Best Known (98−11, 98, s)-Nets in Base 3
(98−11, 98, 318874)-Net over F3 — Constructive and digital
Digital (87, 98, 318874)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 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 (81, 92, 318867)-net over F3, using
- net defined by OOA [i] based on linear OOA(392, 318867, F3, 11, 11) (dual of [(318867, 11), 3507445, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(392, 1594336, F3, 11) (dual of [1594336, 1594244, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(392, 1594336, F3, 11) (dual of [1594336, 1594244, 12]-code), using
- net defined by OOA [i] based on linear OOA(392, 318867, F3, 11, 11) (dual of [(318867, 11), 3507445, 12]-NRT-code), using
- digital (1, 6, 7)-net over F3, using
(98−11, 98, 797184)-Net over F3 — Digital
Digital (87, 98, 797184)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(398, 797184, F3, 2, 11) (dual of [(797184, 2), 1594270, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(398, 1594368, F3, 11) (dual of [1594368, 1594270, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(353, 1594323, F3, 7) (dual of [1594323, 1594270, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-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(10) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(398, 1594368, F3, 11) (dual of [1594368, 1594270, 12]-code), using
(98−11, 98, large)-Net in Base 3 — Upper bound on s
There is no (87, 98, large)-net in base 3, because
- 9 times m-reduction [i] would yield (87, 89, large)-net in base 3, but