Best Known (199−42, 199, s)-Nets in Base 3
(199−42, 199, 688)-Net over F3 — Constructive and digital
Digital (157, 199, 688)-net over F3, using
- 1 times m-reduction [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
(199−42, 199, 1777)-Net over F3 — Digital
Digital (157, 199, 1777)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3199, 1777, F3, 42) (dual of [1777, 1578, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3199, 2192, F3, 42) (dual of [2192, 1993, 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(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- construction X applied to C([0,21]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3199, 2192, F3, 42) (dual of [2192, 1993, 43]-code), using
(199−42, 199, 144109)-Net in Base 3 — Upper bound on s
There is no (157, 199, 144110)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 88544 110227 412340 531021 526860 132468 312520 265543 151176 197462 627142 021001 052972 424816 944586 030677 > 3199 [i]