Best Known (187−34, 187, s)-Nets in Base 3
(187−34, 187, 702)-Net over F3 — Constructive and digital
Digital (153, 187, 702)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (6, 23, 14)-net over F3, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 6 and N(F) ≥ 14, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- digital (130, 164, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 41, 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, 41, 172)-net over F81, using
- digital (6, 23, 14)-net over F3, using
(187−34, 187, 3764)-Net over F3 — Digital
Digital (153, 187, 3764)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3187, 3764, F3, 34) (dual of [3764, 3577, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3187, 6597, F3, 34) (dual of [6597, 6410, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3187, 6597, F3, 34) (dual of [6597, 6410, 35]-code), using
(187−34, 187, 635678)-Net in Base 3 — Upper bound on s
There is no (153, 187, 635679)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 166600 668713 657761 578002 999658 795646 215379 369986 452507 686129 321685 934259 816459 494983 054879 > 3187 [i]