Best Known (144, 144+26, s)-Nets in Base 3
(144, 144+26, 1524)-Net over F3 — Constructive and digital
Digital (144, 170, 1524)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (128, 154, 1514)-net over F3, using
- net defined by OOA [i] based on linear OOA(3154, 1514, F3, 26, 26) (dual of [(1514, 26), 39210, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3154, 19682, F3, 26) (dual of [19682, 19528, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3154, 19682, F3, 26) (dual of [19682, 19528, 27]-code), using
- net defined by OOA [i] based on linear OOA(3154, 1514, F3, 26, 26) (dual of [(1514, 26), 39210, 27]-NRT-code), using
- digital (3, 16, 10)-net over F3, using
(144, 144+26, 11199)-Net over F3 — Digital
Digital (144, 170, 11199)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3170, 11199, F3, 26) (dual of [11199, 11029, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3170, 19744, F3, 26) (dual of [19744, 19574, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(316, 61, F3, 6) (dual of [61, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 80, F3, 6) (dual of [80, 64, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(316, 80, F3, 6) (dual of [80, 64, 7]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3170, 19744, F3, 26) (dual of [19744, 19574, 27]-code), using
(144, 144+26, 4916465)-Net in Base 3 — Upper bound on s
There is no (144, 170, 4916466)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1290 070675 038562 763507 011027 714574 162334 191964 349247 587214 430539 150056 863849 842349 > 3170 [i]