Best Known (227, 227+21, s)-Nets in Base 3
(227, 227+21, 839024)-Net over F3 — Constructive and digital
Digital (227, 248, 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 (189, 210, 838860)-net over F3, using
- net defined by OOA [i] based on linear OOA(3210, 838860, F3, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3210, 8388601, F3, 21) (dual of [8388601, 8388391, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, large, F3, 21) (dual of [large, large−210, 22]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(3210, large, F3, 21) (dual of [large, large−210, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3210, 8388601, F3, 21) (dual of [8388601, 8388391, 22]-code), using
- net defined by OOA [i] based on linear OOA(3210, 838860, F3, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
- digital (28, 38, 164)-net over F3, using
(227, 227+21, 6320633)-Net over F3 — Digital
Digital (227, 248, 6320633)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3248, 6320633, F3, 21) (dual of [6320633, 6320385, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3248, large, F3, 21) (dual of [large, large−248, 22]-code), using
- 38 times code embedding in larger space [i] based on linear OA(3210, large, F3, 21) (dual of [large, large−210, 22]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 38 times code embedding in larger space [i] based on linear OA(3210, large, F3, 21) (dual of [large, large−210, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3248, large, F3, 21) (dual of [large, large−248, 22]-code), using
(227, 227+21, large)-Net in Base 3 — Upper bound on s
There is no (227, 248, large)-net in base 3, because
- 19 times m-reduction [i] would yield (227, 229, large)-net in base 3, but