Best Known (49−14, 49, s)-Nets in Base 3
(49−14, 49, 144)-Net over F3 — Constructive and digital
Digital (35, 49, 144)-net over F3, using
- 31 times duplication [i] based on digital (34, 48, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 16, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- trace code for nets [i] based on digital (2, 16, 48)-net over F27, using
(49−14, 49, 204)-Net over F3 — Digital
Digital (35, 49, 204)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(349, 204, F3, 14) (dual of [204, 155, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(349, 256, F3, 14) (dual of [256, 207, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(346, 243, F3, 14) (dual of [243, 197, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(336, 243, F3, 11) (dual of [243, 207, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(349, 256, F3, 14) (dual of [256, 207, 15]-code), using
(49−14, 49, 3689)-Net in Base 3 — Upper bound on s
There is no (35, 49, 3690)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 239502 726051 030177 508297 > 349 [i]