Best Known (200−42, 200, s)-Nets in Base 3
(200−42, 200, 688)-Net over F3 — Constructive and digital
Digital (158, 200, 688)-net over F3, using
- t-expansion [i] based on digital (157, 200, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 50, 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
- trace code for nets [i] based on digital (7, 50, 172)-net over F81, using
(200−42, 200, 1827)-Net over F3 — Digital
Digital (158, 200, 1827)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3200, 1827, F3, 42) (dual of [1827, 1627, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3200, 2201, F3, 42) (dual of [2201, 2001, 43]-code), using
- construction X applied to C([0,21]) ⊂ C([0,19]) [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]
- linear OA(3183, 2188, F3, 39) (dual of [2188, 2005, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to C([0,21]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3200, 2201, F3, 42) (dual of [2201, 2001, 43]-code), using
(200−42, 200, 151850)-Net in Base 3 — Upper bound on s
There is no (158, 200, 151851)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 265636 566493 952005 241302 110964 522521 194021 912751 540070 002893 231856 224316 335143 286705 081962 330863 > 3200 [i]