MinT - Best Known (26, 26+29, s)-Nets in Base 2

Best Known (26, 26+29, s)-Nets in Base 2

(26, 26+29, 21)-Net over F₂ — Constructive and digital

Digital (26, 55, 21)-net over F₂, using

t-expansion [i] based on digital (21, 55, 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]

(26, 26+29, 24)-Net over F₂ — Digital

Digital (26, 55, 24)-net over F₂, using

t-expansion [i] based on digital (25, 55, 24)-net over F₂, using
- net from sequence [i] based on digital (25, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 25 and N(F) ≥ 24, using
    - Drinfeld modules of rank 1 [i]

(26, 26+29, 61)-Net over F₂ — Upper bound on s (digital)

There is no digital (26, 55, 62)-net over F₂, because

1 times m-reduction [i] would yield digital (26, 54, 62)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵⁴, 62, F₂, 28) (dual of [62, 8, 29]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2⁵⁵, 63, F₂, 28) (dual of [63, 8, 29]-code), but
    - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(26, 26+29, 62)-Net in Base 2 — Upper bound on s

There is no (26, 55, 63)-net in base 2, because

1 times m-reduction [i] would yield (26, 54, 63)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁴, 63, S₂, 28), but
  - the linear programming bound shows that MÂ â‰¥ 9 223372 036854 775808 / 475 > 2⁵⁴ [i]