MinT - Best Known (18, 18+9, s)-Nets in Base 2

Best Known (18, 18+9, s)-Nets in Base 2

(18, 18+9, 40)-Net over F₂ — Constructive and digital

Digital (18, 27, 40)-net over F₂, using

net defined by OOA [i] based on linear OOA(2²⁷, 40, F₂, 9, 9) (dual of [(40, 9), 333, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2²⁷, 40, F₂, 8, 9) (dual of [(40, 8), 293, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁹, 23, F₂, 8, 4) (dual of [(23, 8), 175, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2⁹, 23, F₂, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁹, 23, F₂, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 9, 23)-net over F₂, using
        linear code embedding found in a computer search [i]
    2. linear OOA(2¹⁸, 20, F₂, 8, 9) (dual of [(20, 8), 142, 10]-NRT-code), using
      - extracting embedded OOA [i] based on digital (9, 18, 20)-net over F₂, using
        linear code embedding found in a computer search [i]

(18, 18+9, 194)-Net in Base 2 — Upper bound on s

There is no (18, 27, 195)-net in base 2, because

1 times m-reduction [i] would yield (18, 26, 195)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 67 252056 > 2²⁶ [i]