Best Known (42, 42+17, s)-Nets in Base 3
(42, 42+17, 144)-Net over F3 — Constructive and digital
Digital (42, 59, 144)-net over F3, using
- 1 times m-reduction [i] based on digital (42, 60, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 20, 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, 20, 48)-net over F27, using
(42, 42+17, 213)-Net over F3 — Digital
Digital (42, 59, 213)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(359, 213, F3, 17) (dual of [213, 154, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(359, 256, F3, 17) (dual of [256, 197, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(356, 243, F3, 17) (dual of [243, 187, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(359, 256, F3, 17) (dual of [256, 197, 18]-code), using
(42, 42+17, 5409)-Net in Base 3 — Upper bound on s
There is no (42, 59, 5410)-net in base 3, because
- 1 times m-reduction [i] would yield (42, 58, 5410)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4710 944707 318469 960892 186705 > 358 [i]