Best Known (159, 202, s)-Nets in Base 3
(159, 202, 688)-Net over F3 — Constructive and digital
Digital (159, 202, 688)-net over F3, using
- 32 times duplication [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
(159, 202, 1724)-Net over F3 — Digital
Digital (159, 202, 1724)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3202, 1724, F3, 43) (dual of [1724, 1522, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3202, 2207, F3, 43) (dual of [2207, 2005, 44]-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(35, 19, F3, 3) (dual of [19, 14, 4]-code or 19-cap in PG(4,3)), using
- construction X applied to C([0,21]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3202, 2207, F3, 43) (dual of [2207, 2005, 44]-code), using
(159, 202, 160006)-Net in Base 3 — Upper bound on s
There is no (159, 202, 160007)-net in base 3, because
- 1 times m-reduction [i] would yield (159, 201, 160007)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 796843 377201 327017 184408 971442 503282 545886 750510 971636 044077 978751 422732 130259 783624 517332 064295 > 3201 [i]