MinT - Best Known (72−38, 72, s)-Nets in Base 2

Best Known (72−38, 72, s)-Nets in Base 2

(72−38, 72, 24)-Net over F₂ — Constructive and digital

Digital (34, 72, 24)-net over F₂, using

t-expansion [i] based on digital (33, 72, 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]

(72−38, 72, 28)-Net over F₂ — Digital

Digital (34, 72, 28)-net over F₂, using

t-expansion [i] based on digital (33, 72, 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]

(72−38, 72, 76)-Net over F₂ — Upper bound on s (digital)

There is no digital (34, 72, 77)-net over F₂, because

2 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]

(72−38, 72, 79)-Net in Base 2 — Upper bound on s

There is no (34, 72, 80)-net in base 2, because

2 times m-reduction [i] would yield (34, 70, 80)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁷⁰, 80, S₂, 36), but
  - the linear programming bound shows that MÂ â‰¥ 689465 506498 968201 199616 / 437 > 2⁷⁰ [i]