MinT - Best Known (34, 34+34, s)-Nets in Base 2

Best Known (34, 34+34, s)-Nets in Base 2

(34, 34+34, 24)-Net over F₂ — Constructive and digital

Digital (34, 68, 24)-net over F₂, using

t-expansion [i] based on digital (33, 68, 24)-net over F₂, using
- net from sequence [i] based on digital (33, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(34, 34+34, 28)-Net over F₂ — Digital

Digital (34, 68, 28)-net over F₂, using

t-expansion [i] based on digital (33, 68, 28)-net over F₂, using
- net from sequence [i] based on digital (33, 27)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 28, using
    - Drinfeld modules of rank 1 [i]

(34, 34+34, 78)-Net over F₂ — Upper bound on s (digital)

There is no digital (34, 68, 79)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁶⁸, 79, F₂, 34) (dual of [79, 11, 35]-code), but
- â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(34, 34+34, 81)-Net in Base 2 — Upper bound on s

There is no (34, 68, 82)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁶⁸, 82, S₂, 34), but
- the linear programming bound shows that MÂ â‰¥ 705993 789189 011959 447552 / 2233 > 2⁶⁸ [i]