Best Known (209−44, 209, s)-Nets in Base 3
(209−44, 209, 688)-Net over F3 — Constructive and digital
Digital (165, 209, 688)-net over F3, using
- 31 times duplication [i] based on digital (164, 208, 688)-net over F3, using
- t-expansion [i] based on digital (163, 208, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 52, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 52, 172)-net over F81, using
- t-expansion [i] based on digital (163, 208, 688)-net over F3, using
(209−44, 209, 1865)-Net over F3 — Digital
Digital (165, 209, 1865)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3209, 1865, F3, 44) (dual of [1865, 1656, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3209, 2207, F3, 44) (dual of [2207, 1998, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(39) [i] based on
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [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(43) ⊂ Ce(39) [i] based on
- discarding factors / shortening the dual code based on linear OA(3209, 2207, F3, 44) (dual of [2207, 1998, 45]-code), using
(209−44, 209, 154317)-Net in Base 3 — Upper bound on s
There is no (165, 209, 154318)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5228 645429 381763 492083 289336 317140 750252 467051 340031 236849 554857 206065 831659 936508 836264 527726 678821 > 3209 [i]