MinT - Best Known (14, 31, s)-Nets in Base 2

Best Known (14, 31, s)-Nets in Base 2

(14, 31, 15)-Net over F₂ — Constructive and digital

Digital (14, 31, 15)-net over F₂, using

t-expansion [i] based on digital (13, 31, 15)-net over F₂, using
- net from sequence [i] based on digital (13, 14)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 13 and N(F) ≥ 15, using
    - an explicitly constructive algebraic function field [i]
- Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]

(14, 31, 34)-Net over F₂ — Upper bound on s (digital)

There is no digital (14, 31, 35)-net over F₂, because

1 times m-reduction [i] would yield digital (14, 30, 35)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2³⁰, 35, F₂, 16) (dual of [35, 5, 17]-code), but
  - Griesmer bound [i]

(14, 31, 37)-Net in Base 2 — Upper bound on s

There is no (14, 31, 38)-net in base 2, because

1 times m-reduction [i] would yield (14, 30, 38)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2³⁰, 38, S₂, 16), but
  - the linear programming bound shows that MÂ â‰¥ 10737 418240 / 9 > 2³⁰ [i]