Best Known (125, 125+18, s)-Nets in Base 3
(125, 125+18, 19691)-Net over F3 — Constructive and digital
Digital (125, 143, 19691)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 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 (114, 132, 19683)-net over F3, using
- net defined by OOA [i] based on linear OOA(3132, 19683, F3, 18, 18) (dual of [(19683, 18), 354162, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3132, 177147, F3, 18) (dual of [177147, 177015, 19]-code), using
- 1 times truncation [i] based on linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using
- OA 9-folding and stacking [i] based on linear OA(3132, 177147, F3, 18) (dual of [177147, 177015, 19]-code), using
- net defined by OOA [i] based on linear OOA(3132, 19683, F3, 18, 18) (dual of [(19683, 18), 354162, 19]-NRT-code), using
- digital (2, 11, 8)-net over F3, using
(125, 125+18, 88601)-Net over F3 — Digital
Digital (125, 143, 88601)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3143, 88601, F3, 2, 18) (dual of [(88601, 2), 177059, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3143, 177202, F3, 18) (dual of [177202, 177059, 19]-code), using
- 1 times truncation [i] based on linear OA(3144, 177203, F3, 19) (dual of [177203, 177059, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(389, 177148, F3, 13) (dual of [177148, 177059, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(311, 55, F3, 5) (dual of [55, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- 1 times truncation [i] based on linear OA(3144, 177203, F3, 19) (dual of [177203, 177059, 20]-code), using
- OOA 2-folding [i] based on linear OA(3143, 177202, F3, 18) (dual of [177202, 177059, 19]-code), using
(125, 125+18, large)-Net in Base 3 — Upper bound on s
There is no (125, 143, large)-net in base 3, because
- 16 times m-reduction [i] would yield (125, 127, large)-net in base 3, but