Best Known (58−15, 58, s)-Nets in Base 3
(58−15, 58, 192)-Net over F3 — Constructive and digital
Digital (43, 58, 192)-net over F3, using
- 31 times duplication [i] based on digital (42, 57, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 19, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- trace code for nets [i] based on digital (4, 19, 64)-net over F27, using
(58−15, 58, 339)-Net over F3 — Digital
Digital (43, 58, 339)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(358, 339, F3, 15) (dual of [339, 281, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(358, 368, F3, 15) (dual of [368, 310, 16]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(359, 369, F3, 16) (dual of [369, 310, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(358, 368, F3, 15) (dual of [368, 310, 16]-code), using
(58−15, 58, 12965)-Net in Base 3 — Upper bound on s
There is no (43, 58, 12966)-net in base 3, because
- 1 times m-reduction [i] would yield (43, 57, 12966)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1570 163466 215585 424092 744601 > 357 [i]