Upper Bounds for the Lengths of s-Extremal Codes Over \\BBF 2, \\BBF 4, and \\BBF 2 + u\\BBF 2

Our purpose is to find an upper bound for the length of an s-extremal code over F₂ (resp. F₄) when d equiv 2 (mod 4) (resp. d odd). This question is left open in [A bound for certain s -extremal lattices and codes, Archiv der Mathematik, vol. 89, no. 2, pp. 143-151, 2007] (resp. [s-extremal additive F₄ codes, Advances in Mathematics of Communications, vol. 1, no. 1, pp. 111-130,2007]). More precisely, we show that if there is an [n, n/2, d] s-extremal Type I binary self-dual code with d > 6 and d equiv 2 (mod 4), then n < 21d - 82. Similarly we show that if there is an (n, 2", d) s-extremal Type I additive self-dual code over F4 with d > 1 and d, equiv 1 (mod 2), then n < 13d - 26. We also define s-extremal self-dual codes over F₂ + uF₂ and derive an upper bound for the length of an .s-extremal self-dual code over F₂ + uF₂ using the information on binary s-extremal codes.