Best Known (210−36, 210, s)-Nets in Base 3
(210−36, 210, 1480)-Net over F3 — Constructive and digital
Digital (174, 210, 1480)-net over F3, using
- 32 times duplication [i] based on digital (172, 208, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 52, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 52, 370)-net over F81, using
(210−36, 210, 5766)-Net over F3 — Digital
Digital (174, 210, 5766)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3210, 5766, F3, 36) (dual of [5766, 5556, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 6626, F3, 36) (dual of [6626, 6416, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(27) [i] based on
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(317, 65, F3, 7) (dual of [65, 48, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(36) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3210, 6626, F3, 36) (dual of [6626, 6416, 37]-code), using
(210−36, 210, 1391583)-Net in Base 3 — Upper bound on s
There is no (174, 210, 1391584)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 15684 363132 858664 679423 943635 945519 857279 403980 973766 679881 285985 327508 208283 519670 696060 206759 678785 > 3210 [i]