MinT - Best Known (227−203, 227, s)-Nets in Base 2

Best Known (227−203, 227, s)-Nets in Base 2

(227−203, 227, 21)-Net over F₂ — Constructive and digital

Digital (24, 227, 21)-net over F₂, using

t-expansion [i] based on digital (21, 227, 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]

(227−203, 227, 22)-Net over F₂ — Digital

Digital (24, 227, 22)-net over F₂, using

t-expansion [i] based on digital (23, 227, 22)-net over F₂, using
- net from sequence [i] based on digital (23, 21)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 23 and N(F) ≥ 22, using
    - Drinfeld modules of rank 1 [i]

(227−203, 227, 31)-Net in Base 2 — Upper bound on s

There is no (24, 227, 32)-net in base 2, because

42 times m-reduction [i] would yield (24, 185, 32)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁸⁵, 32, S₂, 6, 161), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1569 275433 846670 190958 947355 801916 604025 588861 116008 628224 / 27 > 2¹⁸⁵ [i]