Best Known (212, 212+21, s)-Nets in Base 3
(212, 212+21, 838892)-Net over F3 — Constructive and digital
Digital (212, 233, 838892)-net over F3, using
- 31 times duplication [i] based on digital (211, 232, 838892)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (12, 22, 32)-net over F3, using
- trace code for nets [i] based on digital (1, 11, 16)-net over F9, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- trace code for nets [i] based on digital (1, 11, 16)-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 (12, 22, 32)-net over F3, using
- (u, u+v)-construction [i] based on
(212, 212+21, 4194335)-Net over F3 — Digital
Digital (212, 233, 4194335)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3233, 4194335, F3, 2, 21) (dual of [(4194335, 2), 8388437, 22]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(323, 34, F3, 2, 10) (dual of [(34, 2), 45, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(323, 68, F3, 10) (dual of [68, 45, 11]-code), using
- a “Glo†code from Brouwer’s database [i]
- OOA 2-folding [i] based on linear OA(323, 68, F3, 10) (dual of [68, 45, 11]-code), using
- linear OOA(3210, 4194301, F3, 2, 21) (dual of [(4194301, 2), 8388392, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3210, 8388602, F3, 21) (dual of [8388602, 8388392, 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 2-folding [i] based on linear OA(3210, 8388602, F3, 21) (dual of [8388602, 8388392, 22]-code), using
- linear OOA(323, 34, F3, 2, 10) (dual of [(34, 2), 45, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
(212, 212+21, large)-Net in Base 3 — Upper bound on s
There is no (212, 233, large)-net in base 3, because
- 19 times m-reduction [i] would yield (212, 214, large)-net in base 3, but