A Unified GF(4)–Symplectic Framework for Quantum Error Correction: A Constructive, Pedagogical Derivation of the Steane [[7,1,3]] Code

This paper presents a complete, constructive derivation of the Steane [[7,1,3]] quantum error-correcting code using a unified framework that bridges GF(4) algebra, binary symplectic representation, and stabilizer formalism. We demonstrate how classical coding theory, finite-field arithmetic, and symplectic geometry naturally converge to form a comprehensive foundation for quantum error correction. Starting from the classical Hamming [7,4,3] code, we provide explicit constructions showing: (1) how GF(4) encodes the Pauli group modulo phases, (2) how the symplectic inner product on F²ⁿ₂ captures commutativity, (3) how syndrome extraction reduces to binary matrix multiplication, and (4) how transversal Clifford gates emerge from symplectic transformations. The step-by-step derivation encompasses stabilizer construction, centralizer analysis, logical operator identification, code distance verification, and fault-tolerant syndrome measurement via flagged circuits. All results are derived using elementary finite-field and binary linear algebra, ensuring the exposition is self-contained and accessible. We further illustrate how this algebraic framework extends naturally to modern quantum LDPC codes. This work serves as both a pedagogical tutorial for students entering quantum error correction and a unified reference for researchers implementing stabilizer codes in practice