Best Known (136−24, 136, s)-Nets in Base 3
(136−24, 136, 692)-Net over F3 — Constructive and digital
Digital (112, 136, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (100, 124, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 31, 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, 31, 172)-net over F81, using
- digital (0, 12, 4)-net over F3, using
(136−24, 136, 3813)-Net over F3 — Digital
Digital (112, 136, 3813)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3136, 3813, F3, 24) (dual of [3813, 3677, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3136, 6593, F3, 24) (dual of [6593, 6457, 25]-code), using
- construction XX applied to Ce(24) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(24) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3136, 6593, F3, 24) (dual of [6593, 6457, 25]-code), using
(136−24, 136, 675613)-Net in Base 3 — Upper bound on s
There is no (112, 136, 675614)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 77356 244185 792330 023345 120310 763627 973453 162704 627126 343415 104729 > 3136 [i]