Extremal self-dual codes over F ₂ × F ₂

In this paper, it is shown that extremal (Hermitian) self-dual codes over F ₂ × F ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over F ₂ × F ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance ≥ 4.