MinT - Best Known (31, 31+57, s)-Nets in Base 2

Best Known (31, 31+57, s)-Nets in Base 2

(31, 31+57, 21)-Net over F₂ — Constructive and digital

Digital (31, 88, 21)-net over F₂, using

t-expansion [i] based on digital (21, 88, 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+57, 27)-Net over F₂ — Digital

Digital (31, 88, 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+57, 62)-Net in Base 2 — Upper bound on s

There is no (31, 88, 63)-net in base 2, because

1 times m-reduction [i] would yield (31, 87, 63)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 181 787845 300111 472692 252177 > 2⁸⁷ [i]