MinT - Best Known (59−30, 59, s)-Nets in Base 2

Best Known (59−30, 59, s)-Nets in Base 2

(59−30, 59, 21)-Net over F₂ — Constructive and digital

Digital (29, 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]

(59−30, 59, 25)-Net over F₂ — Digital

Digital (29, 59, 25)-net over F₂, using

t-expansion [i] based on 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]

(59−30, 59, 67)-Net over F₂ — Upper bound on s (digital)

There is no digital (29, 59, 68)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁵⁹, 68, F₂, 30) (dual of [68, 9, 31]-code), but
- adding a parity check bit [i] would yield linear OA(2⁶⁰, 69, F₂, 31) (dual of [69, 9, 32]-code), but
  - â€œBGVâ€ bound on codes from Brouwerâ€™s database [i]

(59−30, 59, 69)-Net in Base 2 — Upper bound on s

There is no (29, 59, 70)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁵⁹, 70, S₂, 30), but
- the linear programming bound shows that MÂ â‰¥ 23 634890 844440 363008 / 35 > 2⁵⁹ [i]