MinT - Best Known (81−65, 81, s)-Nets in Base 2

Best Known (81−65, 81, s)-Nets in Base 2

(81−65, 81, 17)-Net over F₂ — Constructive and digital

Digital (16, 81, 17)-net over F₂, using

t-expansion [i] based on digital (15, 81, 17)-net over F₂, using
- net from sequence [i] based on digital (15, 16)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(81−65, 81, 24)-Net in Base 2 — Upper bound on s

There is no (16, 81, 25)-net in base 2, because

12 times m-reduction [i] would yield (16, 69, 25)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2⁶⁹, 25, S₂, 3, 53), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 16528 282690 043758 247936 / 27 > 2⁶⁹ [i]