Uniqueness of the Kerdock binary code of block length 256

Determine whether the Kerdock binary code of block length 256, with minimal Hamming distance 120, is unique up to code equivalence (i.e., up to translating and permuting coordinates).

Background

The paper proves uniqueness for block length 64 and non‑uniqueness for block length 1024 (and more generally when 2n−1 is composite), leaving the case of block length 256 unresolved.

Establishing uniqueness or non‑uniqueness for the 256‑length Kerdock code would complete the picture for the small lengths considered.

References

It seems not to be known whether the Kerdock code of block length 256 is unique.

— Optimality of spherical codes via exact semidefinite programming bounds (2403.16874 - Cohn et al., 25 Mar 2024) in Section 5