Best Known (128−20, 128, s)-Nets in Base 3
(128−20, 128, 1973)-Net over F3 — Constructive and digital
Digital (108, 128, 1973)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (98, 118, 1969)-net over F3, using
- net defined by OOA [i] based on linear OOA(3118, 1969, F3, 20, 20) (dual of [(1969, 20), 39262, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3118, 19690, F3, 20) (dual of [19690, 19572, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3118, 19692, F3, 20) (dual of [19692, 19574, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- 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(30, 9, F3, 0) (dual of [9, 9, 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(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3118, 19692, F3, 20) (dual of [19692, 19574, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3118, 19690, F3, 20) (dual of [19690, 19572, 21]-code), using
- net defined by OOA [i] based on linear OOA(3118, 1969, F3, 20, 20) (dual of [(1969, 20), 39262, 21]-NRT-code), using
- digital (0, 10, 4)-net over F3, using
(128−20, 128, 9861)-Net over F3 — Digital
Digital (108, 128, 9861)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3128, 9861, F3, 2, 20) (dual of [(9861, 2), 19594, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3128, 19722, F3, 20) (dual of [19722, 19594, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3128, 19722, F3, 20) (dual of [19722, 19594, 21]-code), using
(128−20, 128, 2897990)-Net in Base 3 — Upper bound on s
There is no (108, 128, 2897991)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 11 790208 475477 867537 292722 944650 438701 218711 313068 773280 990717 > 3128 [i]