MinT - Best Known (68−39, 68, s)-Nets in Base 2

Best Known (68−39, 68, s)-Nets in Base 2

(68−39, 68, 21)-Net over F₂ — Constructive and digital

Digital (29, 68, 21)-net over F₂, using

t-expansion [i] based on digital (21, 68, 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]

(68−39, 68, 25)-Net over F₂ — Digital

Digital (29, 68, 25)-net over F₂, using

t-expansion [i] based on digital (28, 68, 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]

(68−39, 68, 65)-Net over F₂ — Upper bound on s (digital)

There is no digital (29, 68, 66)-net over F₂, because

7 times m-reduction [i] would yield digital (29, 61, 66)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶¹, 66, F₂, 32) (dual of [66, 5, 33]-code), but
  - Griesmer bound [i]

(68−39, 68, 66)-Net in Base 2 — Upper bound on s

There is no (29, 68, 67)-net in base 2, because

1 times m-reduction [i] would yield (29, 67, 67)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 156 671816 140713 462184 > 2⁶⁷ [i]