MinT - Best Known (42−19, 42, s)-Nets in Base 2

Best Known (42−19, 42, s)-Nets in Base 2

(42−19, 42, 22)-Net over F₂ — Constructive and digital

Digital (23, 42, 22)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 15, 12)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (8, 27, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]

(42−19, 42, 23)-Net over F₂ — Digital

Digital (23, 42, 23)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−19, 42, 85)-Net in Base 2 — Upper bound on s

There is no (23, 42, 86)-net in base 2, because

1 times m-reduction [i] would yield (23, 41, 86)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2 357476 167494 > 2⁴¹ [i]