MinT - Best Known (59−29, 59, s)-Nets in Base 2

Best Known (59−29, 59, s)-Nets in Base 2

(59−29, 59, 23)-Net over F₂ — Constructive and digital

Digital (30, 59, 23)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 19, 9)-net over F₂, using
  - net from sequence [i] based on digital (5, 8)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 5 and N(F) ≥ 9, 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]

(59−29, 59, 25)-Net over F₂ — Digital

Digital (30, 59, 25)-net over F₂, using

t-expansion [i] based on digital (28, 59, 25)-net over F₂, using
- net from sequence [i] based on digital (28, 24)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 28 and N(F) ≥ 25, using
    - a curve from the manYPoints database [i]

(59−29, 59, 75)-Net over F₂ — Upper bound on s (digital)

There is no digital (30, 59, 76)-net over F₂, because

1 times m-reduction [i] would yield digital (30, 58, 76)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵⁸, 76, F₂, 28) (dual of [76, 18, 29]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(59−29, 59, 77)-Net in Base 2 — Upper bound on s

There is no (30, 59, 78)-net in base 2, because

1 times m-reduction [i] would yield (30, 58, 78)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁸, 78, S₂, 28), but
  - the linear programming bound shows that MÂ â‰¥ 6382 573449 503504 859136 / 19305 > 2⁵⁸ [i]