MinT - Best Known (8−3, 8, s)-Nets in Base 2

Best Known (8−3, 8, s)-Nets in Base 2

(8−3, 8, 127)-Net over F₂ — Constructive and digital

Digital (5, 8, 127)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁸, 127, F₂, 3, 3) (dual of [(127, 3), 373, 4]-NRT-code), using
- OOA stacking with additional row [i] based on linear OA(2⁸, 128, F₂, 3) (dual of [128, 120, 4]-code or 128-cap in PG(7,2)), using
  - Reedâ€“Muller code RM(1,7) [i]
  - caps in base bÂ =Â 2 [i]

(8−3, 8, 127)-Net in Base 2 — Upper bound on s

There is no (5, 8, 128)-net in base 2, because

1 times m-reduction [i] would yield (5, 7, 128)-net in base 2, but
- mutually orthogonal hypercube bound [i]
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 129 > 2⁷ [i]