Upper Bounds for the length of s - Extremal Codes over F 2 , F 4 , and F 2 + u F 2
نویسندگان
چکیده
Our purpose is to find an upper bound for the length of s-extremal codes over F2 (resp. F4) when d ≡ 2 (mod 4) (resp. d odd). This question is left open in [6], [2]. More precisely, we show that there is no s-extremal binary code of length n ≥ 21d− 82 if d > 6 and d ≡ 2 (mod 4). Similarly we show that there is no s-extremal additive F4 code of length n ≥ 13d− 26 if d > 1 and d is odd. 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.
منابع مشابه
Cyclic Codes and Self-Dual Codes Over F2 + uF2
We introduce linear cyclic codes over the ring F 2 + uF 2 = f0; 1; u; u = u + 1g, where u 2 = 0. This ring shares many properties of Z 4 and F 4 and admits a linear "Gray map". Cyclic codes are described as modules over (F 2 + uF 2) n which may not be free. Self-dual codes of odd length exists as in the case of Z 4-codes. We exhibit some extremal codes of this very interesting family.
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