Best Known (80, 80+11, s)-Nets in Base 3
(80, 80+11, 106297)-Net over F3 — Constructive and digital
Digital (80, 91, 106297)-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 (74, 85, 106290)-net over F3, using
- net defined by OOA [i] based on linear OOA(385, 106290, F3, 11, 11) (dual of [(106290, 11), 1169105, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(385, 531451, F3, 11) (dual of [531451, 531366, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(385, 531453, F3, 11) (dual of [531453, 531368, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 12, F3, 0) (dual of [12, 12, 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
- discarding factors / shortening the dual code based on linear OA(385, 531453, F3, 11) (dual of [531453, 531368, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(385, 531451, F3, 11) (dual of [531451, 531366, 12]-code), using
- net defined by OOA [i] based on linear OOA(385, 106290, F3, 11, 11) (dual of [(106290, 11), 1169105, 12]-NRT-code), using
- digital (1, 6, 7)-net over F3, using
(80, 80+11, 265741)-Net over F3 — Digital
Digital (80, 91, 265741)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(391, 265741, F3, 2, 11) (dual of [(265741, 2), 531391, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(391, 531482, F3, 11) (dual of [531482, 531391, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(391, 531483, F3, 11) (dual of [531483, 531392, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(349, 531441, F3, 7) (dual of [531441, 531392, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-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
- discarding factors / shortening the dual code based on linear OA(391, 531483, F3, 11) (dual of [531483, 531392, 12]-code), using
- OOA 2-folding [i] based on linear OA(391, 531482, F3, 11) (dual of [531482, 531391, 12]-code), using
(80, 80+11, large)-Net in Base 3 — Upper bound on s
There is no (80, 91, large)-net in base 3, because
- 9 times m-reduction [i] would yield (80, 82, large)-net in base 3, but