,\ldots,m-1\}\), allow support only on \(\{t+kQ:k\in F\}\) for each quotient coordinate \(t\), solve the induced restricted kernel problem, and lift the resulting vector back to the full CSS code.

The lifted vector is accepted only after the same algebraic logical check used by the other structural bounds: it must have zero syndrome on the relevant side and must lie outside the opposite row space. This makes the lifted vector an upper-bound witness. The \(P=768\) paper-code row currently has a fiber-quotient \(X\)-logical witness of weight \(24\), obtained with \(m=4\), \(Q=192\), and \(F=\{0,2\}\).

The recovered April 15--16, 2026 fiber-quotient campaign gives the following evaluated rows. The \(X\)- and \(Z\)-side cells list the lifted witness weight followed by the divisor data and the selected fiber pattern. The table value is the smaller of the two side-specific weights.

4. CRT-Compression Upper Bound

The CRT upper bound is an experimental extension of the block-compression method. Instead of requiring a witness to be constant on a single subgroup quotient, we search over coprime stripe subspaces coming from factorizations \(P=q_1q_2\) with \(\gcd(q_1,q_2)=1\). In practice, we restrict to the binary span of residue-class stripes modulo \(q_1\) and modulo \(q_2\), solve the induced quotient kernel problem, and lift the resulting witness back to the full code.

This column is currently used as an auxiliary method rather than as part of the parameter-file comments. It is included in the live table because it can sometimes produce witnesses that are structurally different from ordinary \(m\)-block-constant ones.

5. Direct CSS-Search Upper Bound

The direct CSS upper bound dispenses with quotient structure altogether and searches directly for a low-weight vector in \(\operatorname{Ker}(H_Z)\setminus\operatorname{Row}(H_X)\) or in \(\operatorname{Ker}(H_X)\setminus\operatorname{Row}(H_Z)\). The current implementation is randomized: it samples low-combination vectors in a nullspace basis and rejects those that fall back into the opposite row space.

This method is expensive and therefore still exploratory, but it is the cleanest structure-free witness search among the upper-bound mechanisms we track.

6. Decoder-Fail Upper Bound

This simulation-based bound is obtained from decoder-fail sampling at the physical error rates shown in the live table, currently mainly \(p=0.03\) and \(p=0.04\). For a trial with true error \(e\) and decoder output \(\hat e\), the primary witness is the residual \(r=e+\hat e\). If \(r\) has zero syndrome and is outside the relevant stabilizer row space, then \(r\) is a logical operator. Its Hamming weight \(w=|r|\) gives \(d\le w\).

Current runs also allow a restricted coset-completion step for small residual-syndrome mismatches. In that case we solve a small linear system for a correction \(c\) with the missing syndrome and test \(r+c\) algebraically. The completed vector is accepted only if it has zero syndrome and is verified to be non-stabilizer. Uncompleted mismatches, large residual records, and records that fail the logical check are discarded rather than reported as upper bounds; in the current screening policy, residual records with residual_qubit_weight=10 are treated as rejects.

This information is recorded in decoderlog_x_upper, decoderlog_z_upper, and, when available, the associated fail-log or completion support written into the parameter file. If no checked logical fail has been observed yet, the live table displays -- in the decoder-fail column.

Trial counts in the live table are intended to count work that produced recoverable summaries, not merely jobs that were submitted. For full runs this is the number of completed trials. For interrupted h_rt-limited TSUBAME jobs, append-only live summaries may allow us to report accounted partial trials. Submitted jobs with no recoverable summary are not counted.

In the April 15, 2026 high-\(P\) update, the \(p=0.03\) entries for \(P=1536,1920,2688\) are campaign-ledger totals that include top-per-\(P\) candidate screening for the same lift size. They do not by themselves change the decoder-fail upper-bound column; the detailed ledger is linked from the supplementary page.

7. Use in the Live Table

The live table retains all seven upper-bound columns because they serve different purposes. The latent, \(m\)-block, fiber-quotient, and CRT columns report algebraic upper bounds extracted from the code structure, the direct-CSS column reports structure-free witness searches, and the decoder-fail column reports simulation-based logical-error witnesses. The displayed row bound is therefore updated whenever any one of these mechanisms yields a strictly smaller valid upper bound.

• A row such as \(P=288\) can remain purely structural while the decoder-fail column is still --.
• A row such as \(P=240\) can be tightened by decoder-fail or completion witnesses after the structural bounds have already been computed.
• A row such as \(P=384\) can accumulate accounted TSUBAME trials without improving the row if the only recovered failures are not useful small logical witnesses.
• The separate \(P=768\) paper-code reference row is an exact transcription of the printed Table 1 parameters in arXiv:2601.08824. Recomputing the latent side gives upper bound \(48\), the standalone note gives a 4-block-constant non-latent \(X\) logical operator of weight \(32\), and the fiber-quotient search gives an \(X\)-logical operator of weight \(24\), so the current overall upper bound for that exact row is \(24\).

For a description of the rest of the comment fields inside the parameter files themselves, see How to Read APM Parameter Files.