Best Known (115, 115+17, s)-Nets in Base 3
(115, 115+17, 22152)-Net over F3 — Constructive and digital
Digital (115, 132, 22152)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (105, 122, 22144)-net over F3, using
- net defined by OOA [i] based on linear OOA(3122, 22144, F3, 17, 17) (dual of [(22144, 17), 376326, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3122, 177153, F3, 17) (dual of [177153, 177031, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3122, 177158, F3, 17) (dual of [177158, 177036, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3122, 177158, F3, 17) (dual of [177158, 177036, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3122, 177153, F3, 17) (dual of [177153, 177031, 18]-code), using
- net defined by OOA [i] based on linear OOA(3122, 22144, F3, 17, 17) (dual of [(22144, 17), 376326, 18]-NRT-code), using
- digital (2, 10, 8)-net over F3, using
(115, 115+17, 81427)-Net over F3 — Digital
Digital (115, 132, 81427)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3132, 81427, F3, 2, 17) (dual of [(81427, 2), 162722, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3132, 88595, F3, 2, 17) (dual of [(88595, 2), 177058, 18]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(3128, 88593, F3, 2, 17) (dual of [(88593, 2), 177058, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3128, 177186, F3, 17) (dual of [177186, 177058, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-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(16) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(3128, 177186, F3, 17) (dual of [177186, 177058, 18]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(3128, 88593, F3, 2, 17) (dual of [(88593, 2), 177058, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3132, 88595, F3, 2, 17) (dual of [(88595, 2), 177058, 18]-NRT-code), using
(115, 115+17, large)-Net in Base 3 — Upper bound on s
There is no (115, 132, large)-net in base 3, because
- 15 times m-reduction [i] would yield (115, 117, large)-net in base 3, but