Best Known (164, 164+40, s)-Nets in Base 3
(164, 164+40, 688)-Net over F3 — Constructive and digital
Digital (164, 204, 688)-net over F3, using
- t-expansion [i] based on 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
- 4 times m-reduction [i] based on digital (163, 208, 688)-net over F3, using
(164, 164+40, 2460)-Net over F3 — Digital
Digital (164, 204, 2460)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3204, 2460, F3, 40) (dual of [2460, 2256, 41]-code), using
- 245 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0, 1, 12 times 0, 1, 17 times 0, 1, 21 times 0, 1, 28 times 0, 1, 34 times 0, 1, 41 times 0, 1, 48 times 0) [i] based on linear OA(3184, 2195, F3, 40) (dual of [2195, 2011, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(37) [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]
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(39) ⊂ Ce(37) [i] based on
- 245 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0, 1, 12 times 0, 1, 17 times 0, 1, 21 times 0, 1, 28 times 0, 1, 34 times 0, 1, 41 times 0, 1, 48 times 0) [i] based on linear OA(3184, 2195, F3, 40) (dual of [2195, 2011, 41]-code), using
(164, 164+40, 305411)-Net in Base 3 — Upper bound on s
There is no (164, 204, 305412)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 21 515516 795160 603856 637915 863532 570526 368713 105576 391408 297269 183203 480583 340471 341814 906865 796929 > 3204 [i]