Best Known (231−48, 231, s)-Nets in Base 3
(231−48, 231, 688)-Net over F3 — Constructive and digital
Digital (183, 231, 688)-net over F3, using
- t-expansion [i] based on digital (181, 231, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 58, 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, 58, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
(231−48, 231, 2144)-Net over F3 — Digital
Digital (183, 231, 2144)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3231, 2144, F3, 48) (dual of [2144, 1913, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(3231, 2214, F3, 48) (dual of [2214, 1983, 49]-code), using
- construction X applied to Ce(48) ⊂ Ce(43) [i] based on
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(48) ⊂ Ce(43) [i] based on
- discarding factors / shortening the dual code based on linear OA(3231, 2214, F3, 48) (dual of [2214, 1983, 49]-code), using
(231−48, 231, 191677)-Net in Base 3 — Upper bound on s
There is no (183, 231, 191678)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 164 068960 557993 924317 690869 899699 148681 856280 020148 797006 151125 212874 935432 092505 224224 748521 124340 653526 308849 > 3231 [i]