Best Known (239−40, 239, s)-Nets in Base 3
(239−40, 239, 1480)-Net over F3 — Constructive and digital
Digital (199, 239, 1480)-net over F3, using
- 5 times m-reduction [i] based on digital (199, 244, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 61, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 61, 370)-net over F81, using
(239−40, 239, 8302)-Net over F3 — Digital
Digital (199, 239, 8302)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3239, 8302, F3, 2, 40) (dual of [(8302, 2), 16365, 41]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3239, 9852, F3, 2, 40) (dual of [(9852, 2), 19465, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3239, 19704, F3, 40) (dual of [19704, 19465, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, 19705, F3, 40) (dual of [19705, 19466, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(36) [i] based on
- linear OA(3235, 19683, F3, 40) (dual of [19683, 19448, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(39) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(3239, 19705, F3, 40) (dual of [19705, 19466, 41]-code), using
- OOA 2-folding [i] based on linear OA(3239, 19704, F3, 40) (dual of [19704, 19465, 41]-code), using
- discarding factors / shortening the dual code based on linear OOA(3239, 9852, F3, 2, 40) (dual of [(9852, 2), 19465, 41]-NRT-code), using
(239−40, 239, 2088676)-Net in Base 3 — Upper bound on s
There is no (199, 239, 2088677)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 076420 787309 665020 497988 775984 406888 401359 095255 327023 942094 736241 342378 076268 194460 222562 821760 321641 598217 447441 > 3239 [i]