MinT - Best Known (88−44, 88, s)-Nets in Base 2

Best Known (88−44, 88, s)-Nets in Base 2

(88−44, 88, 33)-Net over F₂ — Constructive and digital

Digital (44, 88, 33)-net over F₂, using

t-expansion [i] based on digital (39, 88, 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]

(88−44, 88, 34)-Net over F₂ — Digital

Digital (44, 88, 34)-net over F₂, using

t-expansion [i] based on digital (43, 88, 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]

(88−44, 88, 97)-Net over F₂ — Upper bound on s (digital)

There is no digital (44, 88, 98)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁸⁸, 98, F₂, 44) (dual of [98, 10, 45]-code), but
- adding a parity check bit [i] would yield linear OA(2⁸⁹, 99, F₂, 45) (dual of [99, 10, 46]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(88−44, 88, 101)-Net in Base 2 — Upper bound on s

There is no (44, 88, 102)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁸⁸, 102, S₂, 44), but
- the linear programming bound shows that MÂ â‰¥ 16 122931 071652 792700 286193 893376 / 51129 > 2⁸⁸ [i]