MinT - Best Known (77−43, 77, s)-Nets in Base 2

Best Known (77−43, 77, s)-Nets in Base 2

(77−43, 77, 24)-Net over F₂ — Constructive and digital

Digital (34, 77, 24)-net over F₂, using

t-expansion [i] based on digital (33, 77, 24)-net over F₂, using
- net from sequence [i] based on digital (33, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(77−43, 77, 28)-Net over F₂ — Digital

Digital (34, 77, 28)-net over F₂, using

t-expansion [i] based on digital (33, 77, 28)-net over F₂, using
- net from sequence [i] based on digital (33, 27)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 28, using
    - Drinfeld modules of rank 1 [i]

(77−43, 77, 76)-Net over F₂ — Upper bound on s (digital)

There is no digital (34, 77, 77)-net over F₂, because

7 times m-reduction [i] would yield digital (34, 70, 77)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁷⁰, 77, F₂, 36) (dual of [77, 7, 37]-code), but
  - residual code [i] would yield linear OA(2³⁴, 40, F₂, 18) (dual of [40, 6, 19]-code), but
    - â€œBMâ€ bound on codes from Brouwerâ€™s database [i]

(77−43, 77, 78)-Net in Base 2 — Upper bound on s

There is no (34, 77, 79)-net in base 2, because

3 times m-reduction [i] would yield (34, 74, 79)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁷⁴, 79, S₂, 40), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 906694 364710 971881 029632 / 41 > 2⁷⁴ [i]