Closed-form distance formula for the even-distance GSE code family

Derive a general closed-form formula for the code distance of the even-distance ring-geometry Generalized Superfast Encoding as a function of the even distance parameter d.

Background

The paper introduces an even-distance Generalized Superfast Encoding in which N fermionic modes are represented using d physical qubits per mode, with d even. The construction has stabilizer generators of weight 4 or 6 and is explicitly verified for the [48,8,6] and [64,8,8] instances.

For these two instances, the authors report the distance empirically rather than proving a formula valid for arbitrary even d. A general closed-form characterization of the distance would establish how the code family scales with its distance parameter and would support systematic finite-size analyses beyond the two demonstrated examples.

References

We report distance empirically for these two instances; a general closed-form distance formula as a function of $d$ is left to future work.

— First fault-tolerant quantum memory demonstration for a generalized superfast encoding  (2609.08957 - Brown et al., 8 Sep 2026) in Section 3, subsection “Stabilizer generators”