Best Known (194, 194+38, s)-Nets in Base 3
(194, 194+38, 1480)-Net over F3 — Constructive and digital
Digital (194, 232, 1480)-net over F3, using
- t-expansion [i] based on digital (193, 232, 1480)-net over F3, using
- 4 times m-reduction [i] based on digital (193, 236, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 59, 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, 59, 370)-net over F81, using
- 4 times m-reduction [i] based on digital (193, 236, 1480)-net over F3, using
(194, 194+38, 9463)-Net over F3 — Digital
Digital (194, 232, 9463)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3232, 9463, F3, 2, 38) (dual of [(9463, 2), 18694, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3232, 9858, F3, 2, 38) (dual of [(9858, 2), 19484, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3232, 19716, F3, 38) (dual of [19716, 19484, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- linear OA(3226, 19683, F3, 38) (dual of [19683, 19457, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- OOA 2-folding [i] based on linear OA(3232, 19716, F3, 38) (dual of [19716, 19484, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(3232, 9858, F3, 2, 38) (dual of [(9858, 2), 19484, 39]-NRT-code), using
(194, 194+38, 2655119)-Net in Base 3 — Upper bound on s
There is no (194, 232, 2655120)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 492 188790 356149 728503 359796 851649 093213 853615 667728 432887 699380 382007 260101 097568 096770 262610 442915 144151 641665 > 3232 [i]