Gilbert–Varshamov-scale size of optimal binary codes

Determine whether the true size of an optimal binary error-correcting code is asymptotically near the Gilbert–Varshamov lower bound.

Background

The Delsarte linear-programming bound is known to be exponentially larger than the Gilbert–Varshamov lower bound. Consequently, substantially stronger relaxations would be needed if optimal binary codes have sizes close to the Gilbert–Varshamov construction.

The paper records this as a conjectural possibility attributed to specialists and does not resolve it. The question concerns the actual asymptotic size of optimal binary codes, not merely the performance of the higher-order hierarchy analyzed in the paper.

References

If the true size of an optimal binary code is actually near the Gilbert--Varshamov bound (as conjectured by some specialists ), then this family of relaxations needs to be substantially strengthened.

— Higher-order Delsarte Dual LPs: Lifting, Constructions and Completeness  (2501.04854 - Coregliano et al., 8 Jan 2025) in Section 1, Introduction