MinT - Best Known (59−31, 59, s)-Nets in Base 2

Best Known (59−31, 59, s)-Nets in Base 2

(59−31, 59, 21)-Net over F₂ — Constructive and digital

Digital (28, 59, 21)-net over F₂, using

t-expansion [i] based on digital (21, 59, 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]

(59−31, 59, 25)-Net over F₂ — Digital

Digital (28, 59, 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]

(59−31, 59, 65)-Net over F₂ — Upper bound on s (digital)

There is no digital (28, 59, 66)-net over F₂, because

3 times m-reduction [i] would yield digital (28, 56, 66)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵⁶, 66, F₂, 28) (dual of [66, 10, 29]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁵⁷, 67, F₂, 29) (dual of [67, 10, 30]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(59−31, 59, 66)-Net in Base 2 — Upper bound on s

There is no (28, 59, 67)-net in base 2, because

1 times m-reduction [i] would yield (28, 58, 67)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁸, 67, S₂, 30), but
  - the linear programming bound shows that MÂ â‰¥ 10 376293 541461 622784 / 35 > 2⁵⁸ [i]