Best Known (90, 90+10, s)-Nets in Base 3
(90, 90+10, 1677748)-Net over F3 — Constructive and digital
Digital (90, 100, 1677748)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 28)-net over F3, using
- digital (81, 91, 1677720)-net over F3, using
- net defined by OOA [i] based on linear OOA(391, 1677720, F3, 10, 10) (dual of [(1677720, 10), 16777109, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(391, 8388600, F3, 10) (dual of [8388600, 8388509, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(391, 8388600, F3, 10) (dual of [8388600, 8388509, 11]-code), using
- net defined by OOA [i] based on linear OOA(391, 1677720, F3, 10, 10) (dual of [(1677720, 10), 16777109, 11]-NRT-code), using
(90, 90+10, 4194329)-Net over F3 — Digital
Digital (90, 100, 4194329)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3100, 4194329, F3, 2, 10) (dual of [(4194329, 2), 8388558, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(39, 28, F3, 2, 5) (dual of [(28, 2), 47, 6]-NRT-code), using
- extracting embedded OOA [i] based on digital (4, 9, 28)-net over F3, using
- linear OOA(391, 4194301, F3, 2, 10) (dual of [(4194301, 2), 8388511, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(391, 8388602, F3, 10) (dual of [8388602, 8388511, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- OOA 2-folding [i] based on linear OA(391, 8388602, F3, 10) (dual of [8388602, 8388511, 11]-code), using
- linear OOA(39, 28, F3, 2, 5) (dual of [(28, 2), 47, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
(90, 90+10, large)-Net in Base 3 — Upper bound on s
There is no (90, 100, large)-net in base 3, because
- 8 times m-reduction [i] would yield (90, 92, large)-net in base 3, but