Best Known (102, 102+33, s)-Nets in Base 3
(102, 102+33, 328)-Net over F3 — Constructive and digital
Digital (102, 135, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (102, 136, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 34, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 34, 82)-net over F81, using
(102, 102+33, 690)-Net over F3 — Digital
Digital (102, 135, 690)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3135, 690, F3, 33) (dual of [690, 555, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 749, F3, 33) (dual of [749, 614, 34]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(3130, 729, F3, 34) (dual of [729, 599, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3112, 729, F3, 29) (dual of [729, 617, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3135, 749, F3, 33) (dual of [749, 614, 34]-code), using
(102, 102+33, 33666)-Net in Base 3 — Upper bound on s
There is no (102, 135, 33667)-net in base 3, because
- 1 times m-reduction [i] would yield (102, 134, 33667)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8596 641498 637942 777863 809335 910374 071213 145968 873163 533689 349345 > 3134 [i]