Best Known (167, 167+10, s)-Nets in Base 3
(167, 167+10, 6723842)-Net over F3 — Constructive and digital
Digital (167, 177, 6723842)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (24, 29, 12962)-net over F3, using
- net defined by OOA [i] based on linear OOA(329, 12962, F3, 5, 5) (dual of [(12962, 5), 64781, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(329, 25925, F3, 5) (dual of [25925, 25896, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(328, 25924, F3, 5) (dual of [25924, 25896, 6]-code), using
- trace code [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(328, 25924, F3, 5) (dual of [25924, 25896, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(329, 25925, F3, 5) (dual of [25925, 25896, 6]-code), using
- net defined by OOA [i] based on linear OOA(329, 12962, F3, 5, 5) (dual of [(12962, 5), 64781, 6]-NRT-code), using
- digital (138, 148, 6710880)-net over F3, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F81, using
- digital (24, 29, 12962)-net over F3, using
(167, 167+10, large)-Net over F3 — Digital
Digital (167, 177, large)-net over F3, using
- t-expansion [i] based on digital (163, 177, large)-net over F3, using
- 1 times m-reduction [i] based on digital (163, 178, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3178, large, F3, 15) (dual of [large, large−178, 16]-code), using
- 28 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 28 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3178, large, F3, 15) (dual of [large, large−178, 16]-code), using
- 1 times m-reduction [i] based on digital (163, 178, large)-net over F3, using
(167, 167+10, large)-Net in Base 3 — Upper bound on s
There is no (167, 177, large)-net in base 3, because
- 8 times m-reduction [i] would yield (167, 169, large)-net in base 3, but