Best Known (35, 35+15, s)-Nets in Base 3
(35, 35+15, 114)-Net over F3 — Constructive and digital
Digital (35, 50, 114)-net over F3, using
- 1 times m-reduction [i] based on digital (35, 51, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 17, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- trace code for nets [i] based on digital (1, 17, 38)-net over F27, using
(35, 35+15, 168)-Net over F3 — Digital
Digital (35, 50, 168)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(350, 168, F3, 15) (dual of [168, 118, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(350, 242, F3, 15) (dual of [242, 192, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(350, 242, F3, 15) (dual of [242, 192, 16]-code), using
(35, 35+15, 3689)-Net in Base 3 — Upper bound on s
There is no (35, 50, 3690)-net in base 3, because
- 1 times m-reduction [i] would yield (35, 49, 3690)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 239502 726051 030177 508297 > 349 [i]