Best Known (56, 63, s)-Nets in Base 4
(56, 63, 2796200)-Net over F4 — Constructive and digital
Digital (56, 63, 2796200)-net over F4, using
- 42 times duplication [i] based on digital (54, 61, 2796200)-net over F4, using
- net defined by OOA [i] based on linear OOA(461, 2796200, F4, 7, 7) (dual of [(2796200, 7), 19573339, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(461, 8388601, F4, 7) (dual of [8388601, 8388540, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(461, 8388601, F4, 7) (dual of [8388601, 8388540, 8]-code), using
- net defined by OOA [i] based on linear OOA(461, 2796200, F4, 7, 7) (dual of [(2796200, 7), 19573339, 8]-NRT-code), using
(56, 63, large)-Net over F4 — Digital
Digital (56, 63, large)-net over F4, using
- 42 times duplication [i] based on digital (54, 61, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
(56, 63, large)-Net in Base 4 — Upper bound on s
There is no (56, 63, large)-net in base 4, because
- 5 times m-reduction [i] would yield (56, 58, large)-net in base 4, but