Best Known (97, 97+10, s)-Nets in Base 3
(97, 97+10, 1677877)-Net over F3 — Constructive and digital
Digital (97, 107, 1677877)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (11, 16, 157)-net over F3, using
- net defined by OOA [i] based on linear OOA(316, 157, F3, 5, 5) (dual of [(157, 5), 769, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(316, 157, F3, 4, 5) (dual of [(157, 4), 612, 6]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(31, 40, F3, 4, 1) (dual of [(40, 4), 159, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(31, s, F3, 4, 1) with arbitrarily large s, using
- appending 3 arbitrary columns [i] based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(31, s, F3, 4, 1) with arbitrarily large s, using
- 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(31, 40, F3, 4, 1) (dual of [(40, 4), 159, 2]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(316, 157, F3, 4, 5) (dual of [(157, 4), 612, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(316, 157, F3, 5, 5) (dual of [(157, 5), 769, 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 (11, 16, 157)-net over F3, using
(97, 97+10, 4194520)-Net over F3 — Digital
Digital (97, 107, 4194520)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3107, 4194520, F3, 2, 10) (dual of [(4194520, 2), 8388933, 11]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(316, 219, F3, 2, 5) (dual of [(219, 2), 422, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(316, 219, F3, 5) (dual of [219, 203, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 242, F3, 5) (dual of [242, 226, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(316, 242, F3, 5) (dual of [242, 226, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(316, 219, F3, 5) (dual of [219, 203, 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(316, 219, F3, 2, 5) (dual of [(219, 2), 422, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
(97, 97+10, large)-Net in Base 3 — Upper bound on s
There is no (97, 107, large)-net in base 3, because
- 8 times m-reduction [i] would yield (97, 99, large)-net in base 3, but