Best Known (240−52, 240, s)-Nets in Base 3
(240−52, 240, 688)-Net over F3 — Constructive and digital
Digital (188, 240, 688)-net over F3, using
- t-expansion [i] based on digital (187, 240, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 60, 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, 60, 172)-net over F81, using
(240−52, 240, 1813)-Net over F3 — Digital
Digital (188, 240, 1813)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3240, 1813, F3, 52) (dual of [1813, 1573, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(3240, 2195, F3, 52) (dual of [2195, 1955, 53]-code), using
- construction X applied to Ce(51) ⊂ Ce(49) [i] based on
- linear OA(3239, 2187, F3, 52) (dual of [2187, 1948, 53]-code), using an extension Ce(51) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,51], and designed minimum distance d ≥ |I|+1 = 52 [i]
- linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 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 Ce(51) ⊂ Ce(49) [i] based on
- discarding factors / shortening the dual code based on linear OA(3240, 2195, F3, 52) (dual of [2195, 1955, 53]-code), using
(240−52, 240, 133775)-Net in Base 3 — Upper bound on s
There is no (188, 240, 133776)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 229581 328609 911257 088719 768205 366720 390644 381640 776018 410185 913720 719717 143213 959955 785508 939736 010120 095605 885025 > 3240 [i]