MinT - Best Known (38, 38+127, s)-Nets in Base 2

Best Known (38, 38+127, s)-Nets in Base 2

(38, 38+127, 24)-Net over F₂ — Constructive and digital

Digital (38, 165, 24)-net over F₂, using

t-expansion [i] based on digital (33, 165, 24)-net over F₂, using
- net from sequence [i] based on digital (33, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(38, 38+127, 30)-Net over F₂ — Digital

Digital (38, 165, 30)-net over F₂, using

t-expansion [i] based on digital (36, 165, 30)-net over F₂, using
- net from sequence [i] based on digital (36, 29)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
    - Drinfeld modules of rank 1 [i]

(38, 38+127, 51)-Net in Base 2 — Upper bound on s

There is no (38, 165, 52)-net in base 2, because

18 times m-reduction [i] would yield (38, 147, 52)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁴⁷, 52, S₂, 3, 109), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 30685 825393 178137 442753 148343 164145 432229 052416 / 165 > 2¹⁴⁷ [i]