MinT - Best Known (229−197, 229, s)-Nets in Base 2

Best Known (229−197, 229, s)-Nets in Base 2

(229−197, 229, 21)-Net over F₂ — Constructive and digital

Digital (32, 229, 21)-net over F₂, using

t-expansion [i] based on digital (21, 229, 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]

(229−197, 229, 27)-Net over F₂ — Digital

Digital (32, 229, 27)-net over F₂, using

t-expansion [i] based on digital (31, 229, 27)-net over F₂, using
- net from sequence [i] based on digital (31, 26)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 31 and N(F) ≥ 27, using
    - Drinfeld modules of rank 1 [i]

(229−197, 229, 40)-Net in Base 2 — Upper bound on s

There is no (32, 229, 41)-net in base 2, because

32 times m-reduction [i] would yield (32, 197, 41)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁹⁷, 41, S₂, 5, 165), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 18 479787 508978 388168 732564 061923 369929 005334 428502 117605 965824 / 83 > 2¹⁹⁷ [i]