Best Known (75, 75+12, s)-Nets in Base 3
(75, 75+12, 9848)-Net over F3 — Constructive and digital
Digital (75, 87, 9848)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- a shift-net [i]
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (68, 80, 9841)-net over F3, using
- net defined by OOA [i] based on linear OOA(380, 9841, F3, 12, 12) (dual of [(9841, 12), 118012, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(380, 59046, F3, 12) (dual of [59046, 58966, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(380, 59049, F3, 12) (dual of [59049, 58969, 13]-code), using
- 1 times truncation [i] based on linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(380, 59049, F3, 12) (dual of [59049, 58969, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(380, 59046, F3, 12) (dual of [59046, 58966, 13]-code), using
- net defined by OOA [i] based on linear OOA(380, 9841, F3, 12, 12) (dual of [(9841, 12), 118012, 13]-NRT-code), using
- digital (1, 7, 7)-net over F3, using
(75, 75+12, 29542)-Net over F3 — Digital
Digital (75, 87, 29542)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(387, 29542, F3, 2, 12) (dual of [(29542, 2), 58997, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(387, 59084, F3, 12) (dual of [59084, 58997, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(387, 59085, F3, 12) (dual of [59085, 58998, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(387, 59085, F3, 12) (dual of [59085, 58998, 13]-code), using
- OOA 2-folding [i] based on linear OA(387, 59084, F3, 12) (dual of [59084, 58997, 13]-code), using
(75, 75+12, large)-Net in Base 3 — Upper bound on s
There is no (75, 87, large)-net in base 3, because
- 10 times m-reduction [i] would yield (75, 77, large)-net in base 3, but