MinT - Best Known (38−21, 38, s)-Nets in Base 2

Best Known (38−21, 38, s)-Nets in Base 2

(38−21, 38, 17)-Net over F₂ — Constructive and digital

Digital (17, 38, 17)-net over F₂, using

t-expansion [i] based on digital (15, 38, 17)-net over F₂, using
- net from sequence [i] based on digital (15, 16)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(38−21, 38, 41)-Net over F₂ — Upper bound on s (digital)

There is no digital (17, 38, 42)-net over F₂, because

1 times m-reduction [i] would yield digital (17, 37, 42)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2³⁷, 42, F₂, 20) (dual of [42, 5, 21]-code), but
  - Griesmer bound [i]

(38−21, 38, 44)-Net in Base 2 — Upper bound on s

There is no (17, 38, 45)-net in base 2, because

1 times m-reduction [i] would yield (17, 37, 45)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2³⁷, 45, S₂, 20), but
  - the linear programming bound shows that MÂ â‰¥ 10 033043 603456 / 55 > 2³⁷ [i]