Best Known (100, 100+19, s)-Nets in Base 3
(100, 100+19, 2194)-Net over F3 — Constructive and digital
Digital (100, 119, 2194)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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 (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 (1, 10, 7)-net over F3, using
(100, 100+19, 9861)-Net over F3 — Digital
Digital (100, 119, 9861)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3119, 9861, F3, 2, 19) (dual of [(9861, 2), 19603, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3119, 19722, F3, 19) (dual of [19722, 19603, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3119, 19723, F3, 19) (dual of [19723, 19604, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [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]
- linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3119, 19723, F3, 19) (dual of [19723, 19604, 20]-code), using
- OOA 2-folding [i] based on linear OA(3119, 19722, F3, 19) (dual of [19722, 19603, 20]-code), using
(100, 100+19, 3735168)-Net in Base 3 — Upper bound on s
There is no (100, 119, 3735169)-net in base 3, because
- 1 times m-reduction [i] would yield (100, 118, 3735169)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 199 667845 389276 234294 926439 180314 714450 181444 824325 976547 > 3118 [i]