MinT - Best Known (111−86, 111, s)-Nets in Base 2

Best Known (111−86, 111, s)-Nets in Base 2

(111−86, 111, 21)-Net over F₂ — Constructive and digital

Digital (25, 111, 21)-net over F₂, using

t-expansion [i] based on digital (21, 111, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(111−86, 111, 24)-Net over F₂ — Digital

Digital (25, 111, 24)-net over F₂, using

net from sequence [i] based on digital (25, 23)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 25 and N(F) ≥ 24, using
  - Drinfeld modules of rank 1 [i]

(111−86, 111, 35)-Net in Base 2 — Upper bound on s

There is no (25, 111, 36)-net in base 2, because

9 times m-reduction [i] would yield (25, 102, 36)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁰², 36, S₂, 3, 77), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 81 129638 414606 681695 789005 144064 / 13 > 2¹⁰² [i]