MinT - Best Known (31, 31+153, s)-Nets in Base 2

Best Known (31, 31+153, s)-Nets in Base 2

(31, 31+153, 21)-Net over F₂ — Constructive and digital

Digital (31, 184, 21)-net over F₂, using

t-expansion [i] based on digital (21, 184, 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]

(31, 31+153, 27)-Net over F₂ — Digital

Digital (31, 184, 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]

(31, 31+153, 40)-Net in Base 2 — Upper bound on s

There is no (31, 184, 41)-net in base 2, because

27 times m-reduction [i] would yield (31, 157, 41)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁵⁷, 41, S₂, 4, 126), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 33 614537 658610 767118 684751 152474 509452 086448 488448 / 127 > 2¹⁵⁷ [i]