Best Known (41, 41+14, s)-Nets in Base 3
(41, 41+14, 192)-Net over F3 — Constructive and digital
Digital (41, 55, 192)-net over F3, using
- 31 times duplication [i] based on digital (40, 54, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 18, 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, 18, 64)-net over F27, using
(41, 41+14, 367)-Net over F3 — Digital
Digital (41, 55, 367)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(355, 367, F3, 2, 14) (dual of [(367, 2), 679, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(355, 734, F3, 14) (dual of [734, 679, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(355, 735, F3, 14) (dual of [735, 680, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(355, 729, F3, 14) (dual of [729, 674, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(349, 729, F3, 13) (dual of [729, 680, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(355, 735, F3, 14) (dual of [735, 680, 15]-code), using
- OOA 2-folding [i] based on linear OA(355, 734, F3, 14) (dual of [734, 679, 15]-code), using
(41, 41+14, 9471)-Net in Base 3 — Upper bound on s
There is no (41, 55, 9472)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 174 564582 352761 370671 365121 > 355 [i]