Best Known (149−15, 149, s)-Nets in Base 4
(149−15, 149, 1198419)-Net over F4 — Constructive and digital
Digital (134, 149, 1198419)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (9, 16, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 8, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 8, 24)-net over F16, using
- digital (118, 133, 1198371)-net over F4, using
- net defined by OOA [i] based on linear OOA(4133, 1198371, F4, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4133, 8388598, F4, 15) (dual of [8388598, 8388465, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4133, 8388598, F4, 15) (dual of [8388598, 8388465, 16]-code), using
- net defined by OOA [i] based on linear OOA(4133, 1198371, F4, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
- digital (9, 16, 48)-net over F4, using
(149−15, 149, large)-Net over F4 — Digital
Digital (134, 149, large)-net over F4, using
- 44 times duplication [i] based on digital (130, 145, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4145, large, F4, 15) (dual of [large, large−145, 16]-code), using
- strength reduction [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- strength reduction [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4145, large, F4, 15) (dual of [large, large−145, 16]-code), using
(149−15, 149, large)-Net in Base 4 — Upper bound on s
There is no (134, 149, large)-net in base 4, because
- 13 times m-reduction [i] would yield (134, 136, large)-net in base 4, but