Best Known (74−7, 74, s)-Nets in Base 4
(74−7, 74, 5592400)-Net over F4 — Constructive and digital
Digital (67, 74, 5592400)-net over F4, using
- trace code for nets [i] based on digital (30, 37, 2796200)-net over F16, using
- net defined by OOA [i] based on linear OOA(1637, 2796200, F16, 7, 7) (dual of [(2796200, 7), 19573363, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1637, 8388601, F16, 7) (dual of [8388601, 8388564, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(1637, large, F16, 7) (dual of [large, large−37, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(1637, large, F16, 7) (dual of [large, large−37, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1637, 8388601, F16, 7) (dual of [8388601, 8388564, 8]-code), using
- net defined by OOA [i] based on linear OOA(1637, 2796200, F16, 7, 7) (dual of [(2796200, 7), 19573363, 8]-NRT-code), using
(74−7, 74, large)-Net over F4 — Digital
Digital (67, 74, large)-net over F4, using
- 42 times duplication [i] based on digital (65, 72, large)-net over F4, using
- t-expansion [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
- t-expansion [i] based on digital (64, 72, large)-net over F4, using
(74−7, 74, large)-Net in Base 4 — Upper bound on s
There is no (67, 74, large)-net in base 4, because
- 5 times m-reduction [i] would yield (67, 69, large)-net in base 4, but