MinT - Best Known (31, 31+29, s)-Nets in Base 2

Best Known (31, 31+29, s)-Nets in Base 2

(31, 31+29, 24)-Net over F₂ — Constructive and digital

Digital (31, 60, 24)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 20, 10)-net over F₂, using
  - net from sequence [i] based on digital (6, 9)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 6 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (11, 40, 14)-net over F₂, using
  - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]

(31, 31+29, 27)-Net over F₂ — Digital

Digital (31, 60, 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+29, 82)-Net in Base 2 — Upper bound on s

There is no (31, 60, 83)-net in base 2, because

1 times m-reduction [i] would yield (31, 59, 83)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁹, 83, S₂, 28), but
  - the linear programming bound shows that MÂ â‰¥ 3 484516 168547 439462 055936 / 5 727645 > 2⁵⁹ [i]