Best Known (10, 10+5, s)-Nets in Base 3
(10, 10+5, 117)-Net over F3 — Constructive and digital
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
(10, 10+5, 151)-Net over F3 — Digital
Digital (10, 15, 151)-net over F3, using
- net defined by OOA [i] based on linear OOA(315, 151, F3, 5, 5) (dual of [(151, 5), 740, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(315, 151, F3, 4, 5) (dual of [(151, 4), 589, 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
- appending kth column [i] based on linear OOA(315, 151, F3, 4, 5) (dual of [(151, 4), 589, 6]-NRT-code), using
(10, 10+5, 1544)-Net in Base 3 — Upper bound on s
There is no (10, 15, 1545)-net in base 3, because
- 1 times m-reduction [i] would yield (10, 14, 1545)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 783321 > 314 [i]