Best Known (187, 187+40, s)-Nets in Base 3
(187, 187+40, 1480)-Net over F3 — Constructive and digital
Digital (187, 227, 1480)-net over F3, using
- 1 times m-reduction [i] based on digital (187, 228, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 57, 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, 57, 370)-net over F81, using
(187, 187+40, 5132)-Net over F3 — Digital
Digital (187, 227, 5132)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 5132, F3, 40) (dual of [5132, 4905, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 6619, F3, 40) (dual of [6619, 6392, 41]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3225, 6617, F3, 40) (dual of [6617, 6392, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(31) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(316, 56, F3, 7) (dual of [56, 40, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to Ce(39) ⊂ Ce(31) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3225, 6617, F3, 40) (dual of [6617, 6392, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 6619, F3, 40) (dual of [6619, 6392, 41]-code), using
(187, 187+40, 1080425)-Net in Base 3 — Upper bound on s
There is no (187, 227, 1080426)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 025501 623671 850625 526803 111684 973959 940246 348614 247332 021170 142323 456647 040649 880310 485547 754042 845106 079561 > 3227 [i]