Best Known (96, 96+10, s)-Nets in Base 3
(96, 96+10, 1677837)-Net over F3 — Constructive and digital
Digital (96, 106, 1677837)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (10, 15, 117)-net over F3, using
- net defined by OOA [i] based on linear OOA(315, 117, F3, 5, 5) (dual of [(117, 5), 570, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(315, 117, F3, 4, 5) (dual of [(117, 4), 453, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(34, 40, F3, 4, 2) (dual of [(40, 4), 156, 3]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(34, 40, F3, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
- appending kth column [i] based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- appending kth column [i] based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- appending 2 arbitrary columns [i] based on linear OOA(34, 40, F3, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
- linear OOA(311, 77, F3, 4, 5) (dual of [(77, 4), 297, 6]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 11, 77)-net over F3, using
- linear OOA(34, 40, F3, 4, 2) (dual of [(40, 4), 156, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(315, 117, F3, 4, 5) (dual of [(117, 4), 453, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(315, 117, F3, 5, 5) (dual of [(117, 5), 570, 6]-NRT-code), 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
- digital (10, 15, 117)-net over F3, using
(96, 96+10, 4194452)-Net over F3 — Digital
Digital (96, 106, 4194452)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3106, 4194452, F3, 2, 10) (dual of [(4194452, 2), 8388798, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(315, 151, F3, 2, 5) (dual of [(151, 2), 287, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(315, 151, F3, 5) (dual of [151, 136, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 168, F3, 5) (dual of [168, 153, 6]-code), using
- a “BZ†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(315, 168, F3, 5) (dual of [168, 153, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(315, 151, F3, 5) (dual of [151, 136, 6]-code), 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(315, 151, F3, 2, 5) (dual of [(151, 2), 287, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
(96, 96+10, large)-Net in Base 3 — Upper bound on s
There is no (96, 106, large)-net in base 3, because
- 8 times m-reduction [i] would yield (96, 98, large)-net in base 3, but