MinT - Best Known (165−108, 165, s)-Nets in Base 2

Best Known (165−108, 165, s)-Nets in Base 2

(165−108, 165, 42)-Net over F₂ — Constructive and digital

Digital (57, 165, 42)-net over F₂, using

t-expansion [i] based on digital (54, 165, 42)-net over F₂, using
- net from sequence [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(165−108, 165, 84)-Net in Base 2 — Upper bound on s

There is no (57, 165, 85)-net in base 2, because

1 times m-reduction [i] would yield (57, 164, 85)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁶⁴, 85, S₂, 2, 107), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 654 752733 524244 507355 250805 056894 792805 857779 253248 / 27 > 2¹⁶⁴ [i]