MinT - Best Known (74−32, 74, s)-Nets in Base 2

Best Known (74−32, 74, s)-Nets in Base 2

(74−32, 74, 34)-Net over F₂ — Constructive and digital

Digital (42, 74, 34)-net over F₂, using

trace code for nets [i] based on digital (5, 37, 17)-net over F₄, using
- net from sequence [i] based on digital (5, 16)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(74−32, 74, 36)-Net over F₂ — Digital

Digital (42, 74, 36)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(74−32, 74, 145)-Net in Base 2 — Upper bound on s

There is no (42, 74, 146)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 19456 495213 635376 951977 > 2⁷⁴ [i]
extracting embedded orthogonal array [i] would yield OA(2⁷⁴, 146, S₂, 32), but
- the linear programming bound shows that MÂ â‰¥ 280 270393 355737 497844 513648 098820 271740 172522 302191 747635 045255 952555 211698 995200 / 14596 329528 207009 222060 595527 173338 787771 222351 404958 268831 > 2⁷⁴ [i]