MinT - Best Known (32, 64, s)-Nets in Base 2

Best Known (32, 64, s)-Nets in Base 2

(32, 64, 23)-Net over F₂ — Constructive and digital

Digital (32, 64, 23)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 21, 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, 43, 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, 64, 27)-Net over F₂ — Digital

Digital (32, 64, 27)-net over F₂, using

t-expansion [i] based on digital (31, 64, 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, 64, 71)-Net over F₂ — Upper bound on s (digital)

There is no digital (32, 64, 72)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁶⁴, 72, F₂, 32) (dual of [72, 8, 33]-code), but
- adding a parity check bit [i] would yield linear OA(2⁶⁵, 73, F₂, 33) (dual of [73, 8, 34]-code), but
  - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(32, 64, 77)-Net in Base 2 — Upper bound on s

There is no (32, 64, 78)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁶⁴, 78, S₂, 32), but
- the linear programming bound shows that MÂ â‰¥ 3 009328 041208 681412 427776 / 153153 > 2⁶⁴ [i]