Best Known (233−20, 233, s)-Nets in Base 3
(233−20, 233, 839005)-Net over F3 — Constructive and digital
Digital (213, 233, 839005)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (27, 37, 145)-net over F3, using
- net defined by OOA [i] based on linear OOA(337, 145, F3, 10, 10) (dual of [(145, 10), 1413, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(337, 725, F3, 10) (dual of [725, 688, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(337, 728, F3, 10) (dual of [728, 691, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(337, 728, F3, 10) (dual of [728, 691, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(337, 725, F3, 10) (dual of [725, 688, 11]-code), using
- net defined by OOA [i] based on linear OOA(337, 145, F3, 10, 10) (dual of [(145, 10), 1413, 11]-NRT-code), 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 (27, 37, 145)-net over F3, using
(233−20, 233, 5329194)-Net over F3 — Digital
Digital (213, 233, 5329194)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3233, 5329194, F3, 20) (dual of [5329194, 5328961, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3233, large, F3, 20) (dual of [large, large−233, 21]-code), using
- 37 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]
- 37 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(3233, large, F3, 20) (dual of [large, large−233, 21]-code), using
(233−20, 233, large)-Net in Base 3 — Upper bound on s
There is no (213, 233, large)-net in base 3, because
- 18 times m-reduction [i] would yield (213, 215, large)-net in base 3, but