Best Known (53−7, 53, s)-Nets in Base 5
(53−7, 53, 2796200)-Net over F5 — Constructive and digital
Digital (46, 53, 2796200)-net over F5, using
- 52 times duplication [i] based on digital (44, 51, 2796200)-net over F5, using
- net defined by OOA [i] based on linear OOA(551, 2796200, F5, 7, 7) (dual of [(2796200, 7), 19573349, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(551, 8388601, F5, 7) (dual of [8388601, 8388550, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(551, 8388601, F5, 7) (dual of [8388601, 8388550, 8]-code), using
- net defined by OOA [i] based on linear OOA(551, 2796200, F5, 7, 7) (dual of [(2796200, 7), 19573349, 8]-NRT-code), using
(53−7, 53, large)-Net over F5 — Digital
Digital (46, 53, large)-net over F5, using
- 51 times duplication [i] based on digital (45, 52, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(552, large, F5, 7) (dual of [large, large−52, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 1 times code embedding in larger space [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(552, large, F5, 7) (dual of [large, large−52, 8]-code), using
(53−7, 53, large)-Net in Base 5 — Upper bound on s
There is no (46, 53, large)-net in base 5, because
- 5 times m-reduction [i] would yield (46, 48, large)-net in base 5, but