MinT - Best Known (27, 27+163, s)-Nets in Base 2

Best Known (27, 27+163, s)-Nets in Base 2

(27, 27+163, 21)-Net over F₂ — Constructive and digital

Digital (27, 190, 21)-net over F₂, using

t-expansion [i] based on digital (21, 190, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(27, 27+163, 24)-Net over F₂ — Digital

Digital (27, 190, 24)-net over F₂, using

t-expansion [i] based on digital (25, 190, 24)-net over F₂, using
- net from sequence [i] based on digital (25, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 25 and N(F) ≥ 24, using
    - Drinfeld modules of rank 1 [i]

(27, 27+163, 35)-Net in Base 2 — Upper bound on s

There is no (27, 190, 36)-net in base 2, because

19 times m-reduction [i] would yield (27, 171, 36)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁷¹, 36, S₂, 5, 144), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 2 351123 529980 772848 125940 605130 015358 852577 305615 663104 / 725 > 2¹⁷¹ [i]