Best Known (196−17, 196, s)-Nets in Base 3
(196−17, 196, 1048739)-Net over F3 — Constructive and digital
Digital (179, 196, 1048739)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (22, 30, 164)-net over F3, using
- trace code for nets [i] based on digital (7, 15, 82)-net over F9, using
- base reduction for projective spaces (embedding PG(7,81) in PG(14,9)) for nets [i] based on digital (0, 8, 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(7,81) in PG(14,9)) for nets [i] based on digital (0, 8, 82)-net over F81, using
- trace code for nets [i] based on digital (7, 15, 82)-net over F9, using
- digital (149, 166, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3166, 1048575, F3, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3166, 8388601, F3, 17) (dual of [8388601, 8388435, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3166, 8388601, F3, 17) (dual of [8388601, 8388435, 18]-code), using
- net defined by OOA [i] based on linear OOA(3166, 1048575, F3, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
- digital (22, 30, 164)-net over F3, using
(196−17, 196, 5120493)-Net over F3 — Digital
Digital (179, 196, 5120493)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3196, 5120493, F3, 17) (dual of [5120493, 5120297, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 17) (dual of [large, large−196, 18]-code), using
- strength reduction [i] based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- strength reduction [i] based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 17) (dual of [large, large−196, 18]-code), using
(196−17, 196, large)-Net in Base 3 — Upper bound on s
There is no (179, 196, large)-net in base 3, because
- 15 times m-reduction [i] would yield (179, 181, large)-net in base 3, but