Best Known (39−11, 39, s)-Nets in Base 3
(39−11, 39, 144)-Net over F3 — Constructive and digital
Digital (28, 39, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 13, 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
(39−11, 39, 207)-Net over F3 — Digital
Digital (28, 39, 207)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(339, 207, F3, 11) (dual of [207, 168, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(339, 256, F3, 11) (dual of [256, 217, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- 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(326, 243, F3, 8) (dual of [243, 217, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(339, 256, F3, 11) (dual of [256, 217, 12]-code), using
(39−11, 39, 5502)-Net in Base 3 — Upper bound on s
There is no (28, 39, 5503)-net in base 3, because
- 1 times m-reduction [i] would yield (28, 38, 5503)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 351265 389270 417143 > 338 [i]