MinT - Best Known (33−18, 33, s)-Nets in Base 2

Best Known (33−18, 33, s)-Nets in Base 2

(33−18, 33, 17)-Net over F₂ — Constructive and digital

Digital (15, 33, 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]

(33−18, 33, 37)-Net over F₂ — Upper bound on s (digital)

There is no digital (15, 33, 38)-net over F₂, because

2 times m-reduction [i] would yield digital (15, 31, 38)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2³¹, 38, F₂, 16) (dual of [38, 7, 17]-code), but
  - residual code [i] would yield linear OA(2¹⁵, 21, F₂, 8) (dual of [21, 6, 9]-code), but
    - residual code [i] would yield linear OA(2⁷, 12, F₂, 4) (dual of [12, 5, 5]-code), but
      - bound by Fontaine and Peterson [i]

(33−18, 33, 39)-Net in Base 2 — Upper bound on s

There is no (15, 33, 40)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2³³, 40, S₂, 18), but
- the linear programming bound shows that MÂ â‰¥ 667867 414528 / 75 > 2³³ [i]