Best Known (42, 42+11, s)-Nets in Base 3
(42, 42+11, 464)-Net over F3 — Constructive and digital
Digital (42, 53, 464)-net over F3, using
- 31 times duplication [i] based on digital (41, 52, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 13, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 13, 116)-net over F81, using
(42, 42+11, 1176)-Net over F3 — Digital
Digital (42, 53, 1176)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(353, 1176, F3, 11) (dual of [1176, 1123, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(353, 2200, F3, 11) (dual of [2200, 2147, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(350, 2187, F3, 11) (dual of [2187, 2137, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(336, 2187, F3, 8) (dual of [2187, 2151, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, 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(353, 2200, F3, 11) (dual of [2200, 2147, 12]-code), using
(42, 42+11, 119357)-Net in Base 3 — Upper bound on s
There is no (42, 53, 119358)-net in base 3, because
- 1 times m-reduction [i] would yield (42, 52, 119358)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 461126 778300 237931 706005 > 352 [i]