Best Known (190, 241, s)-Nets in Base 3
(190, 241, 688)-Net over F3 — Constructive and digital
Digital (190, 241, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (190, 244, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 61, 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, 61, 172)-net over F81, using
(190, 241, 2031)-Net over F3 — Digital
Digital (190, 241, 2031)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3241, 2031, F3, 51) (dual of [2031, 1790, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 2204, F3, 51) (dual of [2204, 1963, 52]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3240, 2203, F3, 51) (dual of [2203, 1963, 52]-code), using
- construction X applied to C([0,25]) ⊂ C([0,24]) [i] based on
- linear OA(3239, 2188, F3, 51) (dual of [2188, 1949, 52]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,25], and minimum distance d ≥ |{−25,−24,…,25}|+1 = 52 (BCH-bound) [i]
- linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,25]) ⊂ C([0,24]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3240, 2203, F3, 51) (dual of [2203, 1963, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 2204, F3, 51) (dual of [2204, 1963, 52]-code), using
(190, 241, 193599)-Net in Base 3 — Upper bound on s
There is no (190, 241, 193600)-net in base 3, because
- 1 times m-reduction [i] would yield (190, 240, 193600)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 229325 873151 826858 605414 517432 286397 146210 754989 663929 335695 029603 944421 001032 558491 885352 552474 119704 682150 036609 > 3240 [i]