MinT - Best Known (97−54, 97, s)-Nets in Base 2

Best Known (97−54, 97, s)-Nets in Base 2

(97−54, 97, 33)-Net over F₂ — Constructive and digital

Digital (43, 97, 33)-net over F₂, using

t-expansion [i] based on digital (39, 97, 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]

(97−54, 97, 34)-Net over F₂ — Digital

Digital (43, 97, 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]

(97−54, 97, 95)-Net over F₂ — Upper bound on s (digital)

There is no digital (43, 97, 96)-net over F₂, because

10 times m-reduction [i] would yield digital (43, 87, 96)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁸⁷, 96, F₂, 44) (dual of [96, 9, 45]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(97−54, 97, 96)-Net in Base 2 — Upper bound on s

There is no (43, 97, 97)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 173782 185839 874181 693969 723040 > 2⁹⁷ [i]