Best Known (234−20, 234, s)-Nets in Base 3
(234−20, 234, 839024)-Net over F3 — Constructive and digital
Digital (214, 234, 839024)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (28, 38, 164)-net over F3, using
- trace code for nets [i] based on digital (9, 19, 82)-net over F9, using
- base reduction for projective spaces (embedding PG(9,81) in PG(18,9)) for nets [i] based on digital (0, 10, 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(9,81) in PG(18,9)) for nets [i] based on digital (0, 10, 82)-net over F81, using
- trace code for nets [i] based on digital (9, 19, 82)-net over F9, using
- digital (176, 196, 838860)-net over F3, using
- net defined by OOA [i] based on linear OOA(3196, 838860, F3, 20, 20) (dual of [(838860, 20), 16777004, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3196, 8388600, F3, 20) (dual of [8388600, 8388404, 21]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3196, 8388600, F3, 20) (dual of [8388600, 8388404, 21]-code), using
- net defined by OOA [i] based on linear OOA(3196, 838860, F3, 20, 20) (dual of [(838860, 20), 16777004, 21]-NRT-code), using
- digital (28, 38, 164)-net over F3, using
(234−20, 234, 5664588)-Net over F3 — Digital
Digital (214, 234, 5664588)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 5664588, F3, 20) (dual of [5664588, 5664354, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, large, F3, 20) (dual of [large, large−234, 21]-code), using
- 38 times code embedding in larger space [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]
- 38 times code embedding in larger space [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(3234, large, F3, 20) (dual of [large, large−234, 21]-code), using
(234−20, 234, large)-Net in Base 3 — Upper bound on s
There is no (214, 234, large)-net in base 3, because
- 18 times m-reduction [i] would yield (214, 216, large)-net in base 3, but