Best Known (186, 186+27, s)-Nets in Base 3
(186, 186+27, 13634)-Net over F3 — Constructive and digital
Digital (186, 213, 13634)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 15, 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 (171, 198, 13626)-net over F3, using
- net defined by OOA [i] based on linear OOA(3198, 13626, F3, 27, 27) (dual of [(13626, 27), 367704, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3198, 177139, F3, 27) (dual of [177139, 176941, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 177146, F3, 27) (dual of [177146, 176948, 28]-code), using
- 1 times truncation [i] based on linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 177146, F3, 27) (dual of [177146, 176948, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3198, 177139, F3, 27) (dual of [177139, 176941, 28]-code), using
- net defined by OOA [i] based on linear OOA(3198, 13626, F3, 27, 27) (dual of [(13626, 27), 367704, 28]-NRT-code), using
- digital (2, 15, 8)-net over F3, using
(186, 186+27, 76717)-Net over F3 — Digital
Digital (186, 213, 76717)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3213, 76717, F3, 2, 27) (dual of [(76717, 2), 153221, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3213, 88603, F3, 2, 27) (dual of [(88603, 2), 176993, 28]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3211, 88602, F3, 2, 27) (dual of [(88602, 2), 176993, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3211, 177204, F3, 27) (dual of [177204, 176993, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3199, 177148, F3, 27) (dual of [177148, 176949, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3155, 177148, F3, 21) (dual of [177148, 176993, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(312, 56, F3, 5) (dual of [56, 44, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(3211, 177204, F3, 27) (dual of [177204, 176993, 28]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3211, 88602, F3, 2, 27) (dual of [(88602, 2), 176993, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3213, 88603, F3, 2, 27) (dual of [(88603, 2), 176993, 28]-NRT-code), using
(186, 186+27, large)-Net in Base 3 — Upper bound on s
There is no (186, 213, large)-net in base 3, because
- 25 times m-reduction [i] would yield (186, 188, large)-net in base 3, but