New Quantum Error Correcting Codes from Finite Euclidean Plane

In this work we introduce new families of quasi-cyclic binary codes (QC) and quantum error-correcting codes (QECC). The QC codes are constructed from the finite Euclidean plane defined over the field $\mathbb{F}_q$, which provides a finite analogue of the classical Euclidean plane. The resulting codes are shown to be self-orthogonal with respect to both Euclidean and symplectic inner products. A decoding method for the proposed QC codes is also presented. Using these orthogonality properties and standard constructions such as the CSS method, we obtain new families of QECC whose parameters are derived from the structure of the associated QC codes.