Best Known (239−16, 239, s)-Nets in Base 3
(239−16, 239, 3145736)-Net over F3 — Constructive and digital
Digital (223, 239, 3145736)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 11)-net over F3, using
- digital (212, 228, 3145725)-net over F3, using
- trace code for nets [i] based on digital (60, 76, 1048575)-net over F27, using
- net defined by OOA [i] based on linear OOA(2776, 1048575, F27, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2776, 8388600, F27, 16) (dual of [8388600, 8388524, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2776, large, F27, 16) (dual of [large, large−76, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(2776, large, F27, 16) (dual of [large, large−76, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2776, 8388600, F27, 16) (dual of [8388600, 8388524, 17]-code), using
- net defined by OOA [i] based on linear OOA(2776, 1048575, F27, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
- trace code for nets [i] based on digital (60, 76, 1048575)-net over F27, using
(239−16, 239, large)-Net over F3 — Digital
Digital (223, 239, large)-net over F3, using
- t-expansion [i] based on digital (221, 239, large)-net over F3, using
- 2 times m-reduction [i] based on digital (221, 241, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3241, large, F3, 20) (dual of [large, large−241, 21]-code), using
- strength reduction [i] based on linear OA(3241, large, F3, 25) (dual of [large, large−241, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- strength reduction [i] based on linear OA(3241, large, F3, 25) (dual of [large, large−241, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3241, large, F3, 20) (dual of [large, large−241, 21]-code), using
- 2 times m-reduction [i] based on digital (221, 241, large)-net over F3, using
(239−16, 239, large)-Net in Base 3 — Upper bound on s
There is no (223, 239, large)-net in base 3, because
- 14 times m-reduction [i] would yield (223, 225, large)-net in base 3, but