MinT - Best Known (68−30, 68, s)-Nets in Base 2

Best Known (68−30, 68, s)-Nets in Base 2

(68−30, 68, 30)-Net over F₂ — Constructive and digital

Digital (38, 68, 30)-net over F₂, using

trace code for nets [i] based on digital (4, 34, 15)-net over F₄, using
- net from sequence [i] based on digital (4, 14)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
    - an explicitly constructive algebraic function field [i]

(68−30, 68, 33)-Net over F₂ — Digital

Digital (38, 68, 33)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−30, 68, 126)-Net in Base 2 — Upper bound on s

There is no (38, 68, 127)-net in base 2, because

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]