Best Known (57−15, 57, s)-Nets in Base 3
(57−15, 57, 192)-Net over F3 — Constructive and digital
Digital (42, 57, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 19, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
(57−15, 57, 311)-Net over F3 — Digital
Digital (42, 57, 311)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(357, 311, F3, 15) (dual of [311, 254, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(357, 364, F3, 15) (dual of [364, 307, 16]-code), using
- the narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(357, 364, F3, 15) (dual of [364, 307, 16]-code), using
(57−15, 57, 11081)-Net in Base 3 — Upper bound on s
There is no (42, 57, 11082)-net in base 3, because
- 1 times m-reduction [i] would yield (42, 56, 11082)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 523 466146 286135 981349 128265 > 356 [i]