MinT - Best Known (41, 81, s)-Nets in Base 2

Best Known (41, 81, s)-Nets in Base 2

(41, 81, 33)-Net over F₂ — Constructive and digital

Digital (41, 81, 33)-net over F₂, using

t-expansion [i] based on digital (39, 81, 33)-net over F₂, using
- net from sequence [i] based on digital (39, 32)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(41, 81, 93)-Net over F₂ — Upper bound on s (digital)

There is no digital (41, 81, 94)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁸¹, 94, F₂, 40) (dual of [94, 13, 41]-code), but
- â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(41, 81, 95)-Net in Base 2 — Upper bound on s

There is no (41, 81, 96)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁸¹, 96, S₂, 40), but
- the linear programming bound shows that MÂ â‰¥ 85727 347720 512584 036764 352512 / 34125 > 2⁸¹ [i]