MinT - Best Known (71−39, 71, s)-Nets in Base 2

Best Known (71−39, 71, s)-Nets in Base 2

(71−39, 71, 21)-Net over F₂ — Constructive and digital

Digital (32, 71, 21)-net over F₂, using

t-expansion [i] based on digital (21, 71, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(71−39, 71, 27)-Net over F₂ — Digital

Digital (32, 71, 27)-net over F₂, using

t-expansion [i] based on digital (31, 71, 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]

(71−39, 71, 71)-Net over F₂ — Upper bound on s (digital)

There is no digital (32, 71, 72)-net over F₂, because

7 times m-reduction [i] would yield digital (32, 64, 72)-net over F₂, but
- 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]

(71−39, 71, 74)-Net in Base 2 — Upper bound on s

There is no (32, 71, 75)-net in base 2, because

1 times m-reduction [i] would yield (32, 70, 75)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁷⁰, 75, S₂, 38), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 18889 465931 478580 854784 / 13 > 2⁷⁰ [i]