Best Known (237−50, 237, s)-Nets in Base 3
(237−50, 237, 688)-Net over F3 — Constructive and digital
Digital (187, 237, 688)-net over F3, using
- 3 times m-reduction [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
(237−50, 237, 2033)-Net over F3 — Digital
Digital (187, 237, 2033)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3237, 2033, F3, 50) (dual of [2033, 1796, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 2207, F3, 50) (dual of [2207, 1970, 51]-code), using
- construction X applied to Ce(49) ⊂ Ce(45) [i] based on
- 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(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(49) ⊂ Ce(45) [i] based on
- discarding factors / shortening the dual code based on linear OA(3237, 2207, F3, 50) (dual of [2207, 1970, 51]-code), using
(237−50, 237, 169684)-Net in Base 3 — Upper bound on s
There is no (187, 237, 169685)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 119607 377068 649756 909578 297664 789198 626021 712100 490808 860212 674548 489363 334764 246949 105892 362543 249911 878223 247307 > 3237 [i]