MinT - Best Known (186−172, 186, s)-Nets in Base 2

Best Known (186−172, 186, s)-Nets in Base 2

(186−172, 186, 15)-Net over F₂ — Constructive and digital

Digital (14, 186, 15)-net over F₂, using

t-expansion [i] based on digital (13, 186, 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]

(186−172, 186, 21)-Net in Base 2 — Upper bound on s

There is no (14, 186, 22)-net in base 2, because

106 times m-reduction [i] would yield (14, 80, 22)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2⁸⁰, 22, S₂, 4, 66), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 83 718113 008313 070348 402688 / 67 > 2⁸⁰ [i]