Unique Extendability of Arcs

• Unique extendability of arcs is a phenomenon where sets of points in projective spaces admit a unique maximal extension under specific combinatorial and algebraic constraints.
• The concept relies on sharp divisibility conditions, such as (s+2)|q, to ensure a single nucleus point exists that unambiguously extends an arc across dimensions.
• Connections to coding theory and representation theory reveal that unique arc extensions correspond to optimal error-correcting codes and well-classified exceptional sequences.

Unique extendability of arcs encompasses a collection of phenomena and theorems wherein certain sets of points (arcs) in algebraic or combinatorial geometries possess a unique, maximally-large extension subject to explicit combinatorial or algebraic constraints. These results connect finite projective geometry, algebra (especially in the context of modules over preprojective algebras), and coding theory, particularly codes of maximal or almost-maximal length. This exposition surveys the major strands of the unique extendability problem for arcs, consolidating core definitions, classical results, high-dimensional generalizations, combinatorial-algebraic techniques, and connections to error-correcting codes.

1. Classical and Higher-Dimensional Definitions

Let $\mathbb{F}_q$ denote a finite field of order $q$.

• In projective geometry, $PG(k-1, q)$ is the projective space of dimension $k-1$, whose points are $1$-dimensional subspaces of $\mathbb{F}_q^k$. Hyperplanes correspond to $(k-2)$-dimensional subspaces.
• An $(n, r)$-arc in $PG(k-1, q)$ is a set $K$ of $n>r$ points such that every hyperplane meets $K$ in at most $r$ points, and at least one hyperplane meets $K$ in exactly $r$ points. One may write $r = k-1+s$ for defect $s \ge 0$.
• Maximal arcs fit the equality $n = (s+1)(q+1) + (k-2)$. Arcs for which $n < (s+1)(q+1) + (k-2)$ are said to be incomplete.

A correspondence arises in algebra: for $\mathbb{F}_q^k$, an arc (in the sense where each $k$-subset is a basis) is the column set of a generator for a linear MDS code. In the representation theory of preprojective algebras of type $A$, arcs are specific combinatorial objects encoding the morphism and extension data between bricks.

2. Unique Extension: The Plane and Barlotti's Theorem

Barlotti established a foundational result for arcs in $PG(2,q)$ with the following content (Alderson, 9 Nov 2025):

• For $0 < s < q-2$ and $(s+2) \mid q$, if $K \subset PG(2, q)$ is a $((s+1)(q+1), s+2)$-arc, then there exists exactly one point $X$ outside $K$ lying on every tangent line and on no secant. Equivalently, $K$ uniquely extends to a maximal $((s+1)(q+1)+1, s+2)$-arc.

This demonstrates sharp combinatorial control: only for very special $n$ and $s$, and under field divisibility conditions, does unique extendability arise. In other situations, extension is either impossible beyond a smaller maximal arc, or multiple non-equivalent extensions may occur.

3. Generalization to Higher Dimensions

Alderson (Alderson, 9 Nov 2025) proves that Barlotti's phenomenon persists in all projective dimensions, under analogous divisibility and size conditions. The generalized theorem states:

Given $k\geq 3$, $s\geq 0$, $q$ a prime power, and $r=k-1+s$:

• If $K\subset PG(k-1,q)$ is an $(n, r)$-arc of size $n = (s+1)(q+1) + k - 3$, $0 < s < q-2$, and $(s+2) \mid q$, then there exists a unique point $X\in PG(k-1,q)\setminus K$ incident with all tangents to $K$ and no secants. Thus $K$ admits a unique extension to the maximal arc of size $n+1 = (s+1)(q+1) + (k-2)$.

Outline of proof:

• Induction on $k$ via quotient geometries reduces the problem to Barlotti's base case $k=3$.
• The divisibility hypothesis $(s+2)\mid q$ ensures a unique "nucleus" point can be defined.
• Copunctality arguments imply this nucleus extends $K$ uniquely to maximality.
• A plausible implication is that the non-existence of multi-arcs and specific behaviour of $(k-3)$-flats are necessary.

This result is sharp: if $n$ exceeds or falls short of the threshold, or the divisibility fails, uniqueness can fail dramatically.

4. Algebraic and Combinatorial Matrices in Unique Completion

Ball (Ball, 2016) introduces a matrix-theoretic approach to the extension problem for arcs in $\mathbb{F}_q^k$, especially for odd $q$:

• For an arc $G\subset \mathbb{F}_q^k$, construct a family of matrices $\mathcal{M}_n$ (rows: $(k-1)$-subsets of $G$, columns: pairs $(A,E)$ with $|E|=|G|-n$, $A\subset E$ of size $k-2$; entries involve products of determinants).
• The smallest $n$ such that $\mathcal{M}_n$ has a weight-1 column-space vector yields an upper bound $B(G) = q+2k+n-1-|G|$ on the size of any arc extending $G$.
• If the rank of $\mathcal{M}_n$ is one less than the maximum possible, the collection of co-secant hyperplanes (containing exactly $k-2$ points of the extension arc and only from $G$) is determined by $G$.

When these co-secant hyperplanes, as cut out by an algebraic hypersurface $\phi_S$, determine a unique arc, then $G$ is uniquely completable:

• If $\phi_S$ is irreducible of correct degree, unique extendability follows.
• If $\phi_S$ factors, or if multiple sets of arcs share co-secant data, uniqueness may fail.

This suggests the singularity and factorization behaviour of $\phi_S$ is key to identifying unique extension.

5. Unique Extendability in Representation Theory: Bricks and Arcs

The theory of preprojective algebras of type $A$ gives a categorical interpretation (Hanson et al., 2023). Bricks correspond bijectively to arcs in a certain combinatorial model, and their extension spaces are classified:

• For two arcs $\alpha$, $\beta$, set $X = \sigma^{-1}(\alpha)$, $Y = \sigma^{-1}(\beta)$. The extension space $\operatorname{Ext}^1(X,Y)$ is one-dimensional if and only if $\alpha$ and $\beta$ have exactly one Ext-crossing (either a contested endpoint or an interior crossing) and no other intersection.
• The general formula is $$\dim_K\operatorname{Ext}^1(X,Y) = \#(\text{contested endpoints}) + \#(\text{nontrivial crossings})$$.
• When exactly one Ext-crossing occurs, the non-split extension is unique up to scalars, and an explicit basis of short exact sequences can be constructed.
• The unique-extendability criterion enables the classification of all weak exceptional sequences in type $A$ via the combinatorics of arcs.
• The combinatorial model encodes $$\operatorname{Ext}^1(X,Y) \cong K^{\#\text{Ext-crossings}(\alpha,\beta)}$$, yielding a perfect dictionary between extension dimension and arc configuration.

6. Connections to Coding Theory: A$^s$MDS Codes

The geometric theory of arc extension translates directly to projective codes of maximal length and prescribed Singleton defect. Given a projective $[n,k,d]_q$ code represented by $n$ columns (points) in $PG(k-1,q)$, with Singleton defect $s$:

• The length-maximal bound for $A^s$MDS codes is $n \le (s+1)(q+1)+(k-2)$.
• The unique-extendability theorem for arcs implies that if $C$ is $A^s$MDS with $n = (s+1)(q+1)+(k-3)$ and $0$(s+2)\mid q$, then $C$ admits a unique extension to a length-maximal $A^s$MDS $[n+1,k,d']_q$ code (Alderson, 9 Nov 2025).
• When these conditions are not met, codes may admit several inequivalent maximal extensions or no extension at all.

This geometric criterion offers a powerful tool for the classification and construction of optimal and almost-optimal linear codes.

7. Examples and Open Problems

Examples

Paper/Setting	Setup	Unique Extension?
$PG(2,4)$ (Alderson, 9 Nov 2025)	5-arcs (non-degenerate conic), $s=0$, $q=4$	Yes, to unique hyperoval
$PG(2,9)$ (Alderson, 9 Nov 2025)	20-arc (Denniston), $s=1$, $q=9$	Yes, to unique 21-arc
$\mathbb{F}_{13}^3$ (Ball, 2016)	$G$ satisfies Property W, $|G|=9$	Yes, to unique conic
$\mathbb{F}_{11}^3$ (Ball, 2016)	$G$ with $|G|=7$, $(G,2)$ has weight-1 col	No, at most size 10

Outstanding Questions

• Can the $(s+2)\mid q$ divisibility condition be removed or relaxed in higher dimensions?
• What are the combinatorial-geometric obstructions to unique extension for arcs of shorter length?
• What is the full classification of length-maximal $A^s$MDS codes for $k>3$?
• In Ball's framework, does co-secant data always suffice, or can distinct extension arcs have identical co-secant hyperplanes in higher dimensions?

The unique extendability of arcs remains at the intersection of combinatorial geometry, coding theory, and representation theory, where the existence and uniqueness of maximal structures are controlled by sharp algebraic and combinatorial invariants.