MinT - Best Known (44, 44+40, s)-Nets in Base 2

Best Known (44, 44+40, s)-Nets in Base 2

(44, 44+40, 33)-Net over F₂ — Constructive and digital

Digital (44, 84, 33)-net over F₂, using

t-expansion [i] based on digital (39, 84, 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]

(44, 44+40, 34)-Net over F₂ — Digital

Digital (44, 84, 34)-net over F₂, using

t-expansion [i] based on digital (43, 84, 34)-net over F₂, using
- net from sequence [i] based on digital (43, 33)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 43 and N(F) ≥ 34, using
    - a curve from the manYPoints database [i]

(44, 44+40, 107)-Net over F₂ — Upper bound on s (digital)

There is no digital (44, 84, 108)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁸⁴, 108, F₂, 40) (dual of [108, 24, 41]-code), but
- adding a parity check bit [i] would yield linear OA(2⁸⁵, 109, F₂, 41) (dual of [109, 24, 42]-code), but
  - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

(44, 44+40, 108)-Net in Base 2 — Upper bound on s

There is no (44, 84, 109)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁸⁴, 109, S₂, 40), but
- the linear programming bound shows that MÂ â‰¥ 116 995237 232782 719020 166432 161792 / 4 994815 > 2⁸⁴ [i]