Best Known (68, 68+7, s)-Nets in Base 4
(68, 68+7, 8388602)-Net over F4 — Constructive and digital
Digital (68, 75, 8388602)-net over F4, using
- net defined by OOA [i] based on linear OOA(475, 8388602, F4, 9, 7) (dual of [(8388602, 9), 75497343, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(475, large, F4, 3, 7), using
- trace code [i] based on linear OOA(6425, 2796201, F64, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- trace code [i] based on linear OOA(6425, 2796201, F64, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(475, large, F4, 3, 7), using
(68, 68+7, large)-Net over F4 — Digital
Digital (68, 75, large)-net over F4, using
- 43 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
(68, 68+7, large)-Net in Base 4 — Upper bound on s
There is no (68, 75, large)-net in base 4, because
- 5 times m-reduction [i] would yield (68, 70, large)-net in base 4, but