An explicit pure ((7,2,4))_4 quantum code and its maximality among distance-four codes containing one codeword

We give an explicit pure ((7,2,4))_4 quantum code (equivalently an additive [[7,1/2,4]]_4 stabilizer code): a binary graph state on 14 physical qubits grouped into seven ququarts, with one reference qubit. Exact binary cut-rank computations show that every error supported on at most three ququarts is detected; an explicit weight-four Pauli operator shows the distance is exactly four. Every code of distance at least four that contains the first codeword has dimension at most two, and hence equals the displayed code. Conditional on the published exclusion of stabilizer [[7,1,4]]_4 codes (Ball–Moreno–Simoens, arXiv:2401.06618), dimension two is the largest additive dimension at these parameters. Nothing is claimed about codes not containing the displayed codeword, other than the additive case. Status: not peer reviewed. The author has reviewed the mathematics personally; no independent review by another party has been carried out. Initial searches of the literature did not turn up this code (standard tables list integer k, so a K=2 code would not appear in them); more extensive checks are being carried out. Priority and novelty are pending verification and are not claimed. Files: note (PDF and TeX), graph file, verification script (python3 pure_7_2_4_4_code_verify.py pure_7_2_4_4_code_graph.json), README.