MinT - Best Known (69−49, 69, s)-Nets in Base 2

Best Known (69−49, 69, s)-Nets in Base 2

(69−49, 69, 20)-Net over F₂ — Constructive and digital

Digital (20, 69, 20)-net over F₂, using

t-expansion [i] based on digital (19, 69, 20)-net over F₂, using
- net from sequence [i] based on digital (19, 19)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
    - an explicitly constructive algebraic function field [i]

(69−49, 69, 33)-Net in Base 2 — Upper bound on s

There is no (20, 69, 34)-net in base 2, because

6 times m-reduction [i] would yield (20, 63, 34)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2⁶³, 34, S₂, 2, 43), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 110 680464 442257 309696 / 11 > 2⁶³ [i]