Best Known (237−36, 237, s)-Nets in Base 3
(237−36, 237, 1500)-Net over F3 — Constructive and digital
Digital (201, 237, 1500)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (11, 29, 20)-net over F3, using
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 9, N(F) = 19, and 1 place with degree 3 [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- 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
- digital (11, 29, 20)-net over F3, using
(237−36, 237, 13842)-Net over F3 — Digital
Digital (201, 237, 13842)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3237, 13842, F3, 36) (dual of [13842, 13605, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 19757, F3, 36) (dual of [19757, 19520, 37]-code), using
- 1 times truncation [i] based on linear OA(3238, 19758, F3, 37) (dual of [19758, 19520, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(27) [i] based on
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(321, 75, F3, 8) (dual of [75, 54, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to Ce(36) ⊂ Ce(27) [i] based on
- 1 times truncation [i] based on linear OA(3238, 19758, F3, 37) (dual of [19758, 19520, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 19757, F3, 36) (dual of [19757, 19520, 37]-code), using
(237−36, 237, 7230952)-Net in Base 3 — Upper bound on s
There is no (201, 237, 7230953)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 119601 745335 229938 072381 387937 160123 447268 133569 389233 656533 020154 249885 171812 677302 506514 973033 450405 988338 103257 > 3237 [i]