Best Known (157, 157+41, s)-Nets in Base 3
(157, 157+41, 688)-Net over F3 — Constructive and digital
Digital (157, 198, 688)-net over F3, using
- 2 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
(157, 157+41, 1943)-Net over F3 — Digital
Digital (157, 198, 1943)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3198, 1943, F3, 41) (dual of [1943, 1745, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 2203, F3, 41) (dual of [2203, 2005, 42]-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(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([0,21]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3198, 2203, F3, 41) (dual of [2203, 2005, 42]-code), using
(157, 157+41, 207912)-Net in Base 3 — Upper bound on s
There is no (157, 198, 207913)-net in base 3, because
- 1 times m-reduction [i] would yield (157, 197, 207913)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 9838 361314 273657 878122 678067 162092 438429 727363 094569 574388 595746 404763 532865 709846 123222 496849 > 3197 [i]