Best Known (223, 223+14, s)-Nets in Base 3
(223, 223+14, 4793727)-Net over F3 — Constructive and digital
Digital (223, 237, 4793727)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (18, 25, 243)-net over F3, using
- net defined by OOA [i] based on linear OOA(325, 243, F3, 7, 7) (dual of [(243, 7), 1676, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(325, 730, F3, 7) (dual of [730, 705, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(325, 730, F3, 7) (dual of [730, 705, 8]-code), using
- net defined by OOA [i] based on linear OOA(325, 243, F3, 7, 7) (dual of [(243, 7), 1676, 8]-NRT-code), using
- digital (198, 212, 4793484)-net over F3, using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8153, 1198371, F81, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(8153, 8388597, F81, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(8153, 8388597, F81, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(8153, 1198371, F81, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F81, using
- digital (18, 25, 243)-net over F3, using
(223, 223+14, large)-Net over F3 — Digital
Digital (223, 237, large)-net over F3, using
- t-expansion [i] based on digital (221, 237, large)-net over F3, using
- 4 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
- 4 times m-reduction [i] based on digital (221, 241, large)-net over F3, using
(223, 223+14, large)-Net in Base 3 — Upper bound on s
There is no (223, 237, large)-net in base 3, because
- 12 times m-reduction [i] would yield (223, 225, large)-net in base 3, but