Weak arcs and applications to the DNA-based storage access problem

Abstract: Weak arcs are point sets in PG$(n-1,q)$ meeting every general hyperplane (those are the hyperplanes not going through one of the points given by the standard basis vectors) in at most $n-1$ points. In this paper, we study weak arcs together with balanced variants which are contained on the sides of the fundamental simplex. We give an upper bound on the size of weak arcs, characterise the largest balanced quasi-arcs in the plane and construct large balanced quasi-arcs in PG$(3, q)$. We then use these configurations to build point sets for the random-access problem in DNA-based storage. The constructions are explicit, work over small fields, and attain recovery expectations matching the best known asymptotic bounds.

• The paper introduces and develops the theoretical understanding of weak arcs in higher-dimensional projective spaces, applying it to the random-access problem in DNA-based storage.
• It demonstrates that for any weak arc in PG(n, q) with q >= n+3, the upper bound |S| <= nq + (n+1) is derived.
• The paper provides explicit upper bounds and constructive confirmation of weak arcs and multiplicities balancing the expectation based on a subset of sides and a weighted Weberbach-null weighting scheme.

Overview

The paper by Van de Voorde and Zullo (2608.19550) develops the finite-geometric theory of weak arcs in projective spaces $\mathrm{PG}(n,q)$ and applies it to the random-access problem in DNA-based storage. A weak arc (with respect to fundamental points $P_1,\dots,P_{n+1}$ in general position) is a point set meeting every hyperplane that avoids all fundamental points — a general hyperplane — in at most $n$ points. The work extends the planar framework of Gruica, Montanucci, and Zullo to higher dimensions, characterises extremal configurations, and produces explicit point multisets whose worst-case expected recovery time $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$ matches or slightly improves the best known asymptotic bounds, while working over arbitrary small fields.

Size bounds for weak arcs

The authors count general hyperplanes through a point according to its support size $j(Q)$ relative to the fundamental points. Via a recurrence on tuples of nonzero field elements summing to zero, they show that non-fundamental points lying in the plane spanned by three fundamental points (but not on an edge) lie in the fewest general hyperplanes, namely $f_3=(q-1)^{n-2}(q-2)$. Double counting then yields the upper bound

$|S|\le nq+(n+1)$

for any weak arc in $\mathrm{PG}(n,q)$ with $q\ge n+3$. The bound is attained by the union of $n$ lines through one fundamental point together with the remaining fundamental points; for $P_1,\dots,P_{n+1}$0 and large $P_1,\dots,P_{n+1}$1, the converse is claimed (citing work in preparation), so the bound is tight in the planar case.

A probabilistic construction of balanced quasi-arcs — weak arcs contained in the edges of the fundamental simplex with equal parameter $P_1,\dots,P_{n+1}$2 per edge — is also given: for $P_1,\dots,P_{n+1}$3 edges, balanced quasi-arcs exist for every $P_1,\dots,P_{n+1}$4 with $P_1,\dots,P_{n+1}$5. The authors are explicit that this random method is only a baseline: it yields sets of order $P_1,\dots,P_{n+1}$6, far below what explicit constructions achieve.

Planar balanced quasi-arcs

The planar problem reduces cleanly to multiplicative combinatorics: parametrising side points as $P_1,\dots,P_{n+1}$7, $P_1,\dots,P_{n+1}$8, $P_1,\dots,P_{n+1}$9, collinearity occurs exactly when $n$0, so a balanced quasi-arc corresponds to sets $n$1 with $n$2 and $n$3.

For odd $n$4, Kneser's theorem gives a complete characterisation: the maximum parameter is $n$5, attained uniquely (up to coset structure) by taking $n$6 to be the subgroup of squares and $n$7 the nonsquares, or all three sets equal to the nonsquares. For even $n$8, no index-$n$9 subgroup exists, and the extremal configuration instead uses unions of consecutive cosets of a proper subgroup. The general answer is expressed through the exact minimum product-set size in cyclic groups,

$$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$0

which yields a sharp formula for $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$1 over divisor pairs $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$2 with $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$3. Notably, the natural guess that a single subgroup is optimal fails: for $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$4, two cosets of the index-$$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$5 subgroup give $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$6, beating the single-subgroup value $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$7, and this is best possible.

Weak caps and weak arcs in three dimensions

In $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$8, the authors construct a weak cap (every line avoiding fundamental points meets it in at most two points) contained in the edges of the fundamental tetrahedron, using a multiplicative subgroup $$M(\mathcal G)=\max_i \mathbb E[\tau_{P_i}(\mathcal G)]$$9: assigning $j(Q)$0 or its complement $j(Q)$1 to the six edges so that every facial collinearity condition forces a product relation contradicting the assignment. For odd $j(Q)$2 with $j(Q)$3 the squares this gives a weak cap of size $j(Q)$4 with $j(Q)$5 points per edge; a symmetric variant using only nonsquares also works since $j(Q)$6.

For $j(Q)$7, selecting appropriate cosets of an index-$j(Q)$8 subgroup on each edge produces a genuine weak arc of size $j(Q)$9, by verifying via coset arithmetic that the three possible coplanarity conditions cannot hold. This gives large balanced quasi-arcs in three dimensions, though the construction requires the congruence condition on $f_3=(q-1)^{n-2}(q-2)$0.

Analysis of recovery expectations

The paper uses the identity $f_3=(q-1)^{n-2}(q-2)$1, where $f_3=(q-1)^{n-2}(q-2)$2 counts $f_3=(q-1)^{n-2}(q-2)$3-subsets whose span misses $f_3=(q-1)^{n-2}(q-2)$4, together with a telescoping evaluation $f_3=(q-1)^{n-2}(q-2)$5.

Three structural results support the heuristic that quasi-arcs perform well:

• Extending a balanced quasi-arc by one point per side strictly beats adding three generic off-side points while preserving the weak-arc property.
• Among weak arcs supported on the sides of the triangle with fixed total size, the balanced configuration minimises $f_3=(q-1)^{n-2}(q-2)$6, with strict inequality otherwise (proved via a nontrivial inequality on a function $f_3=(q-1)^{n-2}(q-2)$7 that is not convex).
• For odd $f_3=(q-1)^{n-2}(q-2)$8, among all balanced point sets on the sides of the triangle, the balanced quasi-arc of parameter $f_3=(q-1)^{n-2}(q-2)$9 is optimal, giving

$|S|\le nq+(n+1)$0

An important caveat is stated plainly: this optimality holds only within the class of side-supported balanced sets. Taking the entire fundamental triangle achieves the same asymptotic limit, and allowing points off the sides with suitable weights does substantially better (planar optimised weights reach approximately $|S|\le nq+(n+1)$1).

Explicit constructions and comparison

Two weighted 3-dimensional constructions are computed exactly. First, the full tetrahedron with fundamental points of multiplicity $|S|\le nq+(n+1)$2 has length $|S|\le nq+(n+1)$3 and normalised expectation tending to $|S|\le nq+(n+1)$4. Second, the weighted weak arc with $|S|\le nq+(n+1)$5 points per edge and fundamental-point multiplicity $|S|\le nq+(n+1)$6 has length $|S|\le nq+(n+1)$7 and the same asymptotic limit, but is uniformly better at fixed size. The latter realises the recovery-complete family $|S|\le nq+(n+1)$8 of Boruchovsky et al. geometrically; the paper notes the equivalence between their graph-theoretic recovery rule (the component containing a cycle) and the geometric span condition, and independently verifies their limiting value. Crucially, the geometric construction needs only a small field: $|S|\le nq+(n+1)$9 is realised explicitly over $\mathrm{PG}(n,q)$0, versus a sufficient field size of $\mathrm{PG}(n,q)$1 in the general construction of Boruchovsky et al. (which the authors note is not claimed minimal).

Third, reinterpreting the Wang–Yaakobi construction stratified by Hamming weight, the paper proposes the explicit integer-multiplicity multiset

$\mathrm{PG}(n,q)$2

of length $\mathrm{PG}(n,q)$3, achieving

$\mathrm{PG}(n,q)$4

which slightly improves the numerical bound $\mathrm{PG}(n,q)$5 recorded in Wang–Yaakobi, while being defined over every finite field. The trade-off is the considerably larger length compared to the sparse weak-arc construction.

Construction	Length	Field	$\mathrm{PG}(n,q)$6
Full tetrahedron, weighted	$\mathrm{PG}(n,q)$7	any $\mathrm{PG}(n,q)$8	$\mathrm{PG}(n,q)$9
Weighted weak arc	$q\ge n+3$0	explicit, small $q\ge n+3$1	$q\ge n+3$2
$q\ge n+3$3 (Boruchovsky et al.)	$q\ge n+3$4	large-field sufficient	$q\ge n+3$5
$q\ge n+3$6	$q\ge n+3$7	any $q\ge n+3$8	$q\ge n+3$9
Wang–Yaakobi	—	large $n$0	$n$1

Limitations and open questions

Several restrictions are acknowledged. The sharp planar classification for even $n$2 does not characterise which configurations attain $n$3, only the value. The three-dimensional weak arc requires $n$4, and no analogue of the planar complete characterisation is given in higher dimension. The optimality results for quasi-arcs are confined to side-supported balanced sets; the paper shows explicitly that quasi-arcs are not globally optimal ($n$5 can be improved by off-side weighting), so the broader question of which geometric configurations minimise $n$6 remains open. The converse to the size bound for weak arcs is established only for $n$7 and large $n$8, citing unpublished work. Finally, the claim that the Wang–Yaakobi optimisation is improved rests on a specific discrete choice of multiplicities; no proof is given that the chosen ratios are optimal within the strata-weighting family.

Conclusion

The paper establishes tight bounds and, in the planar odd-characteristic case, a full classification of largest balanced quasi-arcs, extends these structures to $n$9 via subgroup-coset assignments, and demonstrates through exact computations that the resulting configurations are competitive with state-of-the-art constructions for the DNA random-access problem. Its practical contribution is a family of explicit, small-field constructions attaining the best known asymptotic normalised recovery expectations, together with a precise account of when quasi-arcs are and are not the right geometric object for minimising $P_1,\dots,P_{n+1}$00.