Best Known (118−19, 118, s)-Nets in Base 3
(118−19, 118, 2191)-Net over F3 — Constructive and digital
Digital (99, 118, 2191)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (90, 109, 2187)-net over F3, using
- net defined by OOA [i] based on linear OOA(3109, 2187, F3, 19, 19) (dual of [(2187, 19), 41444, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using
- net defined by OOA [i] based on linear OOA(3109, 2187, F3, 19, 19) (dual of [(2187, 19), 41444, 20]-NRT-code), using
- digital (0, 9, 4)-net over F3, using
(118−19, 118, 9772)-Net over F3 — Digital
Digital (99, 118, 9772)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3118, 9772, F3, 2, 19) (dual of [(9772, 2), 19426, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3118, 9859, F3, 2, 19) (dual of [(9859, 2), 19600, 20]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3117, 9859, F3, 2, 19) (dual of [(9859, 2), 19601, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3117, 19718, F3, 19) (dual of [19718, 19601, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3117, 19718, F3, 19) (dual of [19718, 19601, 20]-code), using
- 31 times duplication [i] based on linear OOA(3117, 9859, F3, 2, 19) (dual of [(9859, 2), 19601, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3118, 9859, F3, 2, 19) (dual of [(9859, 2), 19600, 20]-NRT-code), using
(118−19, 118, 3305952)-Net in Base 3 — Upper bound on s
There is no (99, 118, 3305953)-net in base 3, because
- 1 times m-reduction [i] would yield (99, 117, 3305953)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 66 555975 435623 920228 392584 685379 236639 848355 594490 597027 > 3117 [i]