Upper bounds for the lengths of s-extremal codes over F2, F4, and F2 + uF2

초록

Our purpose is to find an upper bound for the length of an s-extremal code over F-2 (resp. F-4) when d = 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 F 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 2 d = 2 (mod 4), then n < 21d - 82. Similarly we show that if there is an (n, 2(n) , d) s-extremal Type I additive self-dual code over F-4 with d > 1 and d = 1 (mod 2), then n < 13d - 26. We also define s-extremal self-dual codes over F-2 + uF(2) and derive an upper bound for the length of an s-extremal self-dual code over F-2 + uF(2) using the information on binary s-extremal codes.

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