Best Known (69, 69+8, s)-Nets in Base 4
(69, 69+8, 2097160)-Net over F4 — Constructive and digital
Digital (69, 77, 2097160)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 10)-net over F4, using
- digital (64, 72, 2097150)-net over F4, using
- net defined by OOA [i] based on linear OOA(472, 2097150, F4, 8, 8) (dual of [(2097150, 8), 16777128, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(472, 8388600, F4, 8) (dual of [8388600, 8388528, 9]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(472, large, F4, 8) (dual of [large, large−72, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(472, 8388600, F4, 8) (dual of [8388600, 8388528, 9]-code), using
- net defined by OOA [i] based on linear OOA(472, 2097150, F4, 8, 8) (dual of [(2097150, 8), 16777128, 9]-NRT-code), using
(69, 69+8, large)-Net over F4 — Digital
Digital (69, 77, large)-net over F4, using
- 45 times duplication [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
(69, 69+8, large)-Net in Base 4 — Upper bound on s
There is no (69, 77, large)-net in base 4, because
- 6 times m-reduction [i] would yield (69, 71, large)-net in base 4, but