Best Known (85, 85+14, s)-Nets in Base 3
(85, 85+14, 8444)-Net over F3 — Constructive and digital
Digital (85, 99, 8444)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (77, 91, 8437)-net over F3, using
- net defined by OOA [i] based on linear OOA(391, 8437, F3, 14, 14) (dual of [(8437, 14), 118027, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(391, 59059, F3, 14) (dual of [59059, 58968, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(30, 10, F3, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- OA 7-folding and stacking [i] based on linear OA(391, 59059, F3, 14) (dual of [59059, 58968, 15]-code), using
- net defined by OOA [i] based on linear OOA(391, 8437, F3, 14, 14) (dual of [(8437, 14), 118027, 15]-NRT-code), using
- digital (1, 8, 7)-net over F3, using
(85, 85+14, 29543)-Net over F3 — Digital
Digital (85, 99, 29543)-net over F3, using
- 31 times duplication [i] based on digital (84, 98, 29543)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(398, 29543, F3, 2, 14) (dual of [(29543, 2), 58988, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(398, 59086, F3, 14) (dual of [59086, 58988, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(397, 59085, F3, 14) (dual of [59085, 58988, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(13) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(397, 59085, F3, 14) (dual of [59085, 58988, 15]-code), using
- OOA 2-folding [i] based on linear OA(398, 59086, F3, 14) (dual of [59086, 58988, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(398, 29543, F3, 2, 14) (dual of [(29543, 2), 58988, 15]-NRT-code), using
(85, 85+14, large)-Net in Base 3 — Upper bound on s
There is no (85, 99, large)-net in base 3, because
- 12 times m-reduction [i] would yield (85, 87, large)-net in base 3, but