Best Known (84−17, 84, s)-Nets in Base 3
(84−17, 84, 600)-Net over F3 — Constructive and digital
Digital (67, 84, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 21, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
(84−17, 84, 1389)-Net over F3 — Digital
Digital (67, 84, 1389)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(384, 1389, F3, 17) (dual of [1389, 1305, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(384, 2214, F3, 17) (dual of [2214, 2130, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(357, 2187, F3, 13) (dual of [2187, 2130, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(384, 2214, F3, 17) (dual of [2214, 2130, 18]-code), using
(84−17, 84, 167792)-Net in Base 3 — Upper bound on s
There is no (67, 84, 167793)-net in base 3, because
- 1 times m-reduction [i] would yield (67, 83, 167793)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3990 850356 973552 708779 514587 510128 843169 > 383 [i]