Best Known (77, 77+18, s)-Nets in Base 3
(77, 77+18, 640)-Net over F3 — Constructive and digital
Digital (77, 95, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (77, 96, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 24, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 24, 160)-net over F81, using
(77, 77+18, 2146)-Net over F3 — Digital
Digital (77, 95, 2146)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(395, 2146, F3, 18) (dual of [2146, 2051, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(395, 2226, F3, 18) (dual of [2226, 2131, 19]-code), using
- strength reduction [i] based on linear OA(395, 2226, F3, 19) (dual of [2226, 2131, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(357, 2188, F3, 13) (dual of [2188, 2131, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on 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
- strength reduction [i] based on linear OA(395, 2226, F3, 19) (dual of [2226, 2131, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(395, 2226, F3, 18) (dual of [2226, 2131, 19]-code), using
(77, 77+18, 225416)-Net in Base 3 — Upper bound on s
There is no (77, 95, 225417)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2120 968899 698648 709792 248930 423452 644550 749043 > 395 [i]