Best Known (224, 224+14, s)-Nets in Base 3
(224, 224+14, 4793729)-Net over F3 — Constructive and digital
Digital (224, 238, 4793729)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (19, 26, 245)-net over F3, using
- net defined by OOA [i] based on linear OOA(326, 245, F3, 7, 7) (dual of [(245, 7), 1689, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(326, 736, F3, 7) (dual of [736, 710, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(325, 729, F3, 7) (dual of [729, 704, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(319, 729, F3, 5) (dual of [729, 710, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(326, 736, F3, 7) (dual of [736, 710, 8]-code), using
- net defined by OOA [i] based on linear OOA(326, 245, F3, 7, 7) (dual of [(245, 7), 1689, 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 (19, 26, 245)-net over F3, using
(224, 224+14, large)-Net over F3 — Digital
Digital (224, 238, large)-net over F3, using
- t-expansion [i] based on digital (221, 238, large)-net over F3, using
- 3 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
- 3 times m-reduction [i] based on digital (221, 241, large)-net over F3, using
(224, 224+14, large)-Net in Base 3 — Upper bound on s
There is no (224, 238, large)-net in base 3, because
- 12 times m-reduction [i] would yield (224, 226, large)-net in base 3, but