How Can Group-Valued Codes Achieve Full Regular Low-Weight Logical Bases?

Frame width joins distance and check weight

Distance d fixes the minimum weight of a nontrivial logical operator, but it does not tell you how heavy the operators become once you must supply canonical logical pairs for every encoded qubit. The paper introduces frame width f, the smallest weight cutoff that permits a complete canonical logical basis, and treats it as a code parameter alongside distance d and check weight w. The same idea appeared earlier for CSS bases as optimized conjugate-basis width.

The distinction matters because reported canonical bases for high-rate codes can have maximum weights several times the available distance upper bounds, even after stabilizer-coset reduction. A basis organized into regular orbits is attractive because all canonical pairs arise by translating a few seed pairs, so coupling patterns can be reused. The paper's question is whether that organization can be guaranteed while keeping every basis operator's weight small, ideally close to distance.

Odd-order groups force distance two

Full regularity first runs into a structural obstruction. For Abelian groups of odd order, unequal logical multiplicities across symmetry components block full regularity. The paper illustrates this with an LP code whose fully populated monomial constituent produces 16 regular copies accounting for 720 logical pairs, plus 64 additional invariant pairs that make component multiplicities unequal.

For fully populated binary monomial CSS checks over a group of odd order, full regularity forces d = 2 whenever k > 0. The paper notes this obstruction is specific to the binary construction, since nonbinary coefficients can avoid it. For groups of power-of-two order, the criterion flips: full row rank of both check matrices after replacing every group element by 1 and summing coefficients guarantees full regularity. That gives a usable design rule, because zero patterns and coefficients can be chosen to ensure complete regular orbits before searching group labels, subject to CSS commutation.

Rate comes from dimensions, distance from search

The design strategy sets the encoding rate k/n through matrix dimensions: those of the classical constituents for LP codes and of the group-valued check matrices for PP codes. Under the corresponding full-rank augmentation conditions, these dimensions fix the rate and guarantee full regularity over a finite 2-group. For a fixed coefficient field, the group order then sets the blocklength, and the search looks for large distance under prescribed limits on check weight and reported frame width.

Sparse checks do not automatically give sparse logical operators, because solving the check equations and imposing canonical pairing can make logical operators much heavier. The constructions control both steps: X- and Z-type seeds are specified on a common qubit block, their components on other blocks are placed on disjoint sets so both operators commute with every stabilizer while keeping shared support, and sparse corrections keep the seeds low-weight. The LP arrangement gives canonical pairing directly, while PP codes need additional constraints to pair canonically without increasing seed weights. Quaternary coefficients enlarge the design space by letting fully populated matrices satisfy the rank conditions while their cancellation patterns keep logical corrections sparse.

Explicit codes keep frames near distance

At check weight w = 10, the paper reports an [[1088,128,22]] LP code with proved optimal frame width 22 and an [[1024,256,24]] PP code with reported frame width 27. The largest seed weight is the reported frame width, because translation preserves weight, and it supplies an upper bound on f. A star in the code table marks entries where the reported frame width equals f.

Within the same PP length and rate, allowing w = 11 yields either distance 25 with reported frame width 27, or distance 24 with the proved optimum f = 25. These examples show check weight being exchanged for distance or for a narrower canonical frame. The paper also compares its [[1024,256,24]] PP code against a [[975,195,≤24]] mitten code: rate 1/4 rather than 1/5, width 27 rather than 102, and check weight 10 rather than 9.

Distance and frame width can strictly separate

The distance-24, width-25 PP example establishes a strict separation between finding a shortest logical operator and supplying a complete logical basis. Exhaustive search shows that every Z logical of weight at most 24 acts on an even number of qubits within two specified binary blocks, whereas some Z logicals have odd parity there. Multiplication by stabilizers preserves this parity, and products of even-parity operators remain even, so the weight-24 operators cannot generate the missing logical classes.

Although weight-24 logical operators exist, even the collection of all such operators fails to span the logical space, so every complete canonical basis must contain an operator of weight at least 25. The full regular frame attains this bound, proving d = 24 < f = 25. The paper states this as a proposition covering the code defined by its coefficient array and exponents, with f_span = f = f_reg = 25, including for unrestricted mixed-Pauli canonical bases.

Low-weight spanning is not low-weight pairing

A second limitation appears in the distance-20 LP example. Each logical sector separately has a complete regular basis consisting of weight-20 operators, yet these minimum-weight operators cannot be arranged into a full regular canonical frame under the prescribed C16 action, because the required pairings cannot all be satisfied simultaneously. Every such frame has width at least 21, while the construction achieves 22.

Here the obstruction concerns canonical pairing compatible with the specified symmetry, and the unrestricted frame width remains unresolved. The paper draws the general lesson that low-weight spanning and low-weight canonical pairing are separate design requirements, even when full regularity is already guaranteed. Full regularity, low-weight spanning, and low-weight canonical pairing impose distinct constraints on code design.

Certifying distances above a thousand qubits

The paper proposes a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and uses it to establish exact distances for LP examples with more than a thousand qubits. Searches on smaller constituent and quotient codes reject candidates whose distance upper bounds fall below the target, building on logical spectroscopy's distance filters. For the tabulated codes, exhaustive full-code searches exclude every logical operator lighter than a known nontrivial witness.

The construction supplies an explicit canonical frame, the search selects codes with larger distance, and exhaustive exclusions then certify those distances. The companion repository supplies the full construction catalogue, physical frames and witnesses, independent checkers, search sources, and completion records. The paper distinguishes finite certificates from exhaustive execution evidence, and large traversal trees are omitted.

Where the gains hold and where they stop

The gains occur at finite blocklength. The 1088-qubit LP code attains d = f = 22, and the 512-qubit PP examples exhibit the tradeoff between check and frame weights. Matrix dimensions permit higher encoding rates, with distance and frame width to be optimized at each choice. At a few hundred to about a thousand qubits, rate 1/4 gives a useful balance for the PP examples. At small blocklengths, the examples lie below the reported kd^2/n frontier, but they supply explicit full regular canonical bases with controlled weights.

Within the 1024-qubit PP family, attaining d = 24, w = 10, and f = 25 together remains open. The existence criteria also cover non-Abelian 2-groups, but extending these constructions to full regular canonical frames requires a common physical group action preserving both CSS sectors, which remains open for those examples. The paper's own framing is that frame width specifies the support needed to supply a complete canonical logical basis directly, and that the obstructions concern such bases rather than all possible generating sets for logical operations.

The most decision-relevant point is the separation the paper proves rather than the code table alone: for the distance-24 PP construction, minimum-weight logicals fail to span the logical space, so a designer who budgets only for distance 24 will underestimate the support a complete canonical basis requires. The practical threshold is therefore f, not d, whenever the task is to supply canonical logical pairs, and the paper's clearest tradeoff is that at 1024 qubits and rate 1/4, check weight 11 buys either distance 25 with frame width 27 or distance 24 with the proved optimum frame width 25.

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