Best Known (181−41, 181, s)-Nets in Base 4
(181−41, 181, 1044)-Net over F4 — Constructive and digital
Digital (140, 181, 1044)-net over F4, using
- 41 times duplication [i] based on digital (139, 180, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 45, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 45, 261)-net over F256, using
(181−41, 181, 3053)-Net over F4 — Digital
Digital (140, 181, 3053)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4181, 3053, F4, 41) (dual of [3053, 2872, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 4096, F4, 41) (dual of [4096, 3915, 42]-code), using
- an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- discarding factors / shortening the dual code based on linear OA(4181, 4096, F4, 41) (dual of [4096, 3915, 42]-code), using
(181−41, 181, 725629)-Net in Base 4 — Upper bound on s
There is no (140, 181, 725630)-net in base 4, because
- 1 times m-reduction [i] would yield (140, 180, 725630)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 348543 363610 649732 243060 829375 778369 160142 163725 302100 270021 200789 047045 788950 178856 694244 878073 268751 329871 > 4180 [i]