Best Known (56−17, 56, s)-Nets in Base 3
(56−17, 56, 114)-Net over F3 — Constructive and digital
Digital (39, 56, 114)-net over F3, using
- 1 times m-reduction [i] based on digital (39, 57, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 19, 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, 19, 38)-net over F27, using
(56−17, 56, 169)-Net over F3 — Digital
Digital (39, 56, 169)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(356, 169, F3, 17) (dual of [169, 113, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using
(56−17, 56, 3580)-Net in Base 3 — Upper bound on s
There is no (39, 56, 3581)-net in base 3, because
- 1 times m-reduction [i] would yield (39, 55, 3581)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 174 588920 053677 065436 217185 > 355 [i]