Best Known (218−47, 218, s)-Nets in Base 3
(218−47, 218, 688)-Net over F3 — Constructive and digital
Digital (171, 218, 688)-net over F3, using
- 32 times duplication [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
(218−47, 218, 1720)-Net over F3 — Digital
Digital (171, 218, 1720)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3218, 1720, F3, 47) (dual of [1720, 1502, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using
- an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- discarding factors / shortening the dual code based on linear OA(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using
(218−47, 218, 149587)-Net in Base 3 — Upper bound on s
There is no (171, 218, 149588)-net in base 3, because
- 1 times m-reduction [i] would yield (171, 217, 149588)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 34 302287 891978 711620 341613 120848 958998 504252 833421 693416 682430 666241 238624 139184 871529 589680 526910 560049 > 3217 [i]