Best Known (189, 219, s)-Nets in Base 3
(189, 219, 3948)-Net over F3 — Constructive and digital
Digital (189, 219, 3948)-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 (170, 200, 3936)-net over F3, using
- net defined by OOA [i] based on linear OOA(3200, 3936, F3, 30, 30) (dual of [(3936, 30), 117880, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(3200, 59040, F3, 30) (dual of [59040, 58840, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3200, 59049, F3, 30) (dual of [59049, 58849, 31]-code), using
- 1 times truncation [i] based on linear OA(3201, 59050, F3, 31) (dual of [59050, 58849, 32]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, 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(3201, 59050, F3, 31) (dual of [59050, 58849, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3200, 59049, F3, 30) (dual of [59049, 58849, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(3200, 59040, F3, 30) (dual of [59040, 58840, 31]-code), using
- net defined by OOA [i] based on linear OOA(3200, 3936, F3, 30, 30) (dual of [(3936, 30), 117880, 31]-NRT-code), using
- digital (4, 19, 12)-net over F3, using
(189, 219, 29563)-Net over F3 — Digital
Digital (189, 219, 29563)-net over F3, using
- 31 times duplication [i] based on digital (188, 218, 29563)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3218, 29563, F3, 2, 30) (dual of [(29563, 2), 58908, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3218, 59126, F3, 30) (dual of [59126, 58908, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(21) [i] based on
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(317, 77, F3, 7) (dual of [77, 60, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3218, 59126, F3, 30) (dual of [59126, 58908, 31]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3218, 29563, F3, 2, 30) (dual of [(29563, 2), 58908, 31]-NRT-code), using
(189, 219, large)-Net in Base 3 — Upper bound on s
There is no (189, 219, large)-net in base 3, because
- 28 times m-reduction [i] would yield (189, 191, large)-net in base 3, but