Best Known (209, 209+30, s)-Nets in Base 3
(209, 209+30, 11821)-Net over F3 — Constructive and digital
Digital (209, 239, 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, 220, 11809)-net over F3, using
- net defined by OOA [i] based on linear OOA(3220, 11809, F3, 30, 30) (dual of [(11809, 30), 354050, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(3220, 177135, F3, 30) (dual of [177135, 176915, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 177147, F3, 30) (dual of [177147, 176927, 31]-code), using
- 1 times truncation [i] based on linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 177147, F3, 30) (dual of [177147, 176927, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(3220, 177135, F3, 30) (dual of [177135, 176915, 31]-code), using
- net defined by OOA [i] based on linear OOA(3220, 11809, F3, 30, 30) (dual of [(11809, 30), 354050, 31]-NRT-code), using
- digital (4, 19, 12)-net over F3, using
(209, 209+30, 84199)-Net over F3 — Digital
Digital (209, 239, 84199)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3239, 84199, F3, 2, 30) (dual of [(84199, 2), 168159, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3239, 88615, F3, 2, 30) (dual of [(88615, 2), 176991, 31]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3238, 88615, F3, 2, 30) (dual of [(88615, 2), 176992, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3238, 177230, F3, 30) (dual of [177230, 176992, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3237, 177229, F3, 30) (dual of [177229, 176992, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(21) [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(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3237, 177229, F3, 30) (dual of [177229, 176992, 31]-code), using
- OOA 2-folding [i] based on linear OA(3238, 177230, F3, 30) (dual of [177230, 176992, 31]-code), using
- 31 times duplication [i] based on linear OOA(3238, 88615, F3, 2, 30) (dual of [(88615, 2), 176992, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3239, 88615, F3, 2, 30) (dual of [(88615, 2), 176991, 31]-NRT-code), using
(209, 209+30, large)-Net in Base 3 — Upper bound on s
There is no (209, 239, large)-net in base 3, because
- 28 times m-reduction [i] would yield (209, 211, large)-net in base 3, but