MinT - Best Known (69−31, 69, s)-Nets in Base 2

Best Known (69−31, 69, s)-Nets in Base 2

(69−31, 69, 28)-Net over F₂ — Constructive and digital

Digital (38, 69, 28)-net over F₂, using

1 times m-reduction [i] based on digital (38, 70, 28)-net over F₂, using
- trace code for nets [i] based on digital (3, 35, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]

(69−31, 69, 31)-Net over F₂ — Digital

Digital (38, 69, 31)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−31, 69, 126)-Net in Base 2 — Upper bound on s

There is no (38, 69, 127)-net in base 2, because

1 times m-reduction [i] would yield (38, 68, 127)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁸, 127, S₂, 30), but
  - the linear programming bound shows that MÂ â‰¥ 5956 789481 066977 461874 107947 023425 660998 525132 996608 / 18 525527 001375 563554 145376 517395 > 2⁶⁸ [i]