Best Known (196−39, 196, s)-Nets in Base 3
(196−39, 196, 688)-Net over F3 — Constructive and digital
Digital (157, 196, 688)-net over F3, using
- 4 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
(196−39, 196, 2296)-Net over F3 — Digital
Digital (157, 196, 2296)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3196, 2296, F3, 39) (dual of [2296, 2100, 40]-code), using
- 96 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 19 times 0, 1, 25 times 0) [i] based on linear OA(3182, 2186, F3, 39) (dual of [2186, 2004, 40]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using
- 96 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 19 times 0, 1, 25 times 0) [i] based on linear OA(3182, 2186, F3, 39) (dual of [2186, 2004, 40]-code), using
(196−39, 196, 312557)-Net in Base 3 — Upper bound on s
There is no (157, 196, 312558)-net in base 3, because
- 1 times m-reduction [i] would yield (157, 195, 312558)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1093 063618 541326 597155 305742 917754 567859 981404 181908 977283 417402 501654 139990 721413 219281 780129 > 3195 [i]