MinT - Best Known (74−37, 74, s)-Nets in Base 2

Best Known (74−37, 74, s)-Nets in Base 2

(74−37, 74, 25)-Net over F₂ — Constructive and digital

Digital (37, 74, 25)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 26, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (11, 48, 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]

(74−37, 74, 30)-Net over F₂ — Digital

Digital (37, 74, 30)-net over F₂, using

t-expansion [i] based on digital (36, 74, 30)-net over F₂, using
- net from sequence [i] based on digital (36, 29)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
    - Drinfeld modules of rank 1 [i]

(74−37, 74, 85)-Net over F₂ — Upper bound on s (digital)

There is no digital (37, 74, 86)-net over F₂, because

1 times m-reduction [i] would yield digital (37, 73, 86)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁷³, 86, F₂, 36) (dual of [86, 13, 37]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(74−37, 74, 87)-Net in Base 2 — Upper bound on s

There is no (37, 74, 88)-net in base 2, because

1 times m-reduction [i] would yield (37, 73, 88)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁷³, 88, S₂, 36), but
  - the linear programming bound shows that MÂ â‰¥ 793 659800 577004 053194 604544 / 82593 > 2⁷³ [i]