Best Known (59−16, 59, s)-Nets in Base 3
(59−16, 59, 156)-Net over F3 — Constructive and digital
Digital (43, 59, 156)-net over F3, using
- 1 times m-reduction [i] based on digital (43, 60, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 20, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- trace code for nets [i] based on digital (3, 20, 52)-net over F27, using
(59−16, 59, 275)-Net over F3 — Digital
Digital (43, 59, 275)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(359, 275, F3, 16) (dual of [275, 216, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(359, 369, F3, 16) (dual of [369, 310, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(358, 365, F3, 16) (dual of [365, 307, 17]-code), using an extension Ce(15) of the narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(355, 365, F3, 14) (dual of [365, 310, 15]-code), using an extension Ce(13) of the narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 4, F3, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(359, 369, F3, 16) (dual of [369, 310, 17]-code), using
(59−16, 59, 6207)-Net in Base 3 — Upper bound on s
There is no (43, 59, 6208)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 14142 490143 304320 017768 075265 > 359 [i]