Best Known (199−40, 199, s)-Nets in Base 3
(199−40, 199, 688)-Net over F3 — Constructive and digital
Digital (159, 199, 688)-net over F3, using
- t-expansion [i] based on digital (157, 199, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (157, 200, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 50, 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, 50, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (157, 200, 688)-net over F3, using
(199−40, 199, 2265)-Net over F3 — Digital
Digital (159, 199, 2265)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3199, 2265, F3, 40) (dual of [2265, 2066, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3199, 2273, F3, 40) (dual of [2273, 2074, 41]-code), using
- 70 step Varšamov–Edel lengthening with (ri) = (4, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0) [i] based on 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]
- 70 step Varšamov–Edel lengthening with (ri) = (4, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0) [i] based on linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3199, 2273, F3, 40) (dual of [2273, 2074, 41]-code), using
(199−40, 199, 232057)-Net in Base 3 — Upper bound on s
There is no (159, 199, 232058)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 88539 287026 030068 856972 170625 582343 155665 651143 518521 780686 482578 491135 771801 748749 507227 483977 > 3199 [i]