Best Known (196−42, 196, s)-Nets in Base 3
(196−42, 196, 688)-Net over F3 — Constructive and digital
Digital (154, 196, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 49, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
(196−42, 196, 1633)-Net over F3 — Digital
Digital (154, 196, 1633)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3196, 1633, F3, 42) (dual of [1633, 1437, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, 2187, F3, 42) (dual of [2187, 1991, 43]-code), using
- 1 times truncation [i] based on linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, 2187, F3, 42) (dual of [2187, 1991, 43]-code), using
(196−42, 196, 123174)-Net in Base 3 — Upper bound on s
There is no (154, 196, 123175)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3279 273401 141259 669407 862265 687493 928279 448967 552689 537352 477672 212937 109905 410929 222536 592231 > 3196 [i]