MinT - Best Known (43, 43+51, s)-Nets in Base 2

Best Known (43, 43+51, s)-Nets in Base 2

(43, 43+51, 33)-Net over F₂ — Constructive and digital

Digital (43, 94, 33)-net over F₂, using

t-expansion [i] based on digital (39, 94, 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]

(43, 43+51, 34)-Net over F₂ — Digital

Digital (43, 94, 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]

(43, 43+51, 95)-Net over F₂ — Upper bound on s (digital)

There is no digital (43, 94, 96)-net over F₂, because

7 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]

(43, 43+51, 96)-Net in Base 2 — Upper bound on s

There is no (43, 94, 97)-net in base 2, because

3 times m-reduction [i] would yield (43, 91, 97)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁹¹, 97, S₂, 48), but
  - adding a parity check bit [i] would yield OA(2⁹², 98, S₂, 49), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 158456 325028 528675 187087 900672 / 25 > 2⁹² [i]