Best Known (215−18, 215, s)-Nets in Base 3
(215−18, 215, 932231)-Net over F3 — Constructive and digital
Digital (197, 215, 932231)-net over F3, using
- 31 times duplication [i] based on digital (196, 214, 932231)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (25, 34, 164)-net over F3, using
- trace code for nets [i] based on digital (8, 17, 82)-net over F9, using
- base reduction for projective spaces (embedding PG(8,81) in PG(16,9)) for nets [i] based on digital (0, 9, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base reduction for projective spaces (embedding PG(8,81) in PG(16,9)) for nets [i] based on digital (0, 9, 82)-net over F81, using
- trace code for nets [i] based on digital (8, 17, 82)-net over F9, using
- digital (162, 180, 932067)-net over F3, using
- net defined by OOA [i] based on linear OOA(3180, 932067, F3, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- net defined by OOA [i] based on linear OOA(3180, 932067, F3, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
- digital (25, 34, 164)-net over F3, using
- (u, u+v)-construction [i] based on
(215−18, 215, 8184737)-Net over F3 — Digital
Digital (197, 215, 8184737)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3215, 8184737, F3, 18) (dual of [8184737, 8184522, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, large, F3, 18) (dual of [large, large−215, 19]-code), using
- 35 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 35 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, large, F3, 18) (dual of [large, large−215, 19]-code), using
(215−18, 215, large)-Net in Base 3 — Upper bound on s
There is no (197, 215, large)-net in base 3, because
- 16 times m-reduction [i] would yield (197, 199, large)-net in base 3, but