The constructions of Singleton-optimal locally repairable codes with minimum distance 6 and locality 3

In this paper, we present new constructions of $q$-ary Singleton-optimal locally repairable codes (LRCs) with minimum distance $d=6$ and locality $r=3$, based on combinatorial structures from finite geometry. By exploiting the well-known correspondence between a complete set of mutually orthogonal Latin squares (MOLS) of order $q$ and the affine plane $\mathrm{AG}(2,q)$, We systematically construct families of disjoint 4-arcs in the projective plane $\mathrm{PG}(2,q)$, such that the union of any two distinct 4-arcs forms an 8-arc. These 4-arcs form what we call 4-local arcs, and their existence is equivalent to that of the desired codes. For any prime power $q\ge 7$, our construction yields codes of length $n = 2q$, $2q-2$, or $2q-6$ depending on whether $q$ is even, $q\equiv 3 \pmod{4}$, or $q\equiv 1 \pmod{4}$, respectively.

💡 Research Summary

The paper addresses the construction of q‑ary Singleton‑optimal locally repairable codes (LRCs) with minimum distance d = 6 and locality r = 3, a parameter regime that has attracted considerable attention because of its relevance to distributed storage systems. The Singleton‑type bound for LRCs, d ≤ n − k − ⌈k/r⌉ + 2, is tight for the codes sought here. Prior work achieved such optimality only for relatively short lengths (typically n ≤ q + 1 or, with more sophisticated algebraic curves, n ≈ q + 2√q). This work dramatically extends the achievable length to roughly twice the field size, namely n = 2q, 2q − 2, or 2q − 6, depending on the parity and congruence class of q.

The authors’ key insight is to translate the coding problem into a geometric one in the projective plane PG(2,q). They start from a complete set of mutually orthogonal Latin squares (MOLS) of order q, which is known to exist whenever q is a prime power. By interpreting the rows, columns, and symbols of each Latin square as coordinates (x, y) and a slope m, they obtain the classical affine plane AG(2,q): points are ordered pairs (x, y)∈F_q², and each line is defined by the equation L(m,ℓ) = { (i, j) | L_m(i, j) = ℓ }. Adding the points at infinity and the line at infinity yields the projective plane PG(2,q).

A central combinatorial object is a “4‑local arc”: a collection of disjoint 4‑arcs S₁,…,S_m in PG(2,q) such that the union of any two distinct arcs forms an 8‑arc (no line meets the union in three or more points). Lemma 1.1 (cited from earlier work) shows that the existence of a 4‑local arc with m = n/4 is equivalent to the existence of a Singleton‑optimal (n, k, d = 6; r = 3) LRC with disjoint repair groups.

To construct such arcs, the authors exploit transversals of a specially structured Latin square. For q = 2^r (even case) they consider the Cayley table of the additive group (F_{2^r}, +), denoted L(r). This square can be recursively built from L(r‑1) by a block construction. A transversal Γ of L(r‑1) (a set of q/2 positions covering each row, column, and symbol exactly once) exists because L(r‑1) has an orthogonal mate when r ≥ 2 (Lemma 4.1). For each (i, j)∈Γ they define a quartet of affine points Σ_u = { (i, j), (i+α, j), (i, j+α), (i+α, j+α) }, where α = ω^{r‑1} and ω is a primitive element of F_{2^r}. The four points of Σ_u form a 4‑arc in AG(2, q); the choice of α guarantees that no three of them are collinear. Moreover, because Γ is a transversal, any two distinct Σ_u and Σ_{u′} together contain exactly eight points with the property that no line contains three of them, i.e., they form an 8‑arc. Embedding these points into PG(2,q) yields a 4‑local arc of size 2q, which by Lemma 1.1 gives a Singleton‑optimal LRC of length n = 2q.

When q is odd, the same construction works but the parameter α is no longer equal to 1, and the transversal structure depends on the congruence of q modulo 4. By carefully selecting which transversals to keep or discard, the authors obtain codes of length n = 2q − 2 for q ≡ 3 (mod 4) and n = 2q − 6 for q ≡ 1 (mod 4). The resulting codes retain disjoint repair groups of size r + 1 = 4, satisfying the locality requirement.

The paper proceeds as follows. Section 2 reviews the parity‑check matrix representation of LRCs with disjoint repair groups, the definitions of affine and projective planes, and basic facts about Latin squares and MOLS. Section 3 establishes the correspondence between a complete set of MOLS and the affine plane AG(2,q), providing explicit formulas for lines through any two points (Proposition 3.2). Section 4 contains the core constructions: Subsection 4.1 treats the even‑q case, proving Theorem 4.2 that the Σ_u sets form a 4‑local arc; Subsection 4.2 adapts the method to odd q, handling the two congruence classes separately. The main result, Theorem 1.2, follows by counting the number of 4‑arcs and applying Lemma 1.1.

Compared with earlier constructions based on elliptic curves, cyclic or constacyclic codes, or hypergraph methods, the present approach offers several advantages:

1. Explicitness – The code coordinates are given directly by simple algebraic formulas involving field elements, making implementation straightforward.
2. Length scalability – For any prime power q ≥ 7 the length reaches 2q (or only a few symbols short of it), which is asymptotically optimal with respect to the field size.
3. Uniform repair groups – The repair groups are exactly the four points of each Σ_u, guaranteeing disjointness and simplifying repair operations.
4. Geometric insight – By interpreting the code construction as a problem of arranging arcs in PG(2,q), the authors provide a clear geometric picture that may inspire further extensions (e.g., larger locality, higher distance).

The paper concludes with remarks on open problems: extending the method to locality r > 3, to distances d > 6, or to non‑prime‑power fields; improving the length for the q ≡ 1 (mod 4) case; and developing efficient encoding/decoding algorithms that exploit the underlying geometric structure.

In summary, the authors present a novel, geometry‑driven construction of Singleton‑optimal (n, k, d = 6; r = 3) LRCs that substantially enlarges the attainable code length while preserving the desirable property of disjoint repair groups. The work bridges combinatorial design theory, finite geometry, and coding theory, and opens new avenues for constructing high‑performance locally repairable codes.