Best Known (71−7, 71, s)-Nets in Base 4
(71−7, 71, 2801240)-Net over F4 — Constructive and digital
Digital (64, 71, 2801240)-net over F4, using
- net defined by OOA [i] based on linear OOA(471, 2801240, F4, 9, 7) (dual of [(2801240, 9), 25211089, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(471, 2801241, F4, 3, 7) (dual of [(2801241, 3), 8403652, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(410, 5040, F4, 3, 3) (dual of [(5040, 3), 15110, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(410, 5040, F4, 2, 3) (dual of [(5040, 2), 10070, 4]-NRT-code), using
- linear OOA(461, 2796201, F4, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OOA 3-folding [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- linear OOA(410, 5040, F4, 3, 3) (dual of [(5040, 3), 15110, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(471, 2801241, F4, 3, 7) (dual of [(2801241, 3), 8403652, 8]-NRT-code), using
(71−7, 71, large)-Net over F4 — Digital
Digital (64, 71, large)-net over F4, using
- 1 times m-reduction [i] based on digital (64, 72, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(472, large, F4, 8) (dual of [large, large−72, 9]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(472, large, F4, 8) (dual of [large, large−72, 9]-code), using
(71−7, 71, large)-Net in Base 4 — Upper bound on s
There is no (64, 71, large)-net in base 4, because
- 5 times m-reduction [i] would yield (64, 66, large)-net in base 4, but