MinT - Best Known (32, 32+31, s)-Nets in Base 2

Best Known (32, 32+31, s)-Nets in Base 2

(32, 32+31, 24)-Net over F₂ — Constructive and digital

Digital (32, 63, 24)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 21, 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, 42, 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]

(32, 32+31, 27)-Net over F₂ — Digital

Digital (32, 63, 27)-net over F₂, using

t-expansion [i] based on digital (31, 63, 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]

(32, 32+31, 79)-Net over F₂ — Upper bound on s (digital)

There is no digital (32, 63, 80)-net over F₂, because

1 times m-reduction [i] would yield digital (32, 62, 80)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶², 80, F₂, 30) (dual of [80, 18, 31]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(32, 32+31, 81)-Net in Base 2 — Upper bound on s

There is no (32, 63, 82)-net in base 2, because

1 times m-reduction [i] would yield (32, 62, 82)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶², 82, S₂, 30), but
  - the linear programming bound shows that MÂ â‰¥ 97408 032081 223287 308288 / 17043 > 2⁶² [i]