Best Known (57, 57+5, s)-Nets in Base 3
(57, 57+5, large)-Net over F3 — Constructive and digital
Digital (57, 62, large)-net over F3, using
- 31 times duplication [i] based on digital (56, 61, large)-net over F3, using
- net defined by OOA [i] based on linear OOA(361, large, F3, 5, 5), using
- appending kth column [i] based on linear OOA(361, large, F3, 4, 5), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(31, 2391484, F3, 4, 1) (dual of [(2391484, 4), 9565935, 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(314, 2391484, F3, 4, 2) (dual of [(2391484, 4), 9565922, 3]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(314, 2391484, F3, 2, 2) (dual of [(2391484, 2), 4782954, 3]-NRT-code), using
- appending kth column [i] based on linear OA(314, 2391484, F3, 2) (dual of [2391484, 2391470, 3]-code), using
- Hamming code H(14,3) [i]
- appending kth column [i] based on linear OA(314, 2391484, F3, 2) (dual of [2391484, 2391470, 3]-code), using
- appending 2 arbitrary columns [i] based on linear OOA(314, 2391484, F3, 2, 2) (dual of [(2391484, 2), 4782954, 3]-NRT-code), using
- linear OOA(346, 4194301, F3, 4, 5) (dual of [(4194301, 4), 16777158, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(346, large, F3, 5) (dual of [large, large−46, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(346, large, F3, 5) (dual of [large, large−46, 6]-code), using
- linear OOA(31, 2391484, F3, 4, 1) (dual of [(2391484, 4), 9565935, 2]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(361, large, F3, 4, 5), using
- net defined by OOA [i] based on linear OOA(361, large, F3, 5, 5), using
(57, 57+5, large)-Net in Base 3 — Upper bound on s
There is no (57, 62, large)-net in base 3, because
- 3 times m-reduction [i] would yield (57, 59, large)-net in base 3, but