MinT - Best Known (6, 6+3, s)-Nets in Base 2

Best Known (6, 6+3, s)-Nets in Base 2

(6, 6+3, 255)-Net over F₂ — Constructive and digital

Digital (6, 9, 255)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁹, 255, F₂, 3, 3) (dual of [(255, 3), 756, 4]-NRT-code), using
- OOA stacking with additional row [i] based on linear OA(2⁹, 256, F₂, 3) (dual of [256, 247, 4]-code or 256-cap in PG(8,2)), using
  - Reedâ€“Muller code RM(1,8) [i]
  - caps in base bÂ =Â 2 [i]

(6, 6+3, 255)-Net in Base 2 — Upper bound on s

There is no (6, 9, 256)-net in base 2, because

1 times m-reduction [i] would yield (6, 8, 256)-net in base 2, but
- mutually orthogonal hypercube bound [i]
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 257 > 2⁸ [i]