Best Known (204−41, 204, s)-Nets in Base 3
(204−41, 204, 688)-Net over F3 — Constructive and digital
Digital (163, 204, 688)-net over F3, using
- 4 times m-reduction [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
(204−41, 204, 2287)-Net over F3 — Digital
Digital (163, 204, 2287)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3204, 2287, F3, 41) (dual of [2287, 2083, 42]-code), using
- 79 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0) [i] based on linear OA(3190, 2194, F3, 41) (dual of [2194, 2004, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(39) [i] based on
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [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(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(40) ⊂ Ce(39) [i] based on
- 79 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0) [i] based on linear OA(3190, 2194, F3, 41) (dual of [2194, 2004, 42]-code), using
(204−41, 204, 289086)-Net in Base 3 — Upper bound on s
There is no (163, 204, 289087)-net in base 3, because
- 1 times m-reduction [i] would yield (163, 203, 289087)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7 171869 869738 414747 351994 184492 567816 700449 610036 474326 158236 996677 614335 500293 247845 975591 825369 > 3203 [i]