MinT - Best Known (113−55, 113, s)-Nets in Base 2

Best Known (113−55, 113, s)-Nets in Base 2

(113−55, 113, 42)-Net over F₂ — Constructive and digital

Digital (58, 113, 42)-net over F₂, using

t-expansion [i] based on digital (54, 113, 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]

(113−55, 113, 132)-Net in Base 2 — Upper bound on s

There is no (58, 113, 133)-net in base 2, because

1 times m-reduction [i] would yield (58, 112, 133)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹¹², 133, S₂, 54), but
  - the linear programming bound shows that MÂ â‰¥ 665 839379 951072 155772 236727 273868 230656 / 115995 > 2¹¹² [i]