Uniformly proving the quasi-arc advantage for two-symbol recovery

Determine whether the recovery advantage of balanced quasi-arcs for requested sets of two information symbols can be proved uniformly for natural length-matched families, rather than established only through formulae and numerical comparisons.

Background

The paper compares balanced quasi-arcs with systematic MDS and simplex encoders in dimension three. For requested sets of two information symbols, the authors derive explicit expectations for balanced quasi-arcs and report numerical examples in which length-matched quasi-arcs perform slightly better than systematic MDS encoders. However, the paper does not establish this advantage as a general theorem for a natural family of length-matched constructions. The open problem is therefore to determine whether the observed two-symbol advantage persists uniformly and can be proved analytically.

References

Several problems remain open. First, it would be useful to determine whether the quasi-arc advantage for $|I|=2$ can be proved uniformly for natural length-matched families, rather than only observed through formulae and numerical comparisons.

— The Generalized Random Access Problem for Linear Codes  (2608.20152 - Gruica et al., 20 Aug 2026) in Section 6, “Discussion, comparisons, and open directions”

While it is not possible to show that $M$ is optimal when a point set is a quasi-arc (see Remark \ref{rem:k3}), the following two propositions support the heuristic that these point sets are a good choice.

— Weak arcs and applications to the DNA-based storage access problem  (2608.19550 - Voorde et al., 20 Aug 2026) in Section 4, Subsection 4.2 ("Balanced quasi-arcs are a good choice"), immediately before Proposition 4.1; see also Remark 4.1