MinT - Best Known (127−87, 127, s)-Nets in Base 2

Best Known (127−87, 127, s)-Nets in Base 2

(127−87, 127, 33)-Net over F₂ — Constructive and digital

Digital (40, 127, 33)-net over F₂, using

t-expansion [i] based on digital (39, 127, 33)-net over F₂, using
- net from sequence [i] based on digital (39, 32)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(127−87, 127, 61)-Net in Base 2 — Upper bound on s

There is no (40, 127, 62)-net in base 2, because

9 times m-reduction [i] would yield (40, 118, 62)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹¹⁸, 62, S₂, 2, 78), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 31 901471 898837 980949 691369 446728 269824 / 79 > 2¹¹⁸ [i]