Best Known (51, 51+5, s)-Nets in Base 3
(51, 51+5, 4223825)-Net over F3 — Constructive and digital
Digital (51, 56, 4223825)-net over F3, using
- net defined by OOA [i] based on linear OOA(356, 4223825, F3, 5, 5) (dual of [(4223825, 5), 21119069, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(356, 4223825, F3, 4, 5) (dual of [(4223825, 4), 16895244, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(310, 29524, F3, 4, 2) (dual of [(29524, 4), 118086, 3]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(310, 29524, F3, 2, 2) (dual of [(29524, 2), 59038, 3]-NRT-code), using
- appending kth column [i] based on linear OA(310, 29524, F3, 2) (dual of [29524, 29514, 3]-code), using
- Hamming code H(10,3) [i]
- appending kth column [i] based on linear OA(310, 29524, F3, 2) (dual of [29524, 29514, 3]-code), using
- appending 2 arbitrary columns [i] based on linear OOA(310, 29524, F3, 2, 2) (dual of [(29524, 2), 59038, 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(310, 29524, F3, 4, 2) (dual of [(29524, 4), 118086, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(356, 4223825, F3, 4, 5) (dual of [(4223825, 4), 16895244, 6]-NRT-code), using
(51, 51+5, large)-Net over F3 — Digital
Digital (51, 56, large)-net over F3, using
- 310 times duplication [i] based on digital (41, 46, large)-net over F3, using
- net defined by OOA [i] based on linear OOA(346, large, F3, 5, 5), using
- appending kth column [i] based on linear OOA(346, large, F3, 4, 5), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(346, large, F3, 5) (dual of [large, large−46, 6]-code), using
- appending kth column [i] based on linear OOA(346, large, F3, 4, 5), using
- net defined by OOA [i] based on linear OOA(346, large, F3, 5, 5), using
(51, 51+5, large)-Net in Base 3 — Upper bound on s
There is no (51, 56, large)-net in base 3, because
- 3 times m-reduction [i] would yield (51, 53, large)-net in base 3, but