Best Known (240−31, 240, s)-Nets in Base 3
(240−31, 240, 11821)-Net over F3 — Constructive and digital
Digital (209, 240, 11821)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 19, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (190, 221, 11809)-net over F3, using
- net defined by OOA [i] based on linear OOA(3221, 11809, F3, 31, 31) (dual of [(11809, 31), 365858, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3221, 177136, F3, 31) (dual of [177136, 176915, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3221, 177136, F3, 31) (dual of [177136, 176915, 32]-code), using
- net defined by OOA [i] based on linear OOA(3221, 11809, F3, 31, 31) (dual of [(11809, 31), 365858, 32]-NRT-code), using
- digital (4, 19, 12)-net over F3, using
(240−31, 240, 64169)-Net over F3 — Digital
Digital (209, 240, 64169)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3240, 64169, F3, 2, 31) (dual of [(64169, 2), 128098, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3240, 88610, F3, 2, 31) (dual of [(88610, 2), 176980, 32]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3239, 88610, F3, 2, 31) (dual of [(88610, 2), 176981, 32]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3237, 88609, F3, 2, 31) (dual of [(88609, 2), 176981, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3237, 177218, F3, 31) (dual of [177218, 176981, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3166, 177147, F3, 23) (dual of [177147, 176981, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(316, 71, F3, 7) (dual of [71, 55, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(3237, 177218, F3, 31) (dual of [177218, 176981, 32]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3237, 88609, F3, 2, 31) (dual of [(88609, 2), 176981, 32]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3239, 88610, F3, 2, 31) (dual of [(88610, 2), 176981, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3240, 88610, F3, 2, 31) (dual of [(88610, 2), 176980, 32]-NRT-code), using
(240−31, 240, large)-Net in Base 3 — Upper bound on s
There is no (209, 240, large)-net in base 3, because
- 29 times m-reduction [i] would yield (209, 211, large)-net in base 3, but