Best Known (54, 54+6, s)-Nets in Base 3
(54, 54+6, 2796201)-Net over F3 — Constructive and digital
Digital (54, 60, 2796201)-net over F3, using
- net defined by OOA [i] based on linear OOA(360, 2796201, F3, 6, 6) (dual of [(2796201, 6), 16777146, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(360, 2796201, F3, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(360, large, F3, 6) (dual of [large, large−60, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(360, large, F3, 6) (dual of [large, large−60, 7]-code), using
- appending kth column [i] based on linear OOA(360, 2796201, F3, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
(54, 54+6, large)-Net over F3 — Digital
Digital (54, 60, large)-net over F3, using
- net defined by OOA [i] based on linear OOA(360, large, F3, 6, 6), using
- appending kth column [i] based on linear OOA(360, large, F3, 5, 6), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(360, large, F3, 6) (dual of [large, large−60, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(360, large, F3, 6) (dual of [large, large−60, 7]-code), using
- appending kth column [i] based on linear OOA(360, large, F3, 5, 6), using
(54, 54+6, large)-Net in Base 3 — Upper bound on s
There is no (54, 60, large)-net in base 3, because
- 4 times m-reduction [i] would yield (54, 56, large)-net in base 3, but