An Explicit Skew-Hadamard Matrix of Order 1252 via Cyclotomic Unions

We construct a skew-Hadamard matrix of order 1252 = 2(5^4 + 1) using a bordered skew-Hadamard difference family over GF(5^4), with blocks given as unions of cyclotomic classes of order N = 16. This order has been reported as missing in some widely used open-source computational tables; we provide an explicit instance together with verification artifacts. We prove the structural prerequisites for the bordered construction (skew-symmetry of one block and the constant autocorrelation-sum condition), and we compute algebraic invariants to facilitate classification: the associated tournament adjacency matrix has full rank over GF(2), and the matrix has full rank over GF(3) and GF(5). We also exhibit an explicit affine subgroup of the automorphism group of size 24 375. All claims are supported by a reproducible artifact bundle including the explicit matrix and verification logs.

💡 Research Summary

The paper presents a concrete construction of a skew‑Hadamard matrix of order 1252, filling a gap in publicly available databases that list this order as missing. The authors work over the finite field GF(5⁴) (size 625) and exploit cyclotomic classes of order N = 16. Let g be a primitive element of GF(5⁴)×; the 16 cyclotomic classes C₀,…,C₁₅ are defined as C_i = g^i·⟨g¹⁶⟩, each containing 39 non‑zero field elements. Two index sets are chosen: I₀ = {4,5,6,7,8,9,10,11} and I₁ = {0,1,2,3,4,5,6,7}. The blocks of the difference family are defined as D₀ = ⋃{i∈I₀} C_i and D₁ = ⋃{i∈I₁} C_i.

The first requirement of a bordered skew‑Hadamard difference family (SHDF) is that D₀ be skew‑symmetric: for every non‑zero x, x ∈ D₀ iff –x ∉ D₀. This follows because –1 = g^{(5⁴−1)/2}=g^{312} lies in C₈, so multiplication by –1 maps C_i to C_{i+8} (indices modulo 16). Since I₀ is a transversal of the pairing i ↔ i+8, D₀ and –D₀ are disjoint and together cover the whole multiplicative group, establishing skew‑symmetry.

The second SHDF condition is a constant autocorrelation‑sum: for every non‑zero shift w, the periodic autocorrelations satisfy P_{D₀}(w) + P_{D₁}(w) = –2. The authors verify this exhaustively by a deterministic Python script that iterates over all 624 non‑zero elements of GF(5⁴) and records a log confirming the identity for each w.

Having satisfied both conditions, the standard Goethals–Seidel bordered construction (Lemma 1) yields a skew‑Hadamard matrix H of order 2(v + 1) = 2·626 = 1252, where v = |GF(5⁴)| = 625. The explicit matrix is supplied in a text file, and two independent verification runs (a pure Python checker and a SageMath computation) confirm that H·Hᵀ = 1252·I and H + Hᵀ = 2·I.

Beyond existence, the paper computes several algebraic invariants. After normalising H so that its first row consists of all +1 entries, the core S (obtained by deleting the first row and column) is used to form the tournament adjacency matrix M = J – S², where J is the all‑ones matrix. The rank of M over GF(2) is full (1251), and the rank of H over GF(3) and GF(5) is also full (1252). These full‑rank properties indicate strong linear independence of the matrix in various modular settings, which is relevant for applications such as coding theory and cryptographic constructions.

The automorphism group of H is shown to contain a sizable affine subgroup. Because D₀ and D₁ are unions of C₀‑cosets, they are invariant under multipliers x ↦ u·x with u ∈ C₀ (|C₀| = 39). Moreover, the group‑developed nature of the construction makes the translation x ↦ x + a (a ∈ GF(5⁴)) an automorphism as well. Consequently, the subgroup Aff_{C₀}(1,5⁴) = { x ↦ u·x + a | u ∈ C₀, a ∈ GF(5⁴) } embeds in Aut(H), giving a lower bound |Aut(H)| ≥ 39·625 = 24 375. The authors provide a script and log that verify the action of this subgroup on the matrix.

To illustrate practical relevance, the authors implement EdgeSketch‑1252, an embedding‑compression pipeline for edge‑AI scenarios. The transformation y = (1/√1252)·H·x maps a 1252‑dimensional floating‑point embedding x to a lower‑dimensional representation; the top‑k (k≈300) components are retained and quantised to 8 bits. On a synthetic drone‑inspection dataset (N = 100, seed = 42), EdgeSketch‑1252 achieves a compression ratio of 5.52× (from 5008 bytes to 908 bytes) and an F1 score of 0.62, comparable to baselines based on power‑of‑two Hadamard matrices (orders 1024 and 2048). The finer granularity of order 1252 allows more precise bandwidth‑performance trade‑offs, and the deterministic, fully verified transformation ensures reproducibility.

Finally, the authors package all core artifacts (the matrix file, block definitions, a manifest with SHA‑256 digests) and execution logs (verification of SHDF conditions, Gate0 checks, automorphism checks) into a reproducible bundle. The bundle separates immutable core data from run‑time logs, facilitating independent verification and future extensions.

In summary, the paper delivers a fully explicit, verifiable construction of a skew‑Hadamard matrix of order 1252, enriches the catalogue of known orders, demonstrates the utility of cyclotomic‑class unions in difference‑family design, provides detailed algebraic invariants, identifies a substantial automorphism subgroup, and showcases a concrete edge‑AI application, all supported by a transparent, reproducible artifact suite.